A5 is simple

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Statement

The alternating group of degree five (denoted A_5) 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 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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