Cyclic Quotient Group of Center

Theorem

Let $G$ be a group.

Let $Z \left({G}\right)$ be the center of $G$.

Let $G / Z \left({G}\right)$ be the quotient group of $G$ by $Z \left({G}\right)$.

Let $G / Z \left({G}\right)$ be cyclic.

Then $G$ is abelian, so $G = Z \left({G}\right)$.

That is, the group $G / Z \left({G}\right)$ can not be a cyclic group which is not trivial.

Proof

Suppose $G / Z \left({G}\right)$ is cyclic.

Then:

$\exists \tau \in G / Z \left({G}\right): G / Z \left({G}\right) = \left \langle {\tau} \right \rangle$

Thus:

$\exists t \in G: G / Z \left({G}\right) = \left \langle {t Z \left({G}\right)} \right \rangle$

Thus each coset of $Z \left({G}\right)$ in $G$ is equal to $\left({t Z \left({G}\right)}\right)^i = t^i Z \left({G}\right)$ for some $i \in \Z$.

Now let $x, y \in G$.

Suppose $x \in t^m Z \left({G}\right), y \in t^n Z \left({G}\right)$.

Then $x = t^m z_1, y = t^n z_2$ for some $z_1, z_2 \in Z \left({G}\right)$.

Thus:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle x y\)	\(=\)	\(\displaystyle t^m z_1 t^n z_2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle t^m t^n z_1 z_2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	($z_1$ commutes with all $t \in G$)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle t^{m + n} z_1 z_2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Similarly, $y x = t^{m + n} z_1 z_2 = x y$.

This holds for all $x, y \in G$, and thus $G$ is abelian, thereby making $Z \left({G}\right) = G$, and thus $G / Z \left({G}\right)$ has one element and is therefore trivial.

$\blacksquare$

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 47 \epsilon$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 51.1$
• John F. Humphreys: A Course in Group Theory (1996): $\S 10$: Proposition $10.21$