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An extension A subset B of a group, ring, module, field, etc., such that A!=B.

An inner automorphism of a group G is an automorphism of the form phi(g)=h^(-1)gh, where h is a fixed element of G. The automorphism of the symmetric group S_3 that maps the ...

Let p be a prime number, G a finite group, and |G| the order of G. 1. If p divides |G|, then G has a Sylow p-subgroup. 2. In a finite group, all the Sylow p-subgroups are ...

In general, groups are not Abelian. However, there is always a group homomorphism h:G->G^' to an Abelian group, and this homomorphism is called Abelianization. The ...

On a Lie group, exp is a map from the Lie algebra to its Lie group. If you think of the Lie algebra as the tangent space to the identity of the Lie group, exp(v) is defined ...

Let G be a group and S be a topological G-set. Then a closed subset F of S is called a fundamental domain of G in S if S is the union of conjugates of F, i.e., S= union _(g ...

Given a principal bundle pi:A->M, with fiber a Lie group G and base manifold M, and a group representation of G, say phi:G×V->V, then the associated vector bundle is ...

A finite group G has a finite number of conjugacy classes and a finite number of distinct irreducible representations. The group character of a group representation is ...

A positive definite function f on a group G is a function for which the matrix {f(x_ix_j^(-1))} is always positive semidefinite Hermitian.

For a subgroup H of a group G, the index of H, denoted (G:H), is the cardinal number of the set of left cosets of H in G (which is equal to the cardinal number of the set of ...