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Let be a finite-dimensional split semisimple Lie algebra over a field of field characteristic 0, a splitting Cartan subalgebra, and a weight of in a representation of . Thenis also a weight. Furthermore, the reflections with a root, generate a group of linear transformations in called the Weyl group of relative to , where is the algebraic conjugate space of and is the Q-space spanned by the roots (Jacobson 1979, pp. 112, 117, and 119).The Weyl group acts on the roots of a semisimple Lie algebra, and it is a finite group. The animations above illustrate this action for Weyl Group acting on the roots of a homotopy from one Weyl matrix to the next one (i.e., it slides the arrows from to ) in the first two figures, while the third figure shows the Weyl Group acting on the roots of the Cartan matrix of the infinite family of semisimple lie algebras (cf. Dynkin diagram), which is the special linear Lie algebra, ...

For a finite group of elements with an th dimensional th irreducible representation,

The wallpaper groups are the 17 possible plane symmetry groups. They are commonly represented using Hermann-Mauguin-like symbols or in orbifold notation (Zwillinger 1995, p. 260).orbifold notationHermann-Mauguin symbolop12222p2**pmxxpg*2222pmm22*pmg22xpggx*cm2*22cmm442p4*442p4m4*2p4g333p3*333p3ml3*3p3lm632p6*632p6mPatterns created with Artlandia SymmetryWorks for each of these groups are illustrated above.Beautiful patterns can be created by repeating geometric and artistic motifs according to the symmetry of the wallpaper groups, as exemplified in works by M. C. Escher and in the patterns created by I. Bakshee in the Wolfram Language using Artlandia, illustrated above.For a description of the symmetry elements present in each space group, see Coxeter (1969, p. 413)...

is the point group of symmetries of the octahedron having order 48 that includes inversion. It is also the symmetry group of the cube, cuboctahedron, and truncated octahedron. It has conjugacy classes 1, , , , , , , , , and (Cotton 1990). Its multiplication table is illustrated above. The octahedral group is implemented in the Wolfram Language as FiniteGroupData["Octahedral", "PermutationGroupRepresentation"] and as a point group as FiniteGroupData["CrystallographicPointGroup", "Oh", "PermutationGroupRepresentation"].The great rhombicuboctahedron can be generated using the matrix representation of using the basis vector .The octahedral group has a pure rotation subgroup denoted that is isomorphic to the tetrahedral group . is of order 24 and has conjugacy classes 1, , , , and (Cotton 1990, pp. 50 and 434). Its multiplication table is illustrated above. The pure..

The vierergruppe is the Abelian abstract group on four elements that is isomorphic to the finite group C2×C2 and the dihedral group . The multiplication table of one possible representation is illustrated below. It can be generated by the permutations 1, 2, 3, 4, 2, 1, 4, 3, 3, 4, 1, 2, and 4, 3, 2, 1.It has subgroups , , , , and all of which are normal, so it is not a simple group. Each element is in its own conjugacy class.

The O'Nan group is the sporadic group O'Nof order(1)(2)It is implemented in the Wolfram Languageas ONanGroupON[].

The dihedral group gives the group of symmetries of a regular hexagon. The group generators are given by a counterclockwise rotation through radians and reflection in a line joining the midpoints of two opposite edges. If denotes rotation and reflection, we have(1)From this, the group elements can be listed as(2)The conjugacy classes of are given by(3)The set of elements which by themselves make up conjugacy classes are in the center of , denoted , so(4)The commutator subgroup is given by(5)which can be used to find the Abelianization. The set of all left cosets of is given by(6)(7)Thus we appear to have two generators for this group, namely and . Therefore, Abelianization gives .It is also known that where is the symmetric group. Furthermore where is the dihedral group with 6 elements, i.e., the group of symmetries of an equilateral triangle.There are thus two ways to produce the character table, either inducing from and using the orthogonality..

The trivial group, denoted or , sometimes also called the identity group, is the unique (up to isomorphism) group containing exactly one element , the identity element. Examples include the zero group (which is the singleton set with respect to the trivial group structure defined by the addition ), the multiplicative group (where ), the point group , and the integers modulo 1 under addition. When viewed as a permutation group on letters, the trivial group consists of the single element which fixes each letter.The trivial group is (trivially) Abelian and cyclic.The multiplication table for is given below. 111The trivial group has the single conjugacy class and the single subgroup .

The group is one of the two groups of order 10. Unlike the cyclic group , is non-Abelian. The molecule ruthenocene belongs to the group , where the letter indicates invariance under a reflection of the fivefold axis (Arfken 1985, p. 248). has cycle index given byIts multiplication table is illustrated above.The dihedral group has conjugacy classes , , , and . It has 8 subgroups: , , , , , , , and , of which , , and , , , , , , , , , are normal.

