## Algebra & Number Theory

### Involutions, weights and $p$-local structure

Geoffrey Robinson

#### Abstract

We prove that for an odd prime $p$, a finite group $G$ with no element of order $2p$ has a $p$-block of defect zero if it has a non-Abelian Sylow $p$-subgroup or more than one conjugacy class of involutions. For $p=2$, we prove similar results using elements of order $3$ in place of involutions. We also illustrate (for an arbitrary prime $p$) that certain pairs $(Q,y)$, with a $p$-regular element $y$ and $Q$ a maximal $y$-invariant $p$-subgroup, give rise to $p$-blocks of defect zero of $NG(Q)∕Q$, and we give lower bounds for the number of such blocks which arise. This relates to the weight conjecture of J. L. Alperin.

#### Article information

Source
Algebra Number Theory, Volume 5, Number 8 (2011), 1063-1068.

Dates
Received: 9 June 2010
Revised: 22 December 2010
Accepted: 7 June 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729733

Digital Object Identifier
doi:10.2140/ant.2011.5.1063

Mathematical Reviews number (MathSciNet)
MR2948472

Zentralblatt MATH identifier
1246.20010

Subjects
Primary: 20C20: Modular representations and characters

Keywords
block involution

#### Citation

Robinson, Geoffrey. Involutions, weights and $p$-local structure. Algebra Number Theory 5 (2011), no. 8, 1063--1068. doi:10.2140/ant.2011.5.1063. https://projecteuclid.org/euclid.ant/1513729733