A5 is simple

Statement

The alternating group of degree five (denoted $A_{5}$) is a simple group.

Related facts

• A stronger statement about $A_{5}$: A5 is the simple non-Abelian group of smallest order

Proof

Proof using sizes of conjugacy classes

Further information: element structure of alternating group:A5

The conjugacy class sizes are $1,12,12,20,15$. A normal subgroup must contain the conjugacy class of size $1$, 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 $1$. By Lagrange's theorem, the order must also divide the order of the group. But no such sum among these numbers divides $60$, other than $1$ and $60$ themselves.

Proof by listing subgroups

Further information: subgroup structure of alternating group:A5

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