examples counterexamples – Find a property of groups that is inherited by subgroups but not by quotients.

This is Exercise 4.6b of Roman’s “Fundamentals of Group Theory: An Advanced Approach”. Part a is here. According to this search and Approach0, it is new to MSE.

The Details:

From p.116 of Roman’s book,

Definition: Let $mathcal{P}$ be a property of groups. [ . . . ] We write $Gin mathcal{P}$ if $G$ has property $mathcal{P}$. [ . . . ] A property $mathcal{P}$ of groups is inherited by subgroups if $$Gin mathcal{P}text{ and }Hle Gimplies Hin mathcal{P},$$ and $mathcal{P}$ is inherited by quotients if $$Gin mathcal{P}text{ and }Hunlhd Gimplies G/Hin mathcal{P}.$$

The Question:

Find a property of groups that is inherited by subgroups but not by quotients.

Thoughts:

The property cannot be “virtually $mathcal{Q}$” for some property $mathcal{Q}$ by this answer by Arturo Magidin to some other question, where “$G$ is virtually $mathcal{X}$” means that there exists $Hle G$ of property $mathcal{X}$ such that $[G:H]$ is finite. So, in particular, it cannot be finite.

Please help :]