Finite group

The properties of finite groups can thus play a role in subjects such as theoretical physics and chemistry.

(n factorial) possible permutations of a set of n symbols, it follows that the order (the number of elements) of the symmetric group Sn is n!.

[5] An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants.

The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.

Hence each non-Abelian finite simple group has order divisible by at least three distinct primes.

However, a significant difference with respect to the case of integer factorization is that such "building blocks" do not necessarily determine uniquely a group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.

The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.

Given a positive integer n, it is not at all a routine matter to determine how many isomorphism types of groups of order n there are.

If n is a higher power of a prime, then results of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphism types of groups of order n, and the number grows very rapidly as the power increases.

Depending on the prime factorization of n, some restrictions may be placed on the structure of groups of order n, as a consequence, for example, of results such as the Sylow theorems.