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Abstract Algebra

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Moderate

Normal Subgroups of the Alternating Group on $4$ Letters

ABSALG-SR3JNW

Recall that the alternating group on 4 letters is denoted by $A_4$ and is the subgroup of even permutations on 4 letters. We can list its elements as $A_4=\{e, (123), (132), (134), (143), (234), (243), (124), (142), (12)(34), (13)(24), (14)(23)\}$.

Which of the following statements is true about $A_4$ and its subgroups? Select ALL that apply.

A

$A_4$ has no normal subgroups of order 3.

B

$A_4$ has no normal subgroups of index 3.

C

$A_4$ has a subgroup of order 6.

D

$A_4$ has a unique subgroup of order 4.