Permutation representations of nonsplit extensions involving alternating groups

TL;DR

This paper investigates the minimal degree of faithful permutation representations of nonsplit extensions involving alternating groups, confirming Babai's lower bound is nearly optimal and providing improved bounds with new methods.

Contribution

The paper proves Babai's lower bound is sharp up to a constant factor and offers a new proof with an improved constant, using different techniques.

Findings

Existence of nonsplit extensions with permutation degree ~1.5k(k-1)

Confirmation that Babai's lower bound is nearly tight

Improved quadratic lower bound with constant 1

Abstract

L. Babai has shown that a faithful permutation representation of a nonsplit extension of a group by an alternating group $A_k$ must have degree at least $k^2(\frac{1}{2}-o(1))$, and has asked how sharp this lower bound is. We prove that Babai's bound is sharp (up to a constant factor), by showing that there are such nonsplit extensions that have faithful permutation representations of degree $\frac{3}{2}k(k-1)$. We also reprove Babai's quadratic lower bound with the constant $\frac{1}{2}$ improved to 1 (by completely different methods).