Given a triangle with angles (, , ), the resulting symmetry group is called a triangle group (also known as a spherical tessellation). In three dimensions, such groups must satisfyand so the only solutions are , , , and (Ball and Coxeter 1987). The group gives rise to the semiregular planar tessellations of types 1, 2, 5, and 7. The group gives hyperbolic tessellations.

In 1979, Conway and Norton discovered an unexpected intimate connection between the monster group and the j-function. The Fourier expansion of is given by(1)(OEIS A000521), where and is the half-period ratio, and the dimensions of the first few irreducible representations of are 1, 196883, 21296876, 842609326, ... (OEIS A001379).In November 1978, J. McKay noticed that the -coefficient 196884 is exactly one more than the smallest dimension of nontrivial representations of the (Conway and Norton 1979). In fact, it turns out that the Fourier coefficients of can be expressed as linear combinations of these dimensions with small coefficients as follows:(2)(3)(4)(5)Borcherds (1992) later proved this relationship, which became known as monstrous moonshine. Amazingly, there turn out to be yet more deep connections between the monster group and the j-function...

The dihedral group is one of the two non-Abelian groups of the five groups total of group order 8. It is sometimes called the octic group. An example of is the symmetry group of the square.The cycle graph of is shown above. has cycle index given by(1)Its multiplication table is illustrated above. has representation(2)(3)(4)(5)(6)(7)(8)(9)Conjugacy classes include , , , , and . There are 10 subgroups of : , , , , , , , , and , . Of these, , , , , , and are normal

The monster group is the highest order sporadic group . It has group order(1)(2)where the divisors are precisely the 15 supersingularprimes (Ogg 1980).The monster group is also called the friendly giant group. It was constructed in 1982 by Robert Griess as a group of rotations in -dimensional space.It is implemented in the Wolfram Languageas MonsterGroupM[].

The dihedral group is a particular instance of one of the two distinct abstract groups of group order 6. Unlike the cyclic group (which is Abelian), is non-Abelian. In fact, is the non-Abelian group having smallest group order.Examples of include the point groups known as , , , , the symmetry group of the equilateral triangle (Arfken 1985, p. 246), and the permutation group of three objects (Arfken 1985, p. 249).The cycle graph of is shown above. has cycle index given by(1)Its multiplication table is illustrated above and enumerated below, where 1 denotes the identity element. Equivalent but slightly different forms are given by (Arfken 1985, p. 247) and Cotton (1990, p. 12), the latter of which denotes the abstract group of by .11111111Like all dihedral groups, a reducible two-dimensional representation using real matrices has generators given by and , where is a rotation by radians about an axis passing through the..

A modulo multiplication group is a finite group of residue classes prime to under multiplication mod . is Abelian of group order , where is the totient function.A modulo multiplication group can be visualized by constructing its cycle graph. Cycle graphs are illustrated above for some low-order modulo multiplication groups. Such graphs are constructed by drawing labeled nodes, one for each element of the residue class, and connecting cycles obtained by iterating . Each edge of such a graph is bidirected, but they are commonly drawn using undirected edges with double edges used to indicate cycles of length two (Shanks 1993, pp. 85 and 87-92).The following table gives the modulo multiplication groups of small orders, together with their isomorphisms with respect to cyclic groups .groupelements2121, 221, 341, 2, 3, 421, 561, 2, 3, 4, 5, 641, 3, 5, 761, 2, 4, 5, 7, 841, 3, 7, 9101, 2, 3, 4, 5, 6, 7, 8, 9, 1041, 5, 7, 11121, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,..

The Thompson group is the sporadic group Thof order(1)(2)It is implemented in the Wolfram Languageas ThompsonGroupTh[].

The McLaughlin group is the sporadic group McLof order(1)(2)It is implemented in the Wolfram Languageas McLaughlinGroupMcL[].

The dihedral group is the symmetry group of an -sided regular polygon for . The group order of is . Dihedral groups are non-Abelian permutation groups for .The th dihedral group is represented in the Wolfram Language as DihedralGroup[n].One group presentation for the dihedral group is .A reducible two-dimensional representation of using real matrices has generators given by and , where is a rotation by radians about an axis passing through the center of a regular -gon and one of its vertices and is a rotation by about the center of the -gon (Arfken 1985, p. 250).Dihedral groups all have the same multiplication table structure. The table for is illustrated above.The cycle index (in variables , ..., ) for the dihedral group is given by(1)where(2)is the cycle index for the cyclic group , means divides , and is the totient function (Harary 1994, p. 184). The cycle indices for the first few are(3)(4)(5)(6)(7)Renteln and Dundes (2005) give..

