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A solvable group is a group having a normal series such that each normal factor is Abelian. The special case of a solvable finite group is a group whose composition indices ...

A group homomorphism is a map f:G->H between two groups such that the group operation is preserved:f(g_1g_2)=f(g_1)f(g_2) for all g_1,g_2 in G, where the product on the ...

Let L be an extension field of K, denoted L/K, and let G be the set of automorphisms of L/K, that is, the set of automorphisms sigma of L such that sigma(x)=x for every x in ...

The special orthogonal group SO_n(q) is the subgroup of the elements of general orthogonal group GO_n(q) with determinant 1. SO_3 (often written SO(3)) is the rotation group ...

A representation of a group G is a group action of G on a vector space V by invertible linear maps. For example, the group of two elements Z_2={0,1} has a representation phi ...

A free Abelian group is a group G with a subset which generates the group G with the only relation being ab=ba. That is, it has no group torsion. All such groups are a direct ...

Every finite group of order n can be represented as a permutation group on n letters, as first proved by Cayley in 1878 (Rotman 1995).

Let A and B be any sets. Then the product of |A| and |B| is defined as the Cartesian product |A|*|B|=|A×B| (Ciesielski 1997, p. 68; Dauben 1990, p. 173; Moore 1982, p. 37; ...

An extension of a group H by a group N is a group G with a normal subgroup M such that M=N and G/M=H. This information can be encoded into a short exact sequence of groups ...

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, ...