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Abstract Algebra

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Perfect Groups: Definition , Identifying Perfect Groups

ABSALG-LFMBK0

A group $G$ is defined to be perfect if and only if $G=[G,G]$, where $[G,G]$ is the commutator or derived subgroup of $G$.

Which of the following groups are perfect?

The group $S_n$ is the symmetric group on $n$ symbols and $A_n$ is the alternating group, the subgroup of even permutations in $S_n$.

A

Every non-abelian simple group.

B

The symmetric group $S_n$, $n\ge 2$.

C

The group ${\rm GL}_n(F)$ of $n\times n$ invertible matrices with coefficients in a field $F$, $n\ge 2$.

D

The group $[S_4,S_4]$

E

The group $[A_4,A_4]$