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Abstract Algebra

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Moderate

To What Extent Does the Converse of Lagrange's Theorem Fail?

ABSALG-I1NBAB

Consider the direct product $G=A_4\times \mathbb{Z}_3$, which is a group of order 36. Note that 36 has for divisors 1, 2, 3 , 4, 6, 9, 12, 18, 36.

Which of the following statements is FALSE about existence of subgroups of $G$ and the given order?

A

$G$ has a subgroup of order $3$.

B

$G$ has a subgroup of order $4$.

C

$G$ has a subgroup of order $6$.

D

$G$ has a subgroup of order $9$.

E

$G$ has a subgroup of order $18$.