There are interesting theorems about groups as union of proper subgroups. The first result in this subject is the theorem of Scorza(1926): "a groups if union of three proper subgroups if and only it has quotient $C_2\times C_2$." In 1959, Haber and Rosenfeld proved interesting theorems on the groups as union of subgroups. Then, in 1994, J. H. E. Cohn proved some interesting theorems about groups as union of few proper subgroups, and made conjectures.

While reading these three papers, which have large gaps in the publishing years, I couldn't find other initial references on "Groups as union of subgroups".

It will be a great pleasure, if one provides a list of references on the subject "Groups as union of proper subgroups", from 1926 to 1959 and from 1959 to 1994.

Especially, it is known that a non-cyclic $p$-group can not be union of $p$-proper subgroups, and if it is union of $p+1$ proper subgroups, then all the subgroups are maximal, and theire intersection has index $p^2$ in $G$. I would like to get original references for this theorem also.

Thanks in advance!!

  • 3
    $\begingroup$ Do you have access to Math Reviews online (MathSciNet)? You could enter the papers you know about, and follow the trail of references and reviews mentioning those papers. You could also type "union of proper subgroups" into an Anywhere box, to see what turns up. But maybe you've already tried that. $\endgroup$ – Gerry Myerson Jul 25 '13 at 6:38
  • $\begingroup$ @Gerry- MathSciNet is a best website to see list of all mathematics papers and to get link. But, I do not have free access (many times, it asks me ID and passward). $\endgroup$ – Soluble Jul 26 '13 at 5:41
  • $\begingroup$ Does your university library not have access? Could you speak to someone in the math department at your university about access? $\endgroup$ – Gerry Myerson Jul 26 '13 at 6:33
  • $\begingroup$ @Gerry- I am in an institute, and the institute has no access of MathSciNet. $\endgroup$ – Soluble Jul 26 '13 at 6:43
  • $\begingroup$ @RDK, I'll send you papers if you need them (and I can access them). Email me. $\endgroup$ – Nick Gill Jul 26 '13 at 8:16

The mentioned result of Cohn has been further extended. Let us write $σ(G) = n$ whenever $G$ is the union of $n$ proper subgroups, but is not the union of any smaller number of proper sub- groups. Thus, for instance, Scorza’s result asserts that $σ(G) = 3$ if and only if $G$ has a quotient isomorphic to $C_2 × C_2$.

Theorem(Cohn 1994): Let $G$ be a group. Then

(a) $σ(G) = 4$ if and only if $G$ has a quotient isomorphic to $S_3$ or $C_3 × C_3$.
(b) $σ(G) = 5$ if and only if $G$ has a quotient isomorphic to the alternating group $A_4$.
(c) $σ(G) = 6$ if and only if $G$ has a quotient isomorphic to $D_5, C_5 × C_5$, or $W$,where $W$ is the group of order $20$ defined by $a^5 =b^4 ={e},ba=a^2b$.

Furthermore, Tomkinson proved that there is no group $G$ such that $σ(G) = 7$. For more information see the article of Mira Bhargava, "Groups as unions of subgroups". The references also contain papers on the subject from $1964$ to $1997$, e.g., J. Sonn, Groups that are the union of finitely many proper subgroups, Amer. Math. Monthly 83 (1976), no. 4, 263–265.

  • $\begingroup$ The result mentioned in question about $p$-group is due to Cohn? (the first part of this result follows from a theorem of Haber-Rosenfeld (1959); but where (i.e. in which paper) does the second part appears first time? $\endgroup$ – Soluble Jul 26 '13 at 5:48
  • $\begingroup$ Which references are given for this in Berkovich's book (see math.stackexchange.com/questions/360082/…) ? $\endgroup$ – Dietrich Burde Jul 28 '13 at 20:21
  • $\begingroup$ In Berkovich's book, there is long bibliography, but for many theorems (including the stated at last of this question), he do not mention the reference "near the theorem". $\endgroup$ – Soluble Jul 29 '13 at 4:33

I have written some papers on the subject, see the following. You may download their PDF files from my home page


Alireza Abdollahi, M.J. Ataei, S.M. Jafarian Amiri and A. Mohammadi Hassanabadi, Groups with a maximal irredundant 6-cover, Communications in Algebra, 33, No. 9 (2005) 3225-3238.

Alireza Abdollahi and S.M. Jafarian Amiri, On groups with an irredundant 7-cover, Journal of Pure and Applied Algebra, 209 (2007) 291-300.

Alireza Abdollahi, M.J. Ataei and A. Mohammadi Hassanabadi, Minimal blocking sets in PG(n,2) and covering groups by subgroups, Communications in Algebra, 36 No. 2 (2008) 365-380.

Alireza Abdollahi and S.M. Jafarian Amiri, Minimal coverings of completely reducible groups, Publicationes Mathematicae Debrecen, 72/1-2 (2008), 167-172 .

Alireza Abdollahi, Groups with maximal irredundant covers and minimal blocking sets, to appear in Ars Combinatoria.


using the notation of Dietrich Burde, classifying the groups with $\sigma(G)$ "small" was the topic of my master thesis, see here:

M. Garonzi; Finite Groups that are the union of at most 25 proper subgroups, Journal of Algebra and Its Applications Vol. 12, No. 4 (2013) 1350002.

Here we deal with direct products:

M. Garonzi, A. Lucchini; Direct products of fi nite groups as unions of proper subgroups. Arch. Math. (Basel) 95 (2010), no. 3, 201206.


To be complete, there is another reference here.

Theorem (Bruckheimer, Bryan and Muir) A group is the union of three proper subgroups if and only if it has a quotient isomorphic to $V_4$.

The proof appeared in the American Math. Monthly $77$, no. $1 (1970)$. As remarked by @soluble, the theorem seems to be proved earlier by the Italian mathematician Gaetano Scorza, I gruppi che possono pensarsi come somma di tre loro sottogruppi, Boll. Un. Mat. Ital. $5 (1926), 216-218$.


There is the whole bunch of papers relating the Bergman property and cofinality -- about ascending chains of subgroups which exhaust the group:

Macpherson, H. D.; Neumann, Peter M. Subgroups of infinite symmetric groups. J. London Math. Soc. (2) 42 (1990), no. 1, 64–84. (there is a free access, search in google)

There is a paper by Georg Bergman: http://arxiv.org/abs/math.GR/0401304

There are several papers on this by Droste:

Uncountable cofinalities of automorphism groups of linear and partial orders

On full groups of measure-preserving and ergodic transformations with uncountable cofinalities

Uncountable cofinalities of permutation groups

(downloadable at http://www.informatik.uni-leipzig.de/~droste/publications.html)

There is a nice paper by Yves Cornulier http://arxiv.org/abs/math/0411466

Finally, the issue of a group being a union of its chain of proper subgroups appears naturally in the characterisation of groups with (FA)-property, and there is a generalisation of that by Sabine Koppelberg


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