# Extending an infinite simple group

Maybe the question does not fit here.

Yesterday in my logic course, I presented a nice example about an application of model theory to group theory. The example is due to Hodges and as following: For any infinite simple group $G$, there is a countable simple group $H\subseteq G$.

Then a logician asked whether the upward version is true. i.e. $\mathbf{ \mbox{is it true that for every infinite simple group$H$and cardinal }\kappa\geq |H|, \mbox{there is a simple group }G\supseteq H \mbox{ with } |G|=\kappa?}$

To give a positive answer to the question, it is sufficient to prove that every infinite simple group $H$ is a proper subgroup of a simple group $G$.

I was told by an algebraist that the question for finite simple groups has a negative answer.

$\mathbf{Remark}$: The question was already answered by Derek Holt. He proved that one may find such a $G$ with $|G|=2^{\kappa}$ for some cardinal $\kappa$. Then by Skolemization including $H$, one may obtain a full answer.

• Every finite group is a subgroup of $A_n$ for all sufficiently large $n$ so it is certainly true that every finite simple group embeds properly in a larger one. – Derek Holt Oct 30 '15 at 10:10
• In general, if $|H| = \kappa$ with $\kappa$ infinite, then you can embed $H$ in ${\rm Sym}(H)$ by Cayley's Theorem, and this induces an embedding of $H$ into the simple group ${\rm Sym}(H)/K$, where $K$ consists of those permutation with support less than $\kappa$, which has cardinalty $2^\kappa$. So, the answer is yes if you assume GCH. Without that, it could be difficult. – Derek Holt Oct 30 '15 at 10:33
• If this argument works, then by a Skolemization including $H$, it holds for any cardinal $\kappa\geq|H|$. – 喻 良 Oct 30 '15 at 10:44

## 1 Answer

I will make my comment into an answer. It is a standard result (see for example Chapter 8 of Dixon & Mortimer's book on Permutation Groups) that, for an infinite cardinal $\kappa$, the only normal subgroups of the symmetric group ${\rm Sym}(\Omega)$ on a set $\Omega$ of cardinality $\kappa$ are the subgroups ${\rm Sym}(\Omega,\lambda)$, consisting of those permutations of $\Omega$ with support strictly less than $\lambda$, for all cardinals $\lambda$ with $\aleph_0 \le \lambda \le \kappa$, and also the subgroup ${\rm Alt}(\Omega)$ of even permutations with finite support.

So, if $H$ is any infinite group with cardinality $\kappa$ then the standard regular embedding of $G$ into ${\rm Sym}(H)$, in which the image of each non-identity element has support equal to $H$, induces an embedding into the simple group ${\rm Sym(H)}/{\rm Sym}(H,\kappa)$, which has cardinality $2^\kappa$.

• I think this is sufficient to answer my question. – 喻 良 Oct 30 '15 at 11:17