"Is Conjugate To A Subgroup Of" in GAP

Question

Suppose one has finite groups $H$ and $K$ with a common parent group $G$, where $|H|$ divides $|K|$.

How would one test whether $H$ is $G$-conjugate to a subgroup of $K$?

In the case that $|H|=|K|$, of course, one could simply use IsConjugate(G,H,K) to test if $H$ is $G$-conjugate to $K$. But what about the subgroup case?

Answer 1

There isn't a specific function as you ask for, primarily as over >20 years this never was identified as a fundamental function, and as it is actually not clear what the "natural" API would be: True/false? Conjugating elements for one? for all?

There are a number of possibilities that might work for you, but some require a bit of work from your side, and might not always be as fast as you might wish for.

• There is a function IsomorphicSubgroups that finds all monomorphisms from $H$ into $K$. Then do explicit conjugacy tests for all images.
• For permutation groups, you could modify the backtrack search for conjugacy to a (more simplistic) search for the conjugate lying within $K$.
• You could calculate representatives for the double cosets $N_G(H){\setminus} G/N_G(K)$ and for each representative $r$ test whether $H^r\le K$.
• (Somewhat along similar ideas but as canned function): A recent paper of mine (Finding intermediate subgroups, Port. Math 2018, DOI: 10.4171/PM/2002) introduces a function EmbeddedConjugates that will be available in GAP 4.9.2 and determines all conjugates of $H$ that lie in $K$.

Of course either of these approaches can be hopeless in cases that yield large search spaces, e.g if $H$ is extraspecial amd $K$ a larger extraspecial group.