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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.
 
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.==
 
==Regular group of permutations.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300501.png" /> be a finite group acting on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300502.png" />, i.e. a permutation group (group of permutations). The permutation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300503.png" /> is said to be regular if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300505.png" />, the stabilizer subgroup at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300506.png" />, is trivial.
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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 $a \in \Omega$, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300505.png"/>, the stabilizer subgroup at $a$, is trivial.
  
In the older mathematical literature, and in physics, a slightly stronger notion is used: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300507.png" /> is transitive (i.e., for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300508.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r1300509.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005010.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005012.png" /> is the number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005014.png" /> is, of course, the number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005015.png" />. It is easy to see that a transitive regular permutation group satisfies this condition. Inversely, a transitive permutation group for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005016.png" /> is regular.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005017.png" /> is a transitive regular permutation group, then all its elements, regarded as permutations on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005018.png" />, are regular permutations.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005020.png" />-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005021.png" /> of permutations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005022.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005023.png" /> defined by left (respectively, right) translation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005024.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005025.png" />) exhibits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005026.png" /> as a regular permutation group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005027.png" />.
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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.==
 
==Regular group of automorphisms.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005028.png" /> act on a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005029.png" /> by means of automorphisms (i.e., there is given a homomorphism of groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005032.png" />). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005033.png" /> is said to act fixed-point-free if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005034.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005035.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005036.png" />, i.e. there is no other global fixed point except the obvious and necessary one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005037.png" />. There is a conjecture that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005038.png" /> acts fixed-point-free on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005040.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005041.png" /> is solvable, [[#References|[a7]]]; see also [[Fitting length|Fitting length]] for some detailed results in this direction.
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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.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005042.png" /> is said to be a regular group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005043.png" /> if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005044.png" /> only the identity element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005045.png" /> is left fixed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005046.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005047.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005048.png" />. Some authors use the terminology  "fixed-point-free"  for the just this property.
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$G$ 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005049.png" />-group.==
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==Regular $p$-group.==
A [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005050.png" />-group]] is said to be regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005053.png" /> is an element of the commutator subgroup of the subgroup generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005055.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005056.png" /> is a product of iterated commutators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130050/r13005058.png" />. See [[#References|[a5]]].
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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====
 
====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>
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<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>

Revision as of 16:58, 1 July 2020

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=11804
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article