 RegularWreathProduct - Maple Help

Home : Support : Online Help : Mathematics : Group Theory : RegularWreathProduct

GroupTheory

 WreathProduct
 form the wreath product of groups
 RegularWreathProduct
 form the regular wreath product of groups Calling Sequence WreathProduct( G, H, ... ) RegularWreathProduct( G, H, ... ) Parameters

 G,H, ... - two or more permutation groups Description

 • Let $G$ and $H$ be permutation groups. The wreath product G $\wr$ H_ of $G$ by $H$ is a permutation group constructed as a semi-direct product of $d$ copies of $G$ (called the base group), where $d$ is the degree of $H$, and the action of $H$ on the base group is the action of $H$ by permuting the copies of $G$. Thus, the order of G $\wr$ H_ is equal to ${\left|G\right|}^{d}\left|H\right|$, and the degree of the wreath product is the product of the degrees of $G$ and $H$.
 • The regular wreath product of $G$ and $H$ is the wreath product in which $H$ is considered as a regular permutation group on itself. (This is also called the "standard wreath product".)
 • The WreathProduct( G, H ) command returns a permutation group that is the wreath product G $\wr$ H_.
 • If more than two groups are provided as input, then an iterated wreath product is constructed using the left associative rule. For example, WreathProduct( A, B, C, D ) returns ((A $\wr$ B) $\wr$ C) $\wr$ D_.
 • The RegularWreathProduct( G, H ) command returns the regular wreath product of $G$ and $H$. In this case, it is not required that $H$ be a permutation group, as a regular permutation representation of the finite group $H$ is used instead. (Here, $H$ may be either a Cayley table group or a finitely presented finite group, as well as a permutation group, which need not be itself regular.) Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(2\right),\mathrm{CyclicGroup}\left(2\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)⟩$ (1)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{DihedralGroup}\left(4\right)\right)$
 ${\mathrm{true}}$ (2)

Iterated wreath products appear naturally as Sylow subgroups of symmetric groups of prime power degree.

 > $\mathrm{AreIsomorphic}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right)\right),\mathrm{SylowSubgroup}\left(3,\mathrm{Symm}\left(9\right)\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{AreIsomorphic}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(2\right)\$3\right),\mathrm{SylowSubgroup}\left(2,\mathrm{Symm}\left({2}^{3}\right)\right)\right)$
 ${\mathrm{true}}$ (4)

Note that iterated wreath products grow quite rapidly.

 > $\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right)\right)\right)$
 ${81}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right)\right)\right)$
 ${1594323}$ (6)
 > $\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right)\right)\right)$
 ${12157665459056928801}$ (7)
 > $\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right),\mathrm{CyclicGroup}\left(3\right)\right)\right)$
 ${5391030899743293631239539488528815119194426882613553319203}$ (8)
 > $W≔\mathrm{WreathProduct}\left(\mathrm{Alt}\left(4\right),\mathrm{Symm}\left(3\right)\right)$
 ${W}{≔}{\mathrm{< a permutation group on 12 letters with 4 generators >}}$ (9)
 > $\mathrm{GroupOrder}\left(W\right)$
 ${10368}$ (10)

The wreath product construction is not commutative; notice that even the order is different.

 > $\mathrm{GroupOrder}\left(\mathrm{WreathProduct}\left(\mathrm{Symm}\left(3\right),\mathrm{Alt}\left(4\right)\right)\right)$
 ${15552}$ (11)

Note that the regular wreath product is also different in this case, since the second argument here is not the regular permutation representation of the symmetric group.

 > $\mathrm{GroupOrder}\left(\mathrm{RegularWreathProduct}\left(\mathrm{Alt}\left(4\right),\mathrm{Symm}\left(3\right)\right)\right)$
 ${17915904}$ (12)

Since both ${\mathbf{S}}_{3}$ and ${\mathbf{A}}_{4}$ are transitive, so too is their wreath product.

 > $\mathrm{IsTransitive}\left(W\right)$
 ${\mathrm{true}}$ (13)

However, in general, the wreath product is not primitive.

 > $\mathrm{IsPrimitive}\left(W\right)$
 ${\mathrm{false}}$ (14)
 > $\mathrm{BlockSystem}\left(W\right)$
 $\left\{\left\{{1}{,}{2}{,}{3}{,}{4}\right\}{,}\left\{{5}{,}{6}{,}{7}{,}{8}\right\}{,}\left\{{9}{,}{10}{,}{11}{,}{12}\right\}\right\}$ (15)

Here we construct a wreath product with an intransitive second argument.

 > $W≔\mathrm{WreathProduct}\left(\mathrm{Alt}\left(4\right),\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right)\right]\right)\right):$

The resulting group is not transitive.

 > $\mathrm{IsTransitive}\left(W\right)$
 ${\mathrm{false}}$ (16)
 > $\mathrm{map}\left(\mathrm{Elements},\mathrm{Orbits}\left(W\right)\right)$
 $\left[\left\{{1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{6}{,}{7}{,}{8}\right\}{,}\left\{{9}{,}{10}{,}{11}{,}{12}{,}{13}{,}{14}{,}{15}{,}{16}\right\}\right]$ (17)

Here we construct a wreath product with an intransitive first argument.