The tetrahedral group is the point group of symmetries of the tetrahedron including the inversion operation. It is one of the 12 non-Abelian groups of order 24. The tetrahedral group has conjugacy classes 1, , , , and (Cotton 1990, pp. 47 and 434). Its multiplication table is illustrated above. The tetrahedral group is implemented in the Wolfram Language as FiniteGroupData["Tetrahedral", "PermutationGroupRepresentation"] and as a point group as FiniteGroupData["CrystallographicPointGroup", "Td", "PermutationGroupRepresentation"]. has a pure rotational subgroup of order 12 denoted (Cotton 1990, pp. 50 and 433). It is isomorphic to the alternating group and has conjugacy classes 1, , , and . It has 10 subgroups: one of length 1, three of length 2, 4 of length 3, one of length 4, and one of length 12. Of these, only the trivial subgroup, subgroup of order 4, and complete..

The five Mathieu groups , , , , and were the first sporadic groups discovered, having been found in 1861 and 1873 by Mathieu. Frobenius showed that all the Mathieu groups are subgroups of .The sporadic Mathieu groups are implemented in the Wolfram Language as MathieuGroupM11[], MathieuGroupM12[], MathieuGroupM22[], MathieuGroupM23[], and MathieuGroupM24[].All the sporadic Mathieu groups are multiply transitive. The following table summarizes some properties of the Mathieu groups, where indicates the transitivity and is the length of the minimal permutation support (from which the groups derive their designations).grouporderfactorization41179205129504032244352042310200960524244823040The Mathieu groups are most simply defined as automorphism groups of Steiner systems, as summarized in the following table.Mathieu groupSteiner system..

The cyclic group is one of the two Abelian groups of the five groups total of group order 12 (the other order-12 Abelian group being finite group C2×C6). Examples include the modulo multiplication groups of orders and 26 (which are the only modulo multiplication groups isomorphic to ).The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.The numbers of elements satisfying for , 2, ..., 12 are 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12.Because the group is Abelian, each element is in its own conjugacy class. There are six subgroups: , , , and . , and which, because the group is Abelian, are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

The symmetric group of degree is the group of all permutations on symbols. is therefore a permutation group of order and contains as subgroups every group of order .The th symmetric group is represented in the Wolfram Language as SymmetricGroup[n]. Its cycle index can be generated in the Wolfram Language using CycleIndexPolynomial[SymmetricGroup[n], x1, ..., xn]. The number of conjugacy classes of is given , where is the partition function P of . The symmetric group is a transitive group (Holton and Sheehan 1993, p. 27).For any finite group , Cayley's group theorem proves is isomorphic to a subgroup of a symmetric group.The multiplication table for is illustrated above.Let be the usual permutation cycle notation for a given permutation. Then the following table gives the multiplication table for , which has elements.(1)(2)(3)(1)(23)(3)(12)(123)(132)(2)(13)(1)(2)(3)(1)(2)(3)(1)(23)(3)(12)(123)(132)(2)(13)(1)(23)(1)(23)(1)(2)(3)(132)(2)(13)(3)(12)(123)(3)(12)(3)(12)(123)(1)(2)(3)(1)(23)(2)(13)(132)(123)(123)(3)(12)(2)(13)(132)(1)(2)(3)(1)(23)(132)(132)(2)(13)(1)(23)(1)(2)(3)(123)(3)(12)(2)(13)(2)(13)(132)(123)(3)(12)(1)(23)(1)(2)(3)This..

The Lyons group is the sporadic group Lyof order(1)(2)It is implemented in the Wolfram Languageas LyonsGroupLy[].

The cyclic group is unique group of group order 11. An example is the integers modulo 11 under addition (). No modulo multiplication group is isomorphic to . Like all cyclic groups, is Abelian.The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.Because the group is Abelian, each element is in its own conjugacy class. Because it is of prime order, the only subgroups are the trivial group and entire group. is therefore a simple group, as are all cyclic graphs of prime order.

Let be a prime number, a finite group, and the order of . 1. If divides , then has a Sylow p-subgroup. 2. In a finite group, all the Sylow p-subgroups are conjugate for some fixed . 3. The number of Sylow p-subgroups for a fixed is congruent to 1 (mod ).

The cyclic group is the unique Abelian group of group order 10 (the other order-10 group being the non-Abelian ). Examples include the integers modulo 10 under addition () and the modulo multiplication groups and (with no others). Like all cyclic groups, is Abelian.The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.The numbers of elements satisfying for , 2, ..., 10 are 1, 2, 1, 2, 5, 2, 1, 2, 1, 10.Because the group is Abelian, each element is in its own conjugacy class. There are four subgroups: , , , and . Because the group is Abelian, these are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

The cyclic group is one of the two Abelian groups of group order 9 (the other order-9 Abelian group being ; there are no non-Abelian groups of order 9). An example is the integers modulo 9 under addition (). No modulo multiplication group is isomorphic to . Like all cyclic groups, is Abelian.The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.The numbers of elements satisfying for , 2, ..., 9 are 1, 1, 3, 1, 1, 3, 1, 1, 9.Because the group is Abelian, each element is in its own conjugacy class. There are three subgroups: , and . Because the group is Abelian, these are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

The Suzuki group is the sporadic group Suzof order(1)(2)It is implemented in the Wolfram Languageas SuzukiGroupSuz[].

