Algebra & Number Theory

Adequate groups of low degree

Robert Guralnick, Florian Herzig, and Pham Huu Tiep

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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

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

Received: 13 April 2014
Accepted: 14 December 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Artin–Wedderburn theorem irreducible representations automorphic representations Galois representations adequate representations


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.

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