Geodesic
Finite Groups as the Union of Proper Subgroups
Serdica Mathematical Journal, Tome 32 (2006) no. 2-3, pp. 259-268.

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As is known, if a finite solvable group G is an n-sum group then n − 1 is a prime power. It is an interesting problem in group theory to study for which numbers n with n-1 > 1 and not a prime power there exists a finite n-sum group. In this paper we mainly study finite nonsolvable n-sum groups and show that 15 is the first such number. More precisely, we prove that there exist no finite 11-sum or 13-sum groups and there is indeed a finite 15-sum group. Results by J. H. E. Cohn and M. J. Tomkinson are thus extended and further generalizations are possible.
Keywords: Finite Group, Simple Group, Covering Number 
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Zhang, Jiping. Finite Groups as the Union of Proper Subgroups. Serdica Mathematical Journal, Tome 32 (2006) no. 2-3, pp. 259-268. http://geodesic.mathdoc.fr/item/SMJ2_2006_32_2-3_a7/