# A5 is simple

From Groupprops

## Contents

## Statement

The alternating group of degree five (denoted ) is a simple group.

## Related facts

## Proof

### Proof using sizes of conjugacy classes

`Further information: element structure of alternating group:A5`

The conjugacy class sizes are . A normal subgroup must contain the conjugacy class of size , and one or more other conjugacy classes. Thus, the order of any normal subgroup must be a sum of some of these numbers, including the . By Lagrange's theorem, the order must also divide the order of the group. But no such sum among these numbers divides , other than and themselves.

### Proof by listing subgroups

`Further information: subgroup structure of alternating group:A5`

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