Perfectness is not subgroup-closed
This article gives the statement, and possibly proof, of a group property (i.e., perfect group) not satisfying a group metaproperty (i.e., subgroup-closed group property).
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In fact, the following somewhat stronger statement is true: for any nontrivial perfect group , we can find a subgroup that is not perfect. Note that since nontrivial perfect groups do exist (for instance, alternating group:A5) this statement is indeed stronger.
- Perfectness is not characteristic subgroup-closed
- Every finite group is a subgroup of a finite perfect group
- Every group is a subgroup of a perfect group
Given: A nontrivial perfect group
To prove: has a subgroup that is not perfect.
Proof: Take any non-identity element of , and define as the cyclic subgroup generated by that element. is a nontrivial cyclic group, and since cyclic implies abelian, it is a nontrivial abelian group, hence not perfect.