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Suppose that E(G) (the commuting product of all components of G) is simple and G contains a semisimple group involution. Then there is some semisimple group involution x such ...

The underlying set of the fundamental group of X is the set of based homotopy classes from the circle to X, denoted [S^1,X]. For general spaces X and Y, there is no natural ...

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. An outer automorphism of G is an automorphism which cannot ...

Suppose that V is a group representation of G, and W is a group representation of H. Then the vector space tensor product V tensor W is a group representation of the group ...

A characteristic factor is a factor in a particular factorization of the totient function phi(n) such that the product of characteristic factors gives the representation of a ...

For a finite group of h elements with an n_ith dimensional ith irreducible representation, sum_(i)n_i^2=h.

A p-element x of a group G is semisimple if E(C_G(x))!=1, where E(H) is the commuting product of all components of H and C_G(x) is the centralizer of G.

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

A complete set of mutually conjugate group elements. Each element in a group belongs to exactly one class, and the identity element (I=1) is always in its own class. The ...

The cokernel of a group homomorphism f:A-->B of Abelian groups (modules, or abstract vector spaces) is the quotient group (quotient module or quotient space, respectively) ...