Group covered by finitely many cosets
This question appears in my textbook's exercises, who can help me prove it?
If a group $G$ is the set-theoretic union of finitely-many cosets, $$G=x_1S_1\cup\cdots\cup x_nS_n$$ prove that at least one of the subgroups $S_i$ has finite index in $G$.
I think that the intersection of these cosets is either empty or a coset of the intersection of all the $S_i$. I want to start from this point to prove it. So I suppose none of these $S_i$ has finite index, but I don't know how to continue?
For completeness, the Neumann proof is roughly as follows.
Let $r$ be the number of distinct subgroups in $S_1,S_2,\dots,S_n$. We will prove by induction on $r$.
If $r=1$, then $G=\bigcup_{i=1}^n x_iS_1$ and $S_1$ thus has finite index in $G$.
Now assume true for $r-1$ distinct groups, and assume $S_1,\dots,S_n$ has $r>1$ distinct groups, with $S_{m+1}=\cdots=S_n$ the same, and $S_i\neq S_n$ for $i\leq m$.
If $G=\bigcup_{i=m+1}^n x_iS_i$ then $S_n$ has finite index, from the $r=1$ case.
So assume $h\not\in \bigcup_{i=m+1}^n x_iS_i$. Then $hS_n\cap \left(\bigcup_{i=m+1}^n x_iS_i\right) = \emptyset$, since $hS_n$ is a distinct coset of $S_n$.
So $$hS_n \subseteq \bigcup_{i=1}^m x_iS_i$$ So for any $x\in G$, $$xS_n=xh^{-1}hS_n\subseteq \bigcup_{i=1}^m xh^{-1}x_iS_i$$
So we can replace $x_iS_i=x_iS_n$ for $i>m$ with a finite number of cosets of $S_1,\dots,S_m$. And now we have $G$ as a finite union of cosets of the $r-1$ distinct subgroups in $S_1,\dots,S_m$. And, by induction, one of those must have finite index in $G$.
(The harder result, that some $S_i$ has finite index smaller than $n$, does not follow from this argument, since we might increase the number of terms when we do the reduction from $r$ to $r-1$.)
I believe that this is a nontrivial result. It was proven in
B. H. Neumann, Groups covered by finitely many cosets, Publ. Math. Debrecen 3 (1954), 227–242. MR 17, 234.
Unfortunately, I could not find the original argument. Perhaps someone with a sharper Google-fu can retrieve it.
(I am curious to know what book you found this in. It seems like a cruel exercise, at least assuming that your book is an introductory book on group theory.)