Affine Semi-linear Groups with Three Irreducible Character Degrees

Lewis, Mark L.; Riedl, Jeffrey M.

doi:10.1006/jabr.2001.9021

Cited by 6 publications

(3 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Working as in the proof of Theorem 3.1, we can define an action of F * and Gal(F ) on P . Applying Theorem 11 of [9], we can find subgroups K and N of F * Gal(F ) so that |K| = (p n − 1) ρ , |N | = m, cd(N K) = {1, n}, K is cyclic, N is nilpotent, and N K acts Frobeniusly on P . Take G = P N K. We now apply Lemma 2.1 to obtain the conclusion.…”

Section: Sketch Of Proofmentioning

confidence: 99%

Constructing solvable groups with derived length four and four character degrees

Lewis¹

2019

Publ. Math. Debrecen

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Sketch Of Proofmentioning

confidence: 99%

Constructing solvable groups with derived length four and four character degrees

Lewis¹

2019

Publ. Math. Debrecen

Self Cite

View full text Add to dashboard Cite

show abstract

“…So let us consider the case p | (r − 1). Using [23,Lemma 8], we see that (r p c − 1) p = p c (r − 1) p if p > 2 or (r − 1) p > p and (r p c − 1) p = p c (r + 1) p otherwise. In any case, we always have (r…”

Section: Preliminariesmentioning

confidence: 99%

Variations of Landau's theorem for p-regular and p-singular conjugacy classes

Moretó¹,

Nguyen²

2013

Preprint

View full text Add to dashboard Cite

The well-known Landau's theorem states that, for any positive integer k, there are finitely many isomorphism classes of finite groups with exactly k (conjugacy) classes. We study variations of this theorem for p-regular classes as well as p-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of p-regular classes or the number of p-singular classes of the group. In particular, if G is a finite group with O p (G) = 1 then |G/F(G)| p ′ is bounded in terms of the number of p-regular classes of G. However, it is not possible to prove that there are finitely many groups with no nontrivial normal p-subgroup and k p-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).

show abstract

“…One year later, M. Isaacs proved that groups with 3 character degrees are solvable of derived length 3 (see [8] or [9,Theorem 12.15]). Several decades later, the structure of (solvable) groups with 3 character degrees was analyzed in detail in [22] (see also [18]). As the alternating group A 5 shows, it is not possible to prove that G is solvable when |cd(G)| = 4.…”

Section: Introductionmentioning

confidence: 99%