A5 is simple
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:A5PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]