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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 $a \in \Omega$, $G_a = \{g\in G\,|\, ga=a\}$, 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 $\Omega = G$.

==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.

==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 [[#References|[a5]]].

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