# 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.