Solvable Groups Satisfying the Two-Prime Hypothesis II

Abstract

In this paper, we consider solvable groups that satisfy the twoprime hypothesis. We prove that if G is such a group and G has no nonabelian nilpotent quotients, then |cd(G)| ≤ 462,515. Combining this result with the result from part I, we deduce that if G is any such group, then the same bound holds.

Almost simple groups with no product of two primes dividing three character degrees

“…Lewis [10] could show that if G is a solvable group satisfying P 2 2 , then jcd.G/j 6 9. Hamblin and Lewis [6] proved that if G is a solvable group having P 3 2 , then jcd.G/j 6 462, 515. We note that 27 is the largest known value for jcd.G/j, where G is a solvable group satisfying P 3 2 .…”

Groups satisfying the two-prime hypothesis with a composition factor isomorphic to PSL$_2(q)$ for $q\geq7$

“…Throughout this paper, G will be a finite group, Irr(G) will be the set of complex irreducible characters of G, and cd(G) = {χ(1) | χ ∈ Irr(G)} will be the degree set of Irr(G). Following [4], we will say that a group G satisfies the two-prime hypothesis if whenever a, b ∈ cd(G) with a = b, then gcd (a, b) is divisible by at most two primes counting multiplicity. Solvable groups satisfying the two-prime hypothesis were studied in [4,5] where an upper bound was determined for the number of character degrees for such groups.…”

“…Following [4], we will say that a group G satisfies the two-prime hypothesis if whenever a, b ∈ cd(G) with a = b, then gcd(a, b) is divisible by at most two primes counting multiplicity. Solvable groups satisfying the two-prime hypothesis were studied in [4,5] where an upper bound was determined for the number of character degrees for such groups.…”