What is finite group?

Finite Group

In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of solvable groups and nilpotent groups. A complete determination of the structure of all finite groups is too much to hope for; the number of possible structures soon becomes overwhelming. However, the complete classification of the finite simple groups was achieved, meaning that the "building blocks" from which all finite groups can be built are now known, as each finite group has a composition series.

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Some articles on finite group:

Schur Multiplier - Relation To Projective Representations
... to classify projective representations of a group, and the modern formulation of his definition is the second cohomology group H2(G, C×) ... A projective representation is much like a group representation except that instead of a homomorphism into the general linear group GL(n, C), one takes a homomorphism into ... Schur (1904, 1907) showed that every finite group G has associated to it at least one finite group C, called a Schur cover, with the property that every projective representation of G can be ...
Maschke's Theorem - Reformulation and The Meaning
... One of the approaches to representations of finite groups is through module theory ... Representations of a group G are replaced by modules over its group algebra K ... Maschke's theorem addresses the question is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? In the ...
Iwasawa Group
... In mathematics a group is sometimes called an Iwasawa group or M-group or modular group if its lattice of subgroups is modular ... Finite modular groups are also called Iwasawa groups, after (Iwasawa 1941) where they were classified ... Both finite and infinite M-groups are presented in textbook form in (Schmidt 1994, Ch ...
Examples of Hopfian Groups
... Every finite group, by an elementary counting argument ... More generally, every polycyclic-by-finite group ... Any finitely-generated free group ...

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