Some Supersolvability Conditions for Finite Groups

Character degree sums of finite groups

“…Let S be the largest solvable normal subgroup of G. Clearly, S has index less than 256 in G by Lemma 8. Hence G/S is isomorphic to A 5 , S 5 , or PSL (2,7).…”

“…So H is perfect and a central extension of A 5 . This means that H is either A 5 or SL (2,5). In the former case we have G = A 5 × Z, so assume that H ∼ = SL (2,5).…”

A Note on CLT-Groups

“…In this aspect, C. V. Holmes first proved the following result. Some related topics can be found in [1, 3-11, 13-17, 19, 21-25, 27-29, 31, 32] and [30,Chapter,1,4,and 6].…”

On the commuting probability and supersolvability of finite groups

“…This was improved by Guralnick and Robinson in [5,Theorem 11] where they showed that if d(G) > 3/40 then either G is solvable or G ∼ = A 5 × A for some abelian group A. In [1], Barry, MacHale, and Ní Shé proved that G must be supersolvable whenever d(G) > 1/3 and pointed out that, since d(A 4 ) = 1/3, the bound cannot be improved.…”