is central to the study of all finite groups. The finite simple groups are known and classified. The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of cyclic p-groups. This can be extended to a complete classification of all finitely generated abelian groups, that is all abelian groups that are generated by a finite set. The situation is much more complicated for the non-abelian groups. Free group. Given any set , one can define a group as the smallest group containing the free semigroup of . The group consists of the finite strings (words) that can be composed by elements from , together with other elements that are necessary to form a group. Multiplication of strings is defined by concatenation, for instance Every group is basically a factor group of a free group generated by . Please refer to presentation of a group for more explanation. One can then ask algorithmic questions about these presentations, such as: Do these two presentations specify isomorphic groups?; or Does this presentation specify the trivial group? The general case of this is the word problem, and several of these questions are in fact unsolvable by any general algorithm. General linear group, denoted by GL(n, F), is the group of -by- invertible matrices, where the elements of the matrices are taken from a field such as the real numbers or the complex numbers. Group representation (not to be confused with the presentation of a group). A group representation is a homomorphism from a group to a general linear group. One basically tries to "represent" a given abstract group as a concrete group of invertible matrices which is much easier to study. See also Glossary of Lie groups and Lie algebras Glossary of ring theory Composition series Normal series Group theory Wikipedia glossaries using description lists