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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 $G\in \mathcal{P}$ if $G$ has property $\mathcal{P}$. [ . . . ] A property $\mathcal{P}$ of groups is inherited by subgroups if $$G\in \mathcal{P}\text{ and }H\le G\implies H\in \mathcal{P},$$ and $\mathcal{P}$ is inherited by quotients if $$G\in \mathcal{P}\text{ and }H\unlhd G\implies G/H\in \mathcal{P}.$$

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 $H\le G$ of property $\mathcal{X}$ such that $[G:H]$ is finite. So, in particular, it cannot be finite.

Please help :)

Shaun
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1 Answers1

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One of the answers to your other question uses the following general pattern: the condition that a certain type of equation is always solvable in a group (e.g. the condition that every element have a square root) is inherited by quotients but usually not by subgroups.

There's a sort of dual to this: the condition that a certain type of equation is not solvable in a group is inherited by subgroups but usually not quotients. For example, the condition that there exist no nontrivial solutions to $g^n = 1$ (which is exactly the condition of being torsion-free) works.

Qiaochu Yuan
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