0000000000462760
AUTHOR
Jiakuan Lu
On finite groups with many supersoluble subgroups
[EN] The solubility of a finite group with less than 6 non-supersoluble subgroups is confirmed in the paper. Moreover we prove that a finite insoluble group has exactly 6 non-supersoluble subgroups if and only if it is isomorphic to A5 or SL2 (5). Furthermore, it is shown that a finite insoluble group has exactly 22 non-nilpotent subgroups if and only if it is isomorphic to A5 or SL2 (5). This confirms a conjecture of Zarrin (Arch Math (Basel) 99:201 206, 2012).
Notes on the average number of Sylow subgroups of finite groups
We show that if the average number of (nonnormal) Sylow subgroups of a finite group is less than $${{29} \over 4}$$ then G is solvable or G/F(G) ≌ A5. This generalizes an earlier result by the third author.