P-stable linear representation

This article describes a property to be evaluated for a linear representation of a group, i.e. a homomorphism from the group to the general linear group of a vector space over a field

Definition

Let $G$ be a finite group and $p$ be an odd prime number. Suppose $\rho :G\to GL(V)$ is a linear representation of $G$ over a finite field of characteristic $p$. We say that $\rho$ is $p$-stable if for no non-identity $p$-element $g$ of $G$ (i.e., an element whose order is a power of $p$) does $\rho (g)$ satisfy a quadratic minimal polynomial.

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 103, Theorem 8.1, Chapter 3 (Representations of groups), Section 3.8 (p-stable representations), ^{More info}