The cyclic group is one of the three Abelian groups of the five groups total of group order 8. Examples include the integers modulo 8 under addition () and the residue classes modulo 17 which have quadratic residues, i.e., under multiplication modulo 17. No modulo multiplication group is isomorphic to .The cycle graph of is shown above. The cycle index isIts multiplication table is illustrated above.The elements satisfy , four of them satisfy , and two satisfy .Because the group is Abelian, each element is in its own conjugacy class. There are four subgroups: , , , and which, because the group is Abelian, are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

Every finite Abelian group can be written as a group direct product of cyclic groups of prime power group orders. In fact, the number of nonisomorphic Abelian finite groups of any given group order is given by writing aswhere the are distinct prime factors, thenwhere is the partition function. This gives 1, 1, 1, 2, 1, 1, 1, 3, 2, ... (OEIS A000688).More generally, every finitely generated Abelian group is isomorphic to the group direct sum of a finite number of groups, each of which is either cyclic of prime power order or isomorphic to . This extension of Kronecker decomposition theorem is often referred to as the Kronecker basis theorem.

is the cyclic group that is the unique group of group order 7. Examples include the point group and the integers modulo 7 under addition (). No modulo multiplication group is isomorphic to . Like all cyclic groups, is Abelian.The cycle graph is shown above, and the group hascycle index isThe elements of the group satisfy , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below. 111111111Because it is Abelian, the group conjugacy classes are , , , , , , and . Because 7 is prime, the only subgroups are the trivial group and the entire group. is therefore a simple group, as are all cyclic graphs of prime order.

The sporadic groups are the 26 finite simple groups that do not fit into any of the four infinite families of finite simple groups (i.e., the cyclic groups of prime order, alternating groups of degree at least five, Lie-type Chevalley groups, and Lie-type groups). The smallest sporadic group is the Mathieu group , which has order 7920, and the largest is the monster group, which has order .The orders of the sporadic groups given in increasing order are 7920, 95040, 175560, 443520, 604800, 10200960, 44352000, 50232960, ... (OEIS A001228). A summary of sporadic groups, as given by Conway et al. (1985), is given below.nameorderfactorizationMathieu group 7920Mathieu group 95040Janko group 175560Mathieu group 443520Janko group 604800Mathieu group 10200960Higman-Sims group HS44352000Janko group 50232960Mathieu group 244823040McLaughlin group McL898128000Held group He4030387200Rudvalis Group Ru145926144000Suzuki group Suz448345497600O'Nan..

A generalization of the Kronecker decomposition theorem which states that every finitely generated Abelian group is isomorphic to the group direct sum of a finite number of groups, each of which is either cyclic of prime power order or isomorphic to . This decomposition is unique, and the number of direct summands is equal to the group rank of the Abelian group.

is one of the two groups of group order 6 which, unlike , is Abelian. It is also a cyclic. It is isomorphic to . Examples include the point groups and , the integers modulo 6 under addition (), and the modulo multiplication groups , , and (with no others).The cycle graph is shown above and has cycleindexThe elements of the group satisfy , where 1 is the identity element, three elements satisfy , and two elements satisfy .Its multiplication table is illustrated aboveand enumerated below. 11111111Since is Abelian, the conjugacy classes are , , , , , and . There are four subgroups of : , , , and which, because the group is Abelian, are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

Given a ring with identity, the special linear group is the group of matrices with elements in and determinant 1.The special linear group , where is a prime power, the set of matrices with determinant and entries in the finite field . is the corresponding set of complex matrices having determinant . is a subgroup of the general linear group and is a Lie-type group. Both and are genuine Lie groups.

is the unique group of group order 5, which is Abelian. Examples include the point group and the integers mod 5 under addition (). No modulo multiplication group is isomorphic to .The cycle graph is shown above, and the cycleindexThe elements satisfy , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below. 1111111Since is Abelian, the conjugacy classes are , , , , and . Since 5 is prime, there are no subgroups except the trivial group and the entire group. is therefore a simple group, as are all cyclic graphs of prime order.

is one of the two groups of group order 4. Like , it is Abelian, but unlike , it is a cyclic. Examples include the point groups (note that the same notation is used for the abstract cyclic group and the point group isomorphic to it) and , the integers modulo 4 under addition (), and the modulo multiplication groups and (which are the only two modulo multiplication groups isomorphic to it).The cycle graph of is shown above, and the cycle index is given by(1)The multiplication table for this group may be written in three equivalent ways by permuting the symbols used for the group elements (Cotton 1990, p. 11). One such table is illustrated above and enumerated below. 111111The conjugacy classes of are , , , and . In addition to the trivial group and the entire group, also has as a subgroup which, because the group is Abelian, is normal. is therefore not a simple group.Elements of the group satisfy , where 1 is the identity element, and two of the elements satisfy..

