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](http://www.jstor.org/stable/2310634?&Search=yes&searchText=Groups&searchText=Subgroups&searchText=Unions&searchText=Seymour&searchText=Proper&searchText=Haber&list=hide&searchUri=/action/doBasicSearch?Query=Groups%2bas%2bUnions%2bof%2bProper%2bSubgroups%2bSeymour%2bHaber&acc=on&wc=on&fc=off&prevSearch=&item=1&ttl=6&returnArticleService=showFullText) 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!!



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.


