Nonsolvable Groups All of Whose Character Degrees are Odd-Square-Free

Abstract

A finite group G is odd-square-free if no irreducible complex character of G has degree divisible by the square of an odd prime. We determine all odd-square-free groups G satisfying S G Aut(S) for a finite simple group S. More generally, we show that if G is any nonsolvable odd-square-free group, then G has at most two nonabelian chief factors and these must be simple odd-square-free groups. If the alternating group A 7 is involved in G, the structure of G can be further restricted.

Character degrees of extensions of the Suzuki groups $^2B_2(q^2)$

“…Given a finite group G, let cd(G) = {χ(1) | χ ∈ Irr(G)} be the set of degrees of the ordinary complex irreducible characters of G. A common approach in the studying of nonsolvable groups with a given property on irreducible character degrees begins by examining the property on simple and almost simple groups. Among these, depending on the given property, the most work is done generally on simple groups S with few character degrees and even more so, on groups G with S G Aut(S) such that a few characters of S are extendible to G, (see [3,5,6]).…”

Character degrees of extensions of PSL2(q) and SL2(q)

“…Several of these studies required increasingly detailed information about the character degrees of groups H with L 2 .q/ 6 H 6 Aut.L 2 .q// (see [5][6][7][8]). Of course, the character table of L 2 .q/ is well known, as is the automorphism group, and so the character degrees of H are known in principle.…”