Correction to: Structure of p-Solvable Groups with Three p-Regular Classes

Abstract

There is unfortunately an error in the proof of Lemma 3.1 in [1] and so there are missing groups in the list of Theorem B. This was kindly pointed out to me by J. B. Olsson and his student Madsen. Let G be a finite p-nilpotent group with 0 P (G)-{1}. If r p i{G) = 3, then it is clear that |7r(O p /(G)) < 2. In the Theorem below, we give all the isomorphism classes of finite p-nilpotent groups G with r p >(G)-3 under the assumption that |7r(0p'(G))| = 2. By adding six types of groups given in the theorem to the list of Theorem B, we obtain all the finite/?-solvable groups G with 0 P (G)-{1} which have exactly three p-regular classes. The details can be found in the author's paper [2].

Groups with One or Two Super-Brauer Character Theories

“…It is not difficult to see that B 0 , B 1 , and B 2 are the three orbits for T acting on V . It follows that P must have these three orbits, and the result now follows from Theorem B of [16]. The remaining case is that |W 1 | = 9.…”

“…In Theorem B of [16] (5)- (10) give the examples where P acts on Q having exactly three orbits and are included in the following theorem as (1) (a)-(d), (2) when r = 2, and (3). We note that we determined the sixteen 2-groups that are referred to in (1) (c) and appear in Tables 4 and 5 using the computer algebra system Magma [3].…”

“…Thus, P will need to be a subgroup of T that has the same orbits as T . Again, one can appeal to Theorem B of [16] or one can compute these subgroups in Magma. This completes the proof when V has two P -invariant character theories.…”

Principal blocks and p-radical groups

“…Consider the case when l(B 0 ) 2, and we have the following lemma. [18], Theorem A, G/O p ′ ,p (G) has an abelian p-complement. Thus G/O p ′ ,p (G) is p-radical from [23], Proposition 2.…”

“…The proof of Theorem 1.5 relies on Ninomiya's classification theorem (see [18], Theorem A and Theorem B, and [17], Theorem). By Lemma 2.6, B 0 ∼ = k(G/O p ′ (G)).…”