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A quasigroup with an identity element e such that xe=x and ex=x for any x in the quasigroup. All groups are loops. In general, loops are considered to have very little in the ...

The commutator series of a Lie algebra g, sometimes called the derived series, is the sequence of subalgebras recursively defined by g^(k+1)=[g^k,g^k], (1) with g^0=g. The ...

A Steiner system S(t,k,v) is a set X of v points, and a collection of subsets of X of size k (called blocks), such that any t points of X are in exactly one of the blocks. ...

A square matrix A is said to be unipotent if A-I, where I is an identity matrix is a nilpotent matrix (defined by the property that A^n is the zero matrix for some positive ...

The five Mathieu groups M_(11), M_(12), M_(22), M_(23), and M_(24) were the first sporadic groups discovered, having been found in 1861 and 1873 by Mathieu. Frobenius showed ...

L is a subnormal subgroup of H if there is a "normal series" (in the sense of Jordan-Hölder) from L to H.

Given the direct sum of additive Abelian groups A direct sum B, A and B are called direct summands. The map i_1:A-->A direct sum B defined by i(a)=a direct sum 0 is called ...

A program initiated by F. Klein in an 1872 lecture to describe geometric structures in terms of their automorphism groups.

A semigroup with a noncommutative product in which no product can ever be expressed more simply in terms of other elements.

Two groups are isomorphic if the correspondence between them is one-to-one and the "multiplication" table is preserved. For example, the point groups C_2 and D_1 are ...