## Algebra & Number Theory

### Adequate groups of low degree

#### Abstract

The notion of adequate subgroups was introduced by Jack Thorne. It is a weakening of the notion of big subgroups used in generalizations of the Taylor–Wiles method for proving the automorphy of certain Galois representations. Using this idea, Thorne was able to strengthen many automorphy lifting theorems. It was shown by Guralnick, Herzig, Taylor, and Thorne that if the dimension is small compared to the characteristic, then all absolutely irreducible representations are adequate. Here we extend that result by showing that, in almost all cases, absolutely irreducible $kG$-modules in characteristic $p$ whose irreducible $G+$-summands have dimension less than $p$ (where $G+$ denotes the subgroup of $G$ generated by all $p$-elements of $G$) are adequate.

#### Article information

Source
Algebra Number Theory, Volume 9, Number 1 (2015), 77-147.

Dates
Accepted: 14 December 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.ant/1510842260

Digital Object Identifier
doi:10.2140/ant.2015.9.77

Mathematical Reviews number (MathSciNet)
MR3317762

Zentralblatt MATH identifier
1365.20008

Subjects
Primary: 20C20: Modular representations and characters
Secondary: 11F80: Galois representations

#### Citation

Guralnick, Robert; Herzig, Florian; Tiep, Pham Huu. Adequate groups of low degree. Algebra Number Theory 9 (2015), no. 1, 77--147. doi:10.2140/ant.2015.9.77. https://projecteuclid.org/euclid.ant/1510842260

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