2 Possible Generalizations of Cayley's Theorem?

2012-09-26T09:45:56Z

I'm wondering about the following 2 generalizations of Cayley's Theorem (every group embeds in a symmetric group). If these are known to be true/false, references would be appreciated.

1) (Weak Version) Given any finite collection of (not necessarily distinct) finite groups, can we embed them simultaneously in a (finite) group so that they have pairwise-disjoint intersection (i.e., intersect only at the identity)?

2) (Strong Version) Given an arbitrary set of arbitrary groups, can we embed them simultaneously in some group so that they have pairwise-disjoint intersection? It seems like this version may run into set-theoretic difficulties; if so an explanation/reference for those would also be welcome.

Note of course that if such a group exists for a given collection, then we can embed it in a symmetric group by Cayley's Theorem (so these are indeed generalizations).

Edit: To make it more interesting / rule out the obvious answer pointed out in the comments, can we arrange the embeddings so that the groups' normalizers are pairwise disjoint?

Answer by Derek Holt for 2 Possible Generalizations of Cayley's Theorem?

2012-09-27T14:43:54Z

So the question now is, given two nontrivial finite groups $G,H$, can we embed them both in a finite group $X$ such that the normalizers of $G$ and $H$ in $X$ intersect trivially?

I think we can do that as follows. Suppose that we can find a module $V$ for $G \times H$ over a finite field, such that neither $G$ nor $H$ fix any nonzero vectors $v \in V$, and such that there exists a vector $v \in V$ such that no nontrivial element of $G \times H$ stabilizes $v$. Then we let $X$ be the semidirect product of $V$ by $G \times H$. Because of the first property, the normalizers of $G$ and $H$ in $X$ have trivial intersection with $V$, so they are both equal to $G \times H$. The second property ensures that the complements $G \times H$ and $(G \times H)^v$ intersect trivially. So the subgroups $G$ and $H^v$ have the desired property, because their normalizers in $X$ are $G \times H$ and $(G \times H)^v$.

We can contruct such a module $V$ as follows. Choose a prime $p$ not dividing $|G|$ or $|H|$ and let $V_1$ and $V_2$ be the deleted permutation modules for the regular permutation representations of $G$ and $H$ over ${\mathbb F}_p$. (So $V_1$ and $V_2$ have dimensions $|G|-1$ and $|H|-1$.) These have vectors $v_1$ and $v_2$ that are not stabilised by any nontrivial element of $G$ and $H$, respectively. Now let $V$ be the $G \times H$ module $V_1 \otimes V_2$ and let $v =v_1 \otimes v_2$.

2 Possible Generalizations of Cayley's Theorem? - MathOverflow

2 Possible Generalizations of Cayley's Theorem?

Answer by Derek Holt for 2 Possible Generalizations of Cayley's Theorem?