# Finite group

Finite groups occur in the mathematical branch of group theory . A group is called a finite group if it is a finite set , i.e. has a finite number of elements. ${\ displaystyle (G, *)}$${\ displaystyle G}$

## Axioms

The assumption of finitude enables a simplified system of axioms:

A pair with a finite set and an inner two-place relation is called a group if the following axioms are fulfilled: ${\ displaystyle (G, *)}$ ${\ displaystyle G}$ ${\ displaystyle * \ colon G \ times G \ to G}$

• Associativity : Thefollowing appliestoall group elements${\ displaystyle a, b, c}$${\ displaystyle (a * b) * c = a * (b * c)}$
• Reduction rule : offorfollows.${\ displaystyle a * x = a * x '}$${\ displaystyle x * a = x '* a}$${\ displaystyle x = x '}$

From the abbreviation rule it follows that the left and right multiplications are and injective , from which the surjectivity follows because of the finiteness . Hence there is one with which shows the existence of the neutral element and then one with which shows the existence of the inverse elements . ${\ displaystyle x \ mapsto a * x}$${\ displaystyle x \ mapsto x * a}$ ${\ displaystyle x}$${\ displaystyle a * x = a}$ ${\ displaystyle e}$${\ displaystyle x}$${\ displaystyle a * x = e}$

## Finite subgroup

The general condition that a nonempty set is a subgroup of the group , ${\ displaystyle S \ subseteq G}$${\ displaystyle G}$

S1: ${\ displaystyle \ quad a, b \ in S \ Rightarrow a * b \ in S}$
S2: ${\ displaystyle \ quad a \ in S \ Rightarrow a ^ {- 1} \ in S}$

is also simplified, since S2 follows from S1: If is finite, every element of must have a finite order , from which it follows. But that means that it is already in . A non-empty finite subset of any group is a subgroup if and only if in is also for all . ${\ displaystyle S}$${\ displaystyle a}$${\ displaystyle S}$${\ displaystyle n}$${\ displaystyle a ^ {n} = e}$${\ displaystyle a ^ {n-1} = a ^ {- 1}}$${\ displaystyle S}$${\ displaystyle S}$${\ displaystyle a, b \ in S}$${\ displaystyle a * b}$${\ displaystyle S}$

## Simple groups

Every finite group is composed of a finite number of finite simple groups . However, this composition can be complicated. Despite knowing the building blocks (the simple groups) one is still far from knowing all finite groups.

Although the finite simple groups have been classified as fully classified since 1982, mathematicians working with Aschbacher did not complete the classification until 2002 with a 1200-page proof:

## Applications

Symmetries of bodies, especially in molecular physics, are described by point groups ; Symmetries of crystals through 230 different room groups .