# Alternating group:A4

From Groupprops

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## Contents

## Definition

### Permutation definition

The alternating group is defined as the group of all even permutations on a set of 4 elements.

### Presentation

## Group properties

### Solvability

*This particular group is solvable*

The commutator subgroup of is the Klein group of order 4, namely the normal subgroup comprising double transpositions. This is Abelian.

Thus, is solvable of solvable length 2, or in other words, it is a metabelian group.

### Nilpotence

*This particular group is not nilpotent*

### Abelianness

*This particular group is not Abelian*

### Simplicity

*This particular group is not simple*

Since has a proper nontrivial commutator subgroup, it is not simple.