Regular subgroup of a permutation group

A regular subgroup of a permutation is in the group theory , a subset of a permutation which has the property of any two elements of the support amount of the permutation uniquely by a permutation blank into each other from this sub-group transfer.

A classic problem in finite group theory is the determination of all (finite) primitive permutation groups that have a regular subgroup. Liebeck-Praeger-Saxl solved this problem for almost-simple groups.

definition

Let it be a permutation group acting on a set . A sub-group is called regular if for any two elements a unique element with ${\ displaystyle G}$ ${\ displaystyle \ Omega}$ ${\ displaystyle U \ subset G}$ ${\ displaystyle u, v \ in \ Omega}$ ${\ displaystyle g \ in U}$

{\ displaystyle g (u) = v}

gives.

Examples

If the full permutation group is above with , the subgroup is not regular, because for the permutations given in cycle notation and applies ${\ displaystyle G}$ ${\ displaystyle \ Omega = \ {0, \ ldots, n-1 \}}$ ${\ displaystyle n> 2}$ ${\ displaystyle U = G}$ ${\ displaystyle (0,1)}$ ${\ displaystyle (0,1,2)}$

{\ displaystyle (0,1) (0) = 1}

and ,

{\ displaystyle (0,1,2) (0) = 1}

that means there is more than just one with . ${\ displaystyle g \ in U}$ ${\ displaystyle g (0) = 1}$

The subgroup

{\ displaystyle U = \ {(1), (0,1) \}}

is also not regular, because there is no with . ${\ displaystyle g \ in U}$ ${\ displaystyle g (1) = 2}$

The subgroup created by the cyclic permutation is regular because is to ${\ displaystyle (0.1, \ ldots, n-1)}$ ${\ displaystyle U}$ ${\ displaystyle i, j \ in \ Omega}$

{\ displaystyle (0,1, \ ldots, n-1) ^ {ji} \ in U}

which clearly certain elements from that on maps. This becomes immediately clear when you note that all elements are shifted cyclically by positions ${\ displaystyle U}$ ${\ displaystyle i}$ ${\ displaystyle j}$ ${\ displaystyle (0,1, \ ldots, n-1) ^ {k}}$ ${\ displaystyle \ Omega = \ {0, \ ldots, n-1 \}}$ ${\ displaystyle k}$

literature

Liebeck, Martin W .; Praeger, Cheryl E .; Saxl, Jan: Regular subgroups of primitive permutation groups. Mem. Amer. Math. Soc. 203 (2010), no.952, ISBN 978-0-8218-4654-4