Groups as Union of Proper Subgroups: References

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.

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

sci.ui.ac.ir/~a.abdollahi

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 finite groups as unions of proper subgroups. Arch. Math. (Basel) 95 (2010), no. 3, 201206.