Regular group

There are several (different) notions of regularity in group theory. Most are not intrinsic to a group itself, but pertain to a group acting on something.

Regular group of permutations.

Let be a finite group acting on a set , i.e. a permutation group (group of permutations). The permutation group is said to be regular if for all , , the stabilizer subgroup at , is trivial.

In the older mathematical literature, and in physics, a slightly stronger notion is used: is transitive (i.e., for all there is a such that ) and , where is the number of elements of and is, of course, the number of elements of . It is easy to see that a transitive regular permutation group satisfies this condition. Inversely, a transitive permutation group for which is regular.

A permutation is regular if all cycles in its canonical cycle decomposition have the same length. If is a transitive regular permutation group, then all its elements, regarded as permutations on , are regular permutations.

An example of a transitive regular permutation group is the Klein -group of permutations of .

The regular permutation representation of a group defined by left (respectively, right) translation (respectively, ) exhibits as a regular permutation group on .

Regular group of automorphisms.

Let act on a group by means of automorphisms (i.e., there is given a homomorphism of groups , , ). is said to act fixed-point-free if for all there is a such that , i.e. there is no other global fixed point except the obvious and necessary one . There is a conjecture that if acts fixed-point-free on and , then is solvable, [a7]; see also Fitting length for some detailed results in this direction.

is said to be a regular group of automorphisms of if for all only the identity element of is left fixed by , i.e. for all . Some authors use the terminology "fixed-point-free" for the just this property.

Regular -group.

A -group is said to be regular if , where is an element of the commutator subgroup of the subgroup generated by and , i.e. is a product of iterated commutators of and . See [a5].

References

[a1]	K. Doerk, T. Hawkes, "Finite soluble groups" , de Gruyter (1992) pp. 16
[a2]	W. Ledermann, A.J. Weir, "Introduction to group theory" , Longman (1996) pp. 125 (Edition: Second)
[a3]	M. Hall Jr., "The theory of groups" , Macmillan (1963) pp. 183
[a4]	M. Hamermesh, "Group theory and its applications to physical problems" , Dover, reprint (1989) pp. 19
[a5]	R.D. Carmichael, "Groups of finite order" , Dover, reprint (1956) pp. 54ff
[a6]	L. Dornhoff, "Group representation theory. Part A" , M. Dekker (1971) pp. 65
[a7]	B. Huppert, N. Blackburn, "Finite groups III" , Springer (1982) pp. Chap. X