A finite simple group of Lie type has p-blocks with different defects, p ≠ 2

On the Number of Components of a Graph Related to Character Degrees

“…Suppose that G is a nonsolvable group. Then n(T(G)) < m^{n(T(E))} where E runs through the set of the simple nonabelian composition factors of G. Let x e T/t-fi-As (x(l),p(l)) -1 for all <p e 1tt(G/N), we obtain (X(1),\G/N\) = 1 by Michler's Theorem 5.4 of [11]. Thus xjv G lrv(N) and Gallagher's theorem [6,6.17] yields Xf e Irr(G) for all <p e lrr(G/N).…”

“…= («.-1)2(<? + 1) V + g + 1) and tf(l) = \L '■ A2\v = (q~ l)2(l + 1) V -9 + 1) (see [11,Proposition 2.2]). Since |L| = q6(q -l)2(q + l)2(q2 +q + l)(q2 -q+1), the assertion follows.…”

Prime divisors of irreducible character degrees

“…If G is an almost simple group, then G has no normal abelian Sylow subgroup and so by Ito-Michler's Theorem [Mich,Theorem 5.4], ρ(G) = π (G). This fact will be used without any further reference.…”

Irreducible characters of even degree and normal Sylow 2-subgroups

“…Over the past decades, there have been several variations and refinements of this result by considering Brauer characters [17,18,21], nonvanishing elements [5,25], fields of character values [4,25,13], and Frobenius-Schur indicator [19,28]. One primary direction is to weaken the condition that all irreducible characters of G have degree coprime to p, and assume instead only that a subset of characters with a specified field of values has this property, see [4,19] for real-valued characters and [25] for p-rational characters.…”