A5 is simple
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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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