A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. Simple groups include the infinite families of alternating groups of degree , cyclic groups of prime order, Lie-type groups, and the 26 sporadic groups.Since all subgroups of an Abelian group are normal and all cyclic groups are Abelian, the only simple cyclic groups are those which have no subgroups other than the trivial subgroup and the improper subgroup consisting of the entire original group. And since cyclic groups of composite order can be written as a group direct product of factor groups, this means that only prime cyclic groups lack nontrivial subgroups. Therefore, the only simple cyclic groups are the prime cyclic groups. Furthermore, these are the only Abelian simple groups.In fact, the classification theorem of finite groups states that such groups can be classified completely..

The Janko groups are the four sporadic groups , , and . The Janko group is also known as the Hall-Janko group.The Janko groups are implemented in the Wolfram Language as JankoGroupJ1[], JankoGroupJ2[], JankoGroupJ3[], and JankoGroupJ4[].The following table summarized the group orders ofthe Janko groups.grouporderfactorization1755606048005023296086775571046077562880

is the unique group of group order 3. It is both Abelian and cyclic. Examples include the point groups , , and and the integers under addition modulo 3 (). No modulo multiplication groups are isomorphic to .The cycle graph of is shown above, and the cycle index isThe elements of the group satisfy where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below (Cotton 1990, p. 10). 11111Since is Abelian, the conjugacy classes are , , and . The only subgroups of are the trivial group and the entire group, which are both trivially normal. is therefore a simple group, as are all cyclic graphs of prime order.The irreducible representation (character table)is therefore 11111111

The icosahedral group is the group of symmetries of the icosahedron and dodecahedron having order 120, equivalent to the group direct product of the alternating group and cyclic group . The icosahedral group consists of the conjugacy classes 1, , , , , , , , , and (Cotton 1990, pp. 49 and 436). Its multiplication table is illustrated above. The icosahedral group is a subgroup of the special orthogonal group . The icosahedal group is implemented in the Wolfram Language as FiniteGroupData["Icosahedral", "PermutationGroupRepresentation"].Icosahedral symmetry is possible as a rotational group but is not compatible with translational symmetry. As a result, there are no crystals with this symmetry and so, unlike the octahedral group and tetrahedral group , is not one of the 32 point groups.The great rhombicosidodecahedron can be generated using the matrix representation of using the basis vector , where is the golden..

The group is the unique group of group order 2. is both Abelian and cyclic. Examples include the point groups , , and , the integers modulo 2 under addition (), and the modulo multiplication groups , , and (which are the only modulo multiplication groups isomorphic to ).The group is also trivially simple, and forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University mathematics department a capella group "The Klein Four."The cycle graph is shown above, and the cycleindex isThe elements satisfy , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below. 1111The conjugacy classes are and . The only subgroups of are the trivial group and entire group , both of which are trivially normal.The irreducible representation for the group is ...

The Rudvalis group is the sporadic group Ruof order(1)(2)It is implemented in the Wolfram Languageas RudvalisGroupRu[].

A cyclic group is a group that can be generated by a single element (the group generator). Cyclic groups are Abelian.A cyclic group of finite group order is denoted , , , or ; Shanks 1993, p. 75), and its generator satisfies(1)where is the identity element.The ring of integers form an infinite cyclic group under addition, and the integers 0, 1, 2, ..., () form a cyclic group of order under addition (mod ). In both cases, 0 is the identity element.There exists a unique cyclic group of every order , so cyclic groups of the same order are always isomorphic (Scott 1987, p. 34; Shanks 1993, p. 74). Furthermore, subgroups of cyclic groups are cyclic, and all groups of prime group order are cyclic. In fact, the only simple Abelian groups are the cyclic groups of order or a prime (Scott 1987, p. 35).The th cyclic group is represented in the Wolfram Language as CyclicGroup[n].Examples of cyclic groups include , , , ..., and the modulo multiplication..

The quaternion group is one of the two non-Abelian groups of the five total finite groups of order 8. It is formed by the quaternions , , , and , denoted or .1111111111The multiplication table for is illustrated above, where rows and columns are given in the order , , , , 1, , , , as in the table above.The cycle graph of the quaternion group is illustratedabove.The quaternion group has conjugacy classes , , , , and . Its subgroups are , , , , , and , all of which are normal subgroups.

The Held group is the sporadic group Heof order(1)(2)It is implemented in the Wolfram Languageas HeldGroupHe[].

