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

In the older mathematical literature, and in physics, a slightly stronger notion is used: $G$ is transitive (i.e., for all $a , b \in \Omega$ there is a $g \in G$ such that $g a = b$) and $\operatorname{degree}( G , \Omega ) = \operatorname { order } ( G )$, where $\operatorname{degree}( G , \Omega )$ is the number of elements of $\Omega$ and \operatorname{order}( G )$is, of course, the number of elements of$G$. It is easy to see that a transitive regular permutation group satisfies this condition. Inversely, a transitive permutation group for which$\operatorname{degree}( G , \Omega ) = \operatorname { order } ( G )$is regular. A permutation is regular if all cycles in its canonical cycle decomposition have the same length. If$G$is a transitive regular permutation group, then all its elements, regarded as permutations on$\Omega$, are regular permutations. An example of a transitive regular permutation group is the Klein$4$-group$G = V _ { 4 } = \{ ( 1 ) , ( 12 ) ( 34 ) , ( 13 ) ( 24 ) , ( 14 ) ( 23 ) \}$of permutations of$\Omega = \{ 1,2,3,4 \}$. The regular permutation representation of a group$G$defined by left (respectively, right) translation$g : h \mapsto g h$(respectively,$g : h \mapsto h g ^ { - 1 }$) exhibits$G$as a regular permutation group on$\Omega = G$. =='"UNIQ--h-1--QINU"'Regular group of automorphisms.== Let$G$act on a group$A$by means of automorphisms (i.e., there is given a homomorphism of groups$G \rightarrow \operatorname { Aut } ( A )$,$\alpha \mapsto a ^ { g }$,$a \in A$).$G$is said to act fixed-point-free if for all$a \in A$there is a$g \in G$such that$a ^ { g } \neq a$, i.e. there is no other global fixed point except the obvious and necessary one$1 \in A$. There is a conjecture that if$G$acts fixed-point-free on$A$and$( | G | , | A | ) = 1$, then$A$is solvable, [[#References|[a7]]]; see also [[Fitting length|Fitting length]] for some detailed results in this direction.$G$is said to be a regular group of automorphisms of$A$if for all$1 \neq g \in G$only the identity element of$A$is left fixed by$g$, i.e.$C _ { A } ( g ) = \{ a \in A : a ^ { g } = a \} = \{ 1 \}$for all$g \neq 1$. Some authors use the terminology "fixed-point-free" for the just this property. =='"UNIQ--h-2--QINU"'Regular$p$-group.== A [[P-group|$p$-group]] is said to be regular if$( x y ) ^ { p } = x ^ { p } y ^ { p } z$, where$z$is an element of the commutator subgroup of the subgroup generated by$x$and$y$, i.e.$z$is a product of iterated commutators of$x$and$y\$. 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
How to Cite This Entry:
Regular group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_group&oldid=50720
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article