TITLE: covering groups by infinitely many cosets
QUESTION [5 upvotes]: The classical Neumann lemma states that if a group is covered by finitely many cosets, then at least one of these cosets is the coset of a subgroup of finite index.  (Actually, the lemma says more, namely that the group is covered by the cosets of subgroups of finite index.)
I wonder if there is an infinitary version of the lemma in the following sense:  Suppose that $G$ is a group and $G=\bigcup_{i<\alpha}x_iH_i$, where the $H_i$ are subgroups and $\alpha$ is an ordinal less than some big cardinal $\kappa$.  (Maybe even $\kappa$ is strongly inaccessible.)  Is it necessarily the case that there is some $i$ for which the index of $H_i$ in $G$ is less than $\kappa$?

REPLY [3 votes]: So it appears that my question is false.  Here is a counterexample communicated to me by Ehud Hrushovski, although he attributes the counterexample to Macpherson and Neumann:  Fix an uncountable cardinal $\kappa$ and let $G$ be the group of permutations of $\kappa$ with finite support.  Let $G_n$ be the subgroup of $G$ consisting of those permutations which fix $n\in \omega$.  Then $G=\bigcup G_n$ and each $G_n$ has index $\kappa$ in $G$.