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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 ...

C_7 is the cyclic group that is the unique group of group order 7. Examples include the point group C_7 and the integers modulo 7 under addition (Z_7). No modulo ...

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 ...

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).

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, ...

A set of generators (g_1,...,g_n) is a set of group elements such that possibly repeated application of the generators on themselves and each other is capable of producing ...

The term "homology group" usually means a singular homology group, which is an Abelian group which partially counts the number of holes in a topological space. In particular, ...

A cycle of a finite group G is a minimal set of elements {A^0,A^1,...,A^n} such that A^0=A^n=I, where I is the identity element. A diagram of a group showing every cycle in ...