Characters, exponents and defects in p-solvable groups

Methods and questions in character degrees of finite groups

“…Recently, motivated by a problem on p-blocks of p-solvable groups, a dual problem has come up (see [52]). As pointed out in the proof of Lemma 2.1 of [52], it is easy to see that if P is a p-group and A ≤ P is abelian, then χ(1) divides |P : A| for all χ ∈ Irr(P ). In particular, if |P | = p n and the exponent of P is p e , then χ(1) ≤ p n−e for every χ ∈ Irr(P ).…”

“…Notice that s(P ) = 0 if and only if P has an irreducible character that is induced from some cyclic subgroup. These groups were characterized in Theorem B of [52] (when p is odd). Some results on groups with s(P ) > 0 were obtained in Theorem D of [52], which bounds the (minimal) number of generators of an odd p-group in terms of s(P ).…”

“…These groups were characterized in Theorem B of [52] (when p is odd). Some results on groups with s(P ) > 0 were obtained in Theorem D of [52], which bounds the (minimal) number of generators of an odd p-group in terms of s(P ). For the blocktheoretic purposes of [52], these results are not strong enough when p = 3.…”

“…Some results on groups with s(P ) > 0 were obtained in Theorem D of [52], which bounds the (minimal) number of generators of an odd p-group in terms of s(P ). For the blocktheoretic purposes of [52], these results are not strong enough when p = 3. The following includes Questions 1 and 2 of [52]: Question 3.1.…”

“…As discussed in Section 7 of [52], an affirmative answer to these questions on 3groups, would allow to extend the block-theoretic Theorem A of [52] to p = 3.…”