2

Some time ago, I was told the following theorem, attributed to Neumann (I have no idea which one):

If $G$ is an abelian group, and $G$ is the finite union $\bigcup_{i\in I} g_iH_i$ of cosets of subgroups $H_i\leq G$, then it is already the union of those $g_iH_i$ for which $H_i$ has finite index.

It was proved by inclusion-exclusion principle, and back then, I had an idea of an inductive proof. I think in both cases, there didn't seem to be any reason for it to fail if $G$ is nonabelian.

The question is: which Neumann is this theorem due to, and in what form? And if s/he did not prove it in the above form (or an obviously equivalent one) with $G$ nonabelian, who did? I would appreciate some references.

tomasz
  • 35,474

1 Answers1

2

There is a lot of research on groups covered by a finite set of proper subgroups. It all started with a seminal paper of Bernhard Neumann, Groups covered by finitely many cosets, Publ. Math. Debrecen 3 (1954), 227-242, in which he proves that if a group is the union of a finite number of subgroups (or even cosets), you only have to take into account those subgroups having finite index. This implies that a group has a finite covering by subgroups if and only if it has a finite non-cyclic homomorphic image. See also recent work of Mira Bhargava, the mother of the brilliant number theorist Manjul Bhargava.

Community
  • 1
Nicky Hekster
  • 49,281
  • I think this is actually the reference I had in mind, but I could not find it. – James Jan 22 '18 at 19:20
  • Thanks! That seems to answer my question fully. Well, it would be best if I could actually look at the reference, but this is enough to satisfy my curiosity. :) – tomasz Jan 22 '18 at 20:08