Given the statement:

"If $\left| G / Z(G) \right|$ is finite, then $G'$ is finite."

Here, $G$ is a group, $Z(G)$ denotes the center of $G$, and $G'$ represents the commutator subgroup (or derived subgroup) of $G$.

Step-by-step reasoning:

1. $G / Z(G)$ is finite:

   • The quotient $G / Z(G)$ is the group of cosets of $Z(G)$ in $G$. The statement says that this quotient is finite. This means that the number of distinct cosets of $Z(G)$ in $G$ is finite. Hence, $G$ behaves like a finite group modulo its center.

2. $Z(G)$ is the center of $G$:

   • The center of a group $Z(G)$ consists of elements that commute with every element of the group. It is a normal subgroup of $G$.

3. Implication for $G'$:

   • The commutator subgroup $G'$ is the subgroup generated by all commutators $[x, y] = x^{-1}y^{-1}xy$ for $x, y \in G$. The commutator subgroup measures the "non-abelian" nature of a group: if $G'$ is trivial, $G$ is abelian.
   • Now, if $G / Z(G)$ is finite, then $G$ is a "small" group modulo its center. This means that the quotient group $G / Z(G)$ has only finitely many cosets, so the "extra structure" outside the center is finite.
   • Since the commutator subgroup $G'$ is related to the non-abelian behavior of the group, and $G / Z(G)$ is finite, the structure of $G$ outside the center must be constrained. Specifically, this implies that $G'$ must also be finite.

Conclusion:

The statement is correct: If the quotient group $G / Z(G)$ is finite, then the commutator subgroup $G'$ of $G$ must also be finite.

Would you like further explanation on any specific point, or additional examples?

Here are five related questions to expand the understanding:

1. What is the significance of the commutator subgroup $G'$ in understanding the structure of a group?
2. How does the center of a group $Z(G)$ relate to the group's abelianization?
3. What properties of a group $G$ can we infer if $G'$ is finite?
4. What does it mean for a group to be "almost abelian," and how does $G'$ help characterize this?
5. How does the structure of quotient groups influence our understanding of the original group?

Tip: To study group properties more deeply, investigate the relationship between $G'$, the center $Z(G)$, and the abelianization of a group $G$. This trio provides key insights into how "non-abelian" a group is!