# Permutation representations of nonsplit extensions involving alternating groups

Article

First Online:

- 29 Downloads

## 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 1/2 improved to 1 (by completely different methods).

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]L. Babai,
*On the degrees of non-split extensions by an alternating group*, preprint, available at https://doi.org/people.cs.uchicago.edu/~laci/papers. - [2]D. J. Benson,
*Representations and Cohomology. I. Basic Representation Theory of Finite Groups and Associative Algebras*, Cambridge Studies in Advanced Mathematics, Vol. 30, Cambridge University Press, Cambridge, 1998.Google Scholar - [3]W. Bosma, J. Cannon and C. Playoust,
*The Magma algebra system. I. The user language*, Journal of Symbolic Computation**24**(1997), 235–265.MathSciNetCrossRefzbMATHGoogle Scholar - [4]V. P. Burichenko, A. S. Kleshchev and S. Martin,
*On cohomology of dual Specht modules*, Journal of Pure and Applied Algebra**112**(1996), 157–180.MathSciNetCrossRefzbMATHGoogle Scholar - [5]B. Eckmann,
*Cohomology of groups and transfer*, Annals of Mathematics**58**(1953), 481–493.MathSciNetCrossRefzbMATHGoogle Scholar - [6]G. D. James,
*The Representation Theory of the Symmetric Groups*, Lecture Notes in Mathematics, Vol. 682, Springer, Berlin, 1978.Google Scholar - [7]P. B. Kleidman and M. W. Liebeck,
*The Subgroup Structure of the Finite Classical Groups*, London Mathematical Society Lecture Note Series, Vol. 129, Cambridge University Press, Cambridge, 1990.Google Scholar - [8]A. Kleshchev and A. Premet,
*On second degree cohomology of symmetric and alternating groups*, Communications in Algebra**21**(1993), 583–600.MathSciNetCrossRefzbMATHGoogle Scholar - [9]M. W. Liebeck,
*On graphs whose full automorphism group is an alternating group or a finite classical group*, Proceedings of the London Mathematical Society**47**(1983), 337–362.MathSciNetCrossRefzbMATHGoogle Scholar - [10]M. H. Peel,
*Hook representations of the symmetric groups*, Glasgow Mathematical Journal**12**(1971), 136–149.MathSciNetCrossRefzbMATHGoogle Scholar

## Copyright information

© Hebrew University of Jerusalem 2018