Subextensions for a permutation \(\mathrm{PSL}_2(q)\)-module
A. Zavarnitsine
We denote by \(\mathbb{F}_q\) a finite field of order \(q\) and by \(\mathbb{Z}_n\) a cyclic group of order \(n\).
Let \(q\) be an odd prime power and let \(G=\mathrm{PSL}_2(q)\). From the Universal Embedding Theorem [1, Theorem 2.6.A], it follows that the regular wreath product \(\mathbb{Z}_2 \wr G\) contains a subgroup isomorphic to \(\mathrm{SL}_2(q)\).
It is of interest to know if the same is true for a permutation wreath product that is not necessarily regular. In particular, let \(\rho\) be the natural permutation representation of \(G\) of degree \(q+1\) on the projective line over \(\mathbb{F}_q\). The following problem arose in the research [2].
Problem 1.Does the permutation wreath product \(\mathbb{Z}_2 \wr_\rho G\) contain a subgroup isomorphic to \(\mathrm{SL}_2(q)\)?
Although stated in purely group-theoretic terms, this problem is cohomological in nature. We reformulate a generalized version of this question as an assertion about a homomorphism between second cohomology groups of group modules. We then apply some basic cohomology theory to obtain the following.
Theorem 1.
If \(q\equiv -1 \pmod 4\) then the answer to Problem 1 is affirmative.
It seems that the case \(q\equiv 1 \pmod 4\) is more complicated and requires some deeper considerations than those presented here. We put forward
Conjecture 1. If \(q\equiv 1 \pmod 4\) then \(\mathrm{SL}_2(q)\) is not embedded in \(\mathbb{Z}_2 \wr_\rho G\).
For small values \(q=5,9,13,17\), Conjecture 1 was confirmed using a computer. We show how to reduce this problem to the determination of the first cohomology groups \(H^1(G,U_{\pm})\), where \(U_+\) and \(U_-\) are the two nontrivial absolutely irreducible \(G\)-modules in the principal \(2\)-block.
Acknowledgement. The research is partially supported by the Russian Foundation for Basic Research
(projects 11-01-00456, 11-01-91158, 12-01-90006, 13-01-00505);
by the Federal Target Grant "Scientific and
educational personnel of innovative Russia" (contract 14.740.11.0346).
References
J. D. Dixon, B. Mortimer, Permutation groups, Graduate Texts in Mathematics, 163, Springer-Verlag, New York (1996), 346 p.
N. V. Maslova, D. O. Revin, On composition factors of a finite group that is minimal with respect to the prime spectrum. (in preparation)
