Permutation groups where non-trivial elements have few ...€¦ · In my ongoing work with Kay Magaard, I look at situations where non-trivial elements of G have few xed points on

IntroductionTwo fixed points

Three fixed points

Permutation groups where non-trivial elementshave few fixed points

Nikolaus Conference 2013, Aachen

Rebecca Waldecker (MLU Halle-Wittenberg)

6th December 2013

Nikolaus Conference 2013, Aachen Permutation groups where non-trivial elements have few fixed points


Three fixed points

In this talk G denotes a finite group that acts faithfully andtransitively on a set Ω.



Three fixed points

In my ongoing work with Kay Magaard, I look at situations wherenon-trivial elements of G have few fixed points on Ω. Our startingpoint was the following situation:

The action of G on Ω is not regular, and every non-trivial elementof G fixes at most two (at most three) points.



Three fixed points

In my ongoing work with Kay Magaard, I look at situations wherenon-trivial elements of G have few fixed points on Ω. Our startingpoint was the following situation:

The action of G on Ω is not regular, and every non-trivial elementof G fixes at most two (at most three) points.



Three fixed points

ExamplesProperties of GSimple groups

The smallest non-trivial examples for our situation are provided byS3 in its natural action on 1, 2, 3 and by A4 acting on1, 2, 3, 4. These are examples for Frobenius groups, i.e. there isno non-trivial element with two fixed points.

More examples

The groups S4 and A5 provide examples satisfying our mainhypothesis, in their natural action. This time two fixed pointsactually occur.



Three fixed points


The smallest non-trivial examples for our situation are provided byS3 in its natural action on 1, 2, 3 and by A4 acting on1, 2, 3, 4. These are examples for Frobenius groups, i.e. there isno non-trivial element with two fixed points.

More examples

The groups S4 and A5 provide examples satisfying our mainhypothesis, in their natural action. This time two fixed pointsactually occur.



Three fixed points


Lemma

Let α ∈ Ω and H := Gα, the stabiliser of α in G .

(1) If 1 6= X ≤ H, then |NG (X ) : NH(X )| ≤ 2.

(2) If p is an odd prime divisor of |H|, then H contains a Sylowp-subgroup of G .

We see that an interesting special case of (1) occurs if G has oddorder.Then it follows that G is a Frobenius group with point stabilisersas Frobenius complements.



Three fixed points


Lemma


(1) If 1 6= X ≤ H, then |NG (X ) : NH(X )| ≤ 2.





Three fixed points


Lemma


(1) If 1 6= X ≤ H, then |NG (X ) : NH(X )| ≤ 2.





Three fixed points


Lemma


(1) If 1 6= X ≤ H, then |NG (X ) : NH(X )| ≤ 2.





Three fixed points


Theorem A

Suppose that G is non-abelian and simple and that it has trivialthree point stabilisers. Then either G is isomorphic to PSL3(4) orthere exists a prime power q such that G is isomorphic to PSL2(q)or to Sz(q).

In fact we prove a more general result about finite groups(submitted to JPAA).A direct proof for simple groups works as follows: We study thesubgroup structure of G and then use this knowledge and theClassification of Finite Simple Groups.

At the moment I am working on a direct identification of thesimple groups that appear under our hypothesis.



Three fixed points


Theorem A


In fact we prove a more general result about finite groups(submitted to JPAA).

A direct proof for simple groups works as follows: We study thesubgroup structure of G and then use this knowledge and theClassification of Finite Simple Groups.




Three fixed points


Theorem A






Three fixed points


Theorem A






Three fixed points


For the remainder of this talk we look at the situation where allnon-trivial elements of G have at most three fixed points.



Three fixed points


The smallest (non-boring) example where three fixed pointsactually occur is S5 acting naturally on 1, 2, 3, 4, 5.

A classical result says that the only sharply 4-transitive groups areS4,S5,A6,M11. This gives us two new examples, namely A6 (in itsnatural action) and M11 (on 11 points).



Three fixed points


The smallest (non-boring) example where three fixed pointsactually occur is S5 acting naturally on 1, 2, 3, 4, 5.

A classical result says that the only sharply 4-transitive groups areS4,S5,A6,M11. This gives us two new examples, namely A6 (in itsnatural action) and M11 (on 11 points).



Three fixed points


As in the case with at most two fixed points, we begin our analysiswith the local structure of G .

Lemma

Let α ∈ Ω and H := Gα.

(1) If 1 6= X ≤ H, then |NG (X ) : NH(X )| ≤ 3.

(2) If p ≥ 5 is a prime divisor of |H|, then H contains a Sylowp-subgroup of G .

(3) If H contains a non-trivial 2-element that fixes three points,then H contains a Sylow 2-subgroup of G .



Three fixed points



Lemma


(1) If 1 6= X ≤ H, then |NG (X ) : NH(X )| ≤ 3.





Three fixed points



Lemma


(1) If 1 6= X ≤ H, then |NG (X ) : NH(X )| ≤ 3.





Three fixed points



Lemma


(1) If 1 6= X ≤ H, then |NG (X ) : NH(X )| ≤ 3.





Three fixed points



Lemma


(1) If 1 6= X ≤ H, then |NG (X ) : NH(X )| ≤ 3.





Three fixed points


Theorem B

Suppose that G is non-abelian and simple and that it has trivialfour point stabilisers. Then one of the following holds:

1 G ∼= A5, A6, A7 or A8 (with precise action explained).

2 G ∼=PSL2(7), PSL2(11), PSL3(4), PSL4(3), PSU4(3) orPSL4(5) (with precise action described).

3 There is a prime power q such that G ∼=PSL3(q) or PSU3(q).(These are the generic examples.)

4 G ∼= M11 acting on 11 points.

5 G ∼= M22 acting on 27 · 32 · 5 · 11 points.



Three fixed points


Theorem B









Three fixed points


Theorem B









Three fixed points


Theorem B









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Theorem B









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Theorem B









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Comments on the proof

Again our main results are more general, but in the case of simplegroups the strategy is as before.

The analysis of the local structure of G is much harder this timebecause the primes 2 and 3 both play special roles and allow formany different configurations.

Still we are able to collect enough information so that we canapply the Classification of Finite Simple Groups. However, a directidentification of the simple examples seems out of reach at themoment.



Three fixed points



Again our main results are more general, but in the case of simplegroups the strategy is as before.The analysis of the local structure of G is much harder this timebecause the primes 2 and 3 both play special roles and allow formany different configurations.




Three fixed points



Again our main results are more general, but in the case of simplegroups the strategy is as before.The analysis of the local structure of G is much harder this timebecause the primes 2 and 3 both play special roles and allow formany different configurations.




Three fixed points


With applications to Riemann surfaces and Weierstrass points inmind, we will next look at the situation where also four fixedpoints are allowed. We expect a few more simple groups to appearas examples there.



Three fixed points


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Permutation groups where non-trivial elements have few ...€¦ · In my ongoing work with Kay Magaard, I look at situations where non-trivial elements of G have few xed points on