Let denote the number of cycles of length for a permutation expressed as a product of disjoint cycles. The cycle index of a permutation group of order and degree is then the polynomial in variables , , ..., given by the formula(1)The cycle index of a permutation group is implemented as CycleIndexPolynomial[perm, x1, ..., xn], which returns a polynomial in . For any permutation , the numbers satisfy(2)and thus constitutes a partition of the integer . Sets of values are commonly denoted , where ranges over all the -vectors satisfying equation (2).Formulas for the most important permutation groups (the symmetric group , alternating group , cyclic group , dihedral group , and trivial group ) are given by(3)(4)(5)(6)(7)where means divides and is the totient function (Harary 1994, p. 184)...

The projective symplectic group is the group obtained from the symplectic group on factoring by the scalar matrices contained in that group. is simple except for(1)(2)(3)so it is given the simpler name , with .

The projective special unitary group is the group obtained from the special unitary group on factoring by the scalar matrices contained in that group. is simple except for(1)(2)(3)so it is given the simpler name , with .

A Hajós group is a group for which all factorizations of the form (say) have or periodic, where the period is a divisor of . Hajós groups arose after solution of Minkowski's conjecture on about tiling space with nonoverlapping cuboids. The classification of Hajós finite Abelian groups was achieved by Sands in the 1980s.For example, (mod 12), so while the first factor is acyclic, the second factor has period three. Since this turns out to be the case for all tilings of , it is a Hajós group. The smallest case where this is false is , followed by .The cyclic group of order is a Hajós group if is of the form , , , , , or , where , , , and are distinct primes and and arbitrary integers. Non-Hajós groups therefore have orders 72, 108, 120, 144, 168, 180, 200, 216, ... (OEIS A102562)...

The crystallographic point groups are the point groups in which translational periodicity is required (the so-called crystallography restriction). There are 32 such groups, summarized in the following table which organizes them by Schönflies symbol type.typepoint groupsnonaxial, cyclic, , , , cyclic with horizontal planes, , , cyclic with vertical planes, , , dihedral, , , dihedral with horizontal planes, , , dihedral with planes between axes, improper rotation, cubic groups, , , , Note that while the tetrahedral and octahedral point groups are also crystallographic point groups, the icosahedral group is not. The orders, classes, and group operations for these groups can be concisely summarized in their character tables.

The projective special orthogonal group is the group obtained from the special orthogonal group on factoring by the scalar matrices contained in that group. In general, this group is not simple.

The number of elements in a group , denoted . If the order of a group is a finite number, the group is said to be a finite group.The order of an element of a finite group is the smallest power of such that , where is the identity element. In general, finding the order of the element of a group is at least as hard as factoring (Meijer 1996). However, the problem becomes significantly easier if and the factorization of are known. Under these circumstances, efficient algorithms are known (Cohen 1993).The group order can be computed in the WolframLanguage using the function GroupOrder[n].

The automorphism group of the Leech lattice modulo a center of order two is called "the" Conway group. There are 15 exceptional conjugacy classes of the Conway group. This group, combined with the groups and obtained similarly from the Leech lattice by stabilization of the one- and two-dimensional sublattices, are collectively called Conway groups.The Conway groups are sporadic groups. The are implemented in the Wolfram Language as ConwayGroupCo1[], ConwayGroupCo2[], and ConwayGroupCo3[].The following table summarizes some properties of the Conway groups, where indicates the transitivity and is the length of the minimal permutation support.grouporderfactorization227649576665600012300423054213120001982804157776806543360000

The projective special linear group is the group obtained from the special linear group on factoring by the scalar matrices contained in that group. It is simple for except for(1)(2)and is therefore also denoted .

The general unitary group is the subgroup of all elements of the general linear group that fix a given nonsingular Hermitian form. This is equivalent, in the canonical case, to the definition of as the group of unitary matrices.

The general orthogonal group is the subgroup of all elements of the projective general linear group that fix the particular nonsingular quadratic form . The determinant of such an element is .

The finite simple groups of Lie-type. They include four families of linear simple groups: (the projective special linear group), (the projective special unitary group), (the projective symplectic group), and .The following table lists exceptional (untwisted) Chevalley groups.grouporder

Given a ring with identity, the general linear group is the group of invertible matrices with elements in .The general linear group is the set of matrices with entries in the field which have nonzero determinant.

A characteristic factor is a factor in a particular factorization of the totient function such that the product of characteristic factors gives the representation of a corresponding abstract group as a group direct product. By computing the characteristic factors, any Abelian group can be expressed as a group direct product of cyclic subgroups, for example, the finite group C2×C4 or the finite group C2×C2×C2. There is a simple algorithm for determining the characteristic factors of modulo multiplication groups.

Let be an extension field of , denoted , and let be the set of automorphisms of , that is, the set of automorphisms of such that for every , so that is fixed. Then is a group of transformations of , called the Galois group of . The Galois group of is denoted or .Let be a rational polynomial of degree and let be the splitting field of over , i.e., the smallest subfield of containing all the roots of . Then each element of the Galois group permutes the roots of in a unique way. Thus can be identified with a subgroup of the symmetric group , the group of permutations of the roots of . If is irreducible, then is a transitive subgroup of , i.e., given two roots and of , there exists an element of such that .The roots of are solvable by radicals iff is a solvable group. Since all subgroups of with are solvable, the roots of all polynomials of degree up to 4 are solvable by radicals. However, polynomials of degree 5 or greater are generally not solvable by radicals since (and the alternating..

