## Can a group be a finite union of (left) cosets of infinite-index subgroups?

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

To be more precise (but less snappy): is there an example of a group G with finitely many infinite-index subgroups H_1, ..., H_n and elements k_1, ..., k_n such that G is the union of the left cosets k_1 H_1 , ..., k_n H_n? And what if we relax the requirement that these all be left cosets, and ask: can G be the union of finitely many such cosets, some being left cosets, others being right cosets?

If G is amenable then this can't happen, since any coset of an infinite-index subgroup must have measure 0. So this immediately rules out any abelian group G.

I've tried playing around with the only non-amenable groups that I'm comfortable with, the free groups on two or more generators. A few months ago I thought I found a simple counterexample in the free group on $\aleph_0$ generators, but now I've lost my notes and am beginning to doubt I ever had such an example.

(This question was asked to me by a friend who's interested in some kind of application to model theory, but I think it's interesting as a stand-alone puzzle.)

-

## 1 Answer

No. This follows from a beautiful theorem of B.H. Neumann:

Let $G$ be a group. If `$\{x_iH_i\}_{i=1}^n$` is a covering of $G$ by cosets of proper subgroups, then $n \geq \min_{i} [G:H_i]$.

Explicitly, this is Lemma 4.1 in

http://www.math.uga.edu/~pete/Neumann54.pdf

As Neumann remarks, the identity $gH = (g H g^{-1}) g$ shows that it is no loss of generality to restrict to coverings by left cosets.

-

That was quick!  Thanks, actually I read about this a long time ago and forgot that it applied to any group G... – John Goodrick Mar 7 2010 at 17:37
You're quite welcome.  I have had a side interest in covering problems of various sorts for a couple of years now, and Neumann's result is one of the nicest in this area.   – Pete L. Clark Mar 7 2010 at 17:42
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 5, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9418299198150635, "perplexity_flag": "head"}