FINITE GROUPS THAT ARE THE UNION OF AT MOST 25 PROPER SUBGROUPS

For a finite group G let σ(G) (the “sum” of G) be the least number of proper subgroups of G whose set-theoretical union is equal to G, and σ(G) = ∞ if G is cyclic. We say that a group G is σ-elementary if for every non-trivial normal subgroup N of G we have σ(G) < σ(G/N). In this article we produce the list of all the σ-elementary groups of sum up to 25. We also show that σ(Aut(PSL(2, 8))) = 29.

For a finite group G let sigma(G) (the "sum" of G) be the least number of proper subgroups of G whose set-theoretical union is equal to G, and sigma(G) = infinity if G is cyclic. We say that a group G is sigma-elementary if for every non-trivial normal subgroup N of G we have sigma(G) < sigma(G/N). In this article we produce the list of all the sigma-elementary groups of sum up to 25. We also show that sigma(Aut(PSL(2,8))) = 29.