A finite group has a finite number of conjugacy classes and a finite number of distinct irreducible representations. The group character of a group representation is constant on a conjugacy class. Hence, the values of the characters can be written as an array, known as a character table. Typically, the rows are given by the irreducible representations and the columns are given the conjugacy classes.A character table often contains enough information to identify a given abstract group and distinguish it from others. However, there exist nonisomorphic groups which nevertheless have the same character table, for example (the symmetry group of the square) and (the quaternion group).For example, the symmetric group on three letters has three conjugacy classes, represented by the permutations , , and . It also has three irreducible representations; two are one-dimensional and the third is two-dimensional: 1. The trivial representation ...

The Fischer groups are the three sporadic groups , , and . These groups were discovered during the investigation of 3-transposition groups.The Fischer groups are implemented in the Wolfram Language as FischerGroupFi22[], FischerGroupFi23[], and FischerGroupFi24Prime[].The following table summarizes the orders of the Fischer groups.grouporderfactorization6456175165440040894704732930048001255205709190661721292800The baby monster group is sometimes called Fischer'sbaby monster group.

The finite group is one of the three non-Abelian groups of order 12 (out of a total of fives groups of order 12), the other two being the alternating group and the dihedral group . However, it is highly unfortunate that the symbol is used to refer this particular group, since the symbol is also used to denote the point group that constitutes the pure rotational subgroup of the full tetrahedral group and is isomorphic to . Thus, of the three distinct non-Abelian groups of order 12, two different ones are each known as under some circumstances. Extreme caution is therefore needed. is the semidirect product of by by the map given by , where is the automorphism . The group can be constructed from the generators(1)(2)where as the group elements 1, , , , , , , , , , , and . The multiplication table is illustrated above. has conjugacy classes , , , , , and . There are 8 subgroups, and their lengths are 1, 2, 3, 4, 4, 4, 6, and 12. Of these, the following five are normal: , , , , and..

The finite group is the finite group of order 12 that is the group direct product of the cyclic group C2 and cyclic group C6. It is one of the two Abelian groups of order 12, the other being the cyclic group C12. Examples include the modulo multiplication groups , , , and (and no other modulo multiplication groups).The multiplication table is illustrated above. The numbers of elements for which for , 2, ..., 12 are 1, 4, 3, 4, 1, 12, 1, 4, 3, 4, 1, 12.Each element of is in its own conjugacy class. There are 10 subgroups: the trivial subgroup, 3 of length 2, 1 of length 3, 1 of length 4, 3 of length 6, and the improper subgroup consisting of the entire group. Since is Abelian, all its subgroups are normal. Since it has normal subgroups other than the trivial subgroup and the improper subgroup, is not a simple group...

The Burnside problem originated with Burnside (1902), who wrote, "A still undecided point in the theory of discontinuous groups is whether the group order of a group may be not finite, while the order of every operation it contains is finite." This question would now be phrased, "Can a finitely generated group be infinite while every element in the group has finite order?" (Vaughan-Lee 1993). This question was answered by Golod (1964) when he constructed finitely generated infinite p-groups. These groups, however, do not have a finite exponent.Let be the free group of group rank and let be the normal subgroup generated by the set of th powers . Then is a normal subgroup of . Define to be the quotient group. We call the -generator Burnside group of exponent . It is the largest -generator group of exponent , in the sense that every other such group is a homomorphic image of . The Burnside problem is usually stated as, "For which..

A permutation group is a finite group whose elements are permutations of a given set and whose group operation is composition of permutations in . Permutation groups have orders dividing .Two permutations form a group only if one is the identity element and the other is a permutation involution, i.e., a permutation which is its own inverse (Skiena 1990, p. 20). Every permutation group with more than two elements can be written as a product of transpositions.Permutation groups are represented in the Wolfram Language as a set of permutation cycles with PermutationGroup. A set of permutations may be tested to see if it forms a permutation group using PermutationGroupQ[l] in the Wolfram Language package Combinatorica` .Conjugacy classes of elements which are interchangedin a permutation group are called permutation cycles.Examples of permutation groups include the symmetric group (of order ), the alternating group (of order for ),..

is one of the three Abelian groups of group order 8 (the other two being non-Abelian). Examples include the modulo multiplication groups , , , and (and no others).The elements of this group satisfy , where 1 is the identity element, and four of the elements satisfy . The cycle graph is shown above.Its multiplication table is illustrated above.Since the group is Abelian, each element is in its own conjugacyclass.The subgroups are , , , , , , , and , A, B, C, D, E, F, . Since the group is Abelian, all of these are normal.

