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Abstract Algebra

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Moderate

What Does the Quotient Group say about the Group?

ABSALG-SFUYY2

Which of the following statements is TRUE about quotient groups?

A

If $G$ has a proper normal subgroup $H$ such that $G/H$ is cyclic, then $G$ is cyclic.

B

If $G$ has a proper normal subgroup $H$ such that $G/H$ is Abelian, then $G$ is Abelian.

C

If $G$ is cyclic, then for every normal subgroup $H$, the quotient group $G/H$ is Abelian.

D

If $G$ has a normal subgroup $H$ and $aH$ is an element of finite order in $G/H$, then $a$ has finite order in $G$.

E

If $G$ has a normal subgroup $H$, then for every element $a\in G$ of finite order, $aH$ has finite order in $G/H$ and $|a|=|aH|$.