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Abstract Algebra

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Moderate

Commutator Subgroups, Symmetric, Alternating

ABSALG-RWFJNE

Let $G$ be a finite group and let $G^{(1)}=[G,G]$, the commutator subgroup of $G$.

Let $G^{(i)}=[G^{(i-1)},G^{(i-1)}]$, $i\ge 2$.

Which of the following is FALSE?

A

$[S_2,S_2]$ is isomorphic to $[A_3,A_3]$

B

$S_4^{(2)}=A_4$

C

$S_n^{(1)}=A_n$, for all $n\ge 2$

D

$S_n^{(3)}=A_n$ for all $n\ge 5$

E

$S_4^{(5)}=\{(1)\}$.