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Abstract Algebra

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Moderate

Quotient by the Center: What Does It Hide?

ABSALG-JR@MOY

Recall that for every group $G$, the center $Z(G)$ of $G$ is the set of elements in $G$ that commute with every element in $G$, that is:

$$Z(G)=\{x\in G:xg=gx\; \; \text{for all}\; \; g\in G\}$$

It is clear that $Z(G)$ is a normal subgroup of $G$.
Which of the following facts is true about the quotient group $G/Z(G)$?

A

For every group $G$, the group $G/Z(G)$ is abelian.

B

For every group $G$, the group $G/Z(G)$ is cyclic.

C

For every group $G$, if $G/Z(G)$ is cyclic, then $G$ is abelian.

D

For every group $G$, if $G/Z(G)$ is cyclic, then $G$ is cyclic.

E

For every group $G$, if $G/Z(G)$ is abelian, then $G$ is abelian.