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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 class formed by sets in R^n which have essentially the same structure, regardless of size, shape and dimension. The "essential structure" is what a set keeps when it is ...

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) ...

Every finite simple group (that is not cyclic) has even group order, and the group order of every finite simple noncommutative group is doubly even, i.e., divisible by 4 ...

The subset {0} of a ring. It trivially fulfils the definition of ideal since it is a group (specifically, the zero group), and it is closed under multiplication by any ...

The polynomial giving the number of colorings with m colors of a structure defined by a permutation group.

A group of five elements, also called a quintuplet or pentad.