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Given a group G, the algebra CG is a vector space CG={suma_ig_i|a_i in C,g_i in G} of finite sums of elements of G, with multiplication defined by g·h=gh, the group ...

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 ...

Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. ...

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.