covering groups by infinitely many cosets 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 $\kappa$?
 A: The statement as you have written it is not true for any
uncountable cardinal $\kappa$. To see this, let $G$ be any
group of size $\aleph_{\beta+\omega}$, where
$\kappa=\aleph_\beta$. Every such group is the union of a
countable increasing family of subgroups $G=\bigcup_n G_n$,
where $G_n$ has size $\aleph_{\beta+n}$. This is simply
because every set of size $\aleph_{\beta+\omega}$ is the
union of sets of size $\aleph_{\beta+n}$. But each $G_n$,
being so small, must have index $\aleph_{\beta+\omega}$ in
$G$, which is much larger than $\kappa$. Thus, $G$ is the
union of countably many subgroups, each of index far larger
than $\kappa$. And so it is a counterexample to the
statement you make.
(But perhaps the statement could become true if you
consider only groups $G$ of size $\kappa$, in the case when
$\kappa$ is a large cardinal or at least regular, and
perhaps this is what you meant.)
A: 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 $G$.
A: Edit: I assume that the cardinality of the group is much bigger than the cardinality of the set of cosets. 
Look at Tomkinson, M. J.,
Groups covered by finitely many cosets or subgroups.
Comm. Algebra 15 (1987), no. 4, 845–859. It gives a quantitative version of Neumann's result. It's methods, I think, generalize to the infinite case (at least to "very large" cardinals). 