The pair group of a group is the group that acts on the 2-subsets of whose permutations are induced by . Pair groups can be calculated using PairGroup[g] in the Wolfram Language package Combinatorica` .The cycle index for the pair group induced by is(1)(Harary 1994, p. 185). Here, is the floor function, is a binomial coefficient, LCM is the least common multiple, GCD is the greatest common divisor, the sum is over all exponent vectors of the cycle index of the symmetric group , and is the coefficient of the term with exponent vector in . The first few values of are(2)(3)(4)(5)(6)These can be given by PairGroup[SymmetricGroup[n], x] in the Wolfram Language package Combinatorica` .

The group is one of the three Abelian groups of order 8 (the other two groups are non-Abelian). An example is the modulo multiplication group (which is the only modulo multiplication group isomorphic to ).The cycle graph is shown above. The elements of this group satisfy , where 1 is the identity element.Its multiplication table is illustrated above.Each element is in its own conjugacy class. The subgroups are given by , , , , , , , , , , , , , , , and . Since the group is Abelian, all of these are normal subgroups.

The finite group is one of the two distinct groups of group order 4. The name of this group derives from the fact that it is a group direct product of two subgroups. Like the group , is an Abelian group. Unlike , however, it is not cyclic.The abstract group corresponding to is called the vierergruppe. Examples of the group include the point groups , , and , and the modulo multiplication groups and (and no other modulo multiplication groups). That , the residue classes prime to 8 given by , are a group of type can be shown by verifying that(1)and(2) is therefore a modulo multiplication group.The cycle graph is shown above. In addition to satisfying for each element , it also satisfies , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below (Cotton 1990, p. 11). 111111Since the group is Abelian, the conjugacy classes are , , , and . Nontrivial proper subgroups of are , , and .Now explicitly consider the elements..

The baby monster group, also known as Fischer's baby monster group, is the second-largest sporadic group. It is denoted and has group order(1)(2)It is implemented in the Wolfram Languageas BabyMonsterGroupB[].

A finite group is a group having finite group order. Examples of finite groups are the modulo multiplication groups, point groups, cyclic groups, dihedral groups, symmetric groups, alternating groups, and so on.Properties of finite groups are implemented in the Wolfram Language as FiniteGroupData[group, prop].The classification theorem of finite groups states that the finite simple groups can be classified completely into one of five types.A convenient way to visualize groups is using so-called cycle graphs, which show the cycle structure of a given abstract group. For example, cycle graphs of the 5 nonisomorphic groups of order 8 are illustrated above (Shanks 1993, p. 85).Frucht's theorem states that every finite group is the graph automorphism group of a finite undirected graph.The finite (cyclic) group forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University..

An alternating group is a group of even permutations on a set of length , denoted or Alt() (Scott 1987, p. 267). Alternating groups are therefore permutation groups.The th alternating group is represented in the Wolfram Language as AlternatingGroup[n].An alternating group is a normal subgroup of the permutation group, and has group order , the first few values of which for , 3, ... are 1, 3, 12, 60, 360, 2520, ... (OEIS A001710). The alternating group is -transitive.Amazingly, the pure rotational subgroup of the icosahedral group is isomorphic to . The full icosahedral group is isomorphic to the group direct product , where is the cyclic group on two elements.Alternating groups with are simple groups (Scott 1987, p. 295), i.e., their only normal subgroups are the trivial subgroup and the entire group .The number of conjugacy classes in the alternating groups for , 3, ... are 1, 3, 4, 5, 7, 9, ... (OEIS A000702). is the only nontrivial..

For a finite group , let be the subgroup generated by all the Sylow p-subgroups of . If is a projective curve in characteristic , and if , ..., are points of (for ), then a necessary and sufficient condition that occur as the Galois group of a finite covering of , branched only at the points , ..., , is that the quotient group has generators.Raynaud (1994) solved the Abhyankar problem in the crucial case of the affine line (i.e., the projective line with a point deleted), and Harbater (1994) proved the full Abhyankar conjecture by building upon this special solution.

The classification theorem of finite simple groups, also known as the "enormous theorem," which states that the finite simple groups can be classified completely into 1. Cyclic groups of prime group order, 2. Alternating groups of degree at least five, 3. Lie-type Chevalley groups given by , , , and , 4. Lie-type (twisted Chevalley groups or the Tits group) , , , , , , , , , 5. Sporadic groups , , , , , , Suz, HS, McL, , , , He, , , , HN, Th, , , , O'N, , Ly, Ru, . The "proof" of this theorem is spread throughout the mathematical literature and is estimated to be approximately pages in length.

The group of all nonsingular stochastic matrices over a field . It is denoted . If is prime and is the finite field of order , is written instead of . Particular examples include(1)(2)(3)(4)(5)where is an Abelian group, are symmetric groups on elements, and denotes the semidirect product with (Poole 1995).

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