 > $W≔\mathrm{WreathProduct}\left(\mathrm{Group}\left(\left[\mathrm{Perm}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right)\right]\right),\mathrm{Symm}\left(3\right)\right):$

Again, the result is an intransitive group.

 > $\mathrm{IsTransitive}\left(W\right)$
 ${\mathrm{false}}$ (18)
 > $\mathrm{map}\left(\mathrm{Elements},\mathrm{Orbits}\left(W\right)\right)$
 $\left[\left\{{1}{,}{2}{,}{5}{,}{6}{,}{9}{,}{10}\right\}{,}\left\{{3}{,}{4}{,}{7}{,}{8}{,}{11}{,}{12}\right\}\right]$ (19)
 > $W≔\mathrm{RegularWreathProduct}\left(\mathrm{CyclicGroup}\left(2\right),⟨⟨a,b⟩|⟨{a}^{2},{b}^{3},{\left(\mathrm{.}\left(a,b\right)\right)}^{5}=1⟩⟩\right)$
 ${W}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{3}\right)\left({2}{,}{4}\right)\left({5}{,}{13}\right)\left({6}{,}{14}\right)\left({7}{,}{15}\right)\left({8}{,}{16}\right)\left({9}{,}{17}\right)\left({10}{,}{18}\right)\left({11}{,}{19}\right)\left({12}{,}{20}\right)\left({21}{,}{37}\right)\left({22}{,}{38}\right)\left({23}{,}{39}\right)\left({24}{,}{40}\right)\left({25}{,}{41}\right)\left({26}{,}{42}\right)\left({27}{,}{43}\right)\left({28}{,}{44}\right)\left({29}{,}{45}\right)\left({30}{,}{46}\right)\left({31}{,}{47}\right)\left({32}{,}{48}\right)\left({33}{,}{49}\right)\left({34}{,}{50}\right)\left({35}{,}{51}\right)\left({36}{,}{52}\right)\left({53}{,}{73}\right)\left({54}{,}{74}\right)\left({55}{,}{75}\right)\left({56}{,}{76}\right)\left({57}{,}{59}\right)\left({58}{,}{60}\right)\left({61}{,}{77}\right)\left({62}{,}{78}\right)\left({63}{,}{79}\right)\left({64}{,}{80}\right)\left({65}{,}{81}\right)\left({66}{,}{82}\right)\left({67}{,}{69}\right)\left({68}{,}{70}\right)\left({71}{,}{83}\right)\left({72}{,}{84}\right)\left({85}{,}{107}\right)\left({86}{,}{108}\right)\left({87}{,}{89}\right)\left({88}{,}{90}\right)\left({91}{,}{109}\right)\left({92}{,}{110}\right)\left({93}{,}{111}\right)\left({94}{,}{112}\right)\left({95}{,}{97}\right)\left({96}{,}{98}\right)\left({99}{,}{101}\right)\left({100}{,}{102}\right)\left({103}{,}{113}\right)\left({104}{,}{114}\right)\left({105}{,}{115}\right)\left({106}{,}{116}\right)\left({117}{,}{119}\right)\left({118}{,}{120}\right){,}\left({1}{,}{5}{,}{7}\right)\left({2}{,}{6}{,}{8}\right)\left({3}{,}{9}{,}{11}\right)\left({4}{,}{10}{,}{12}\right)\left({13}{,}{21}{,}{23}\right)\left({14}{,}{22}{,}{24}\right)\left({15}{,}{25}{,}{27}\right)\left({16}{,}{26}{,}{28}\right)\left({17}{,}{29}{,}{31}\right)\left({18}{,}{30}{,}{32}\right)\left({19}{,}{33}{,}{35}\right)\left({20}{,}{34}{,}{36}\right)\left({37}{,}{51}{,}{53}\right)\left({38}{,}{52}{,}{54}\right)\left({39}{,}{55}{,}{57}\right)\left({40}{,}{56}{,}{58}\right)\left({41}{,}{59}{,}{61}\right)\left({42}{,}{60}{,}{62}\right)\left({43}{,}{63}{,}{45}\right)\left({44}{,}{64}{,}{46}\right)\left({47}{,}{65}{,}{67}\right)\left({48}{,}{66}{,}{68}\right)\left({49}{,}{69}{,}{71}\right)\left({50}{,}{70}{,}{72}\right)\left({73}{,}{85}{,}{87}\right)\left({74}{,}{86}{,}{88}\right)\left({75}{,}{89}{,}{91}\right)\left({76}{,}{90}{,}{92}\right)\left({77}{,}{93}{,}{95}\right)\left({78}{,}{94}{,}{96}\right)\left({79}{,}{97}{,}{99}\right)\left({80}{,}{98}{,}{100}\right)\left({81}{,}{101}{,}{103}\right)\left({82}{,}{102}{,}{104}\right)\left({83}{,}{105}{,}{107}\right)\left({84}{,}{106}{,}{108}\right)\left({109}{,}{117}{,}{111}\right)\left({110}{,}{118}{,}{112}\right)\left({113}{,}{119}{,}{115}\right)\left({114}{,}{120}{,}{116}\right)⟩$ (20)
 > $\mathrm{IsPrimitive}\left(W\right)$
 ${\mathrm{false}}$ (21)
 > $\mathrm{BlockSystem}\left(W\right)$
 $\left\{\left\{{1}{,}{2}\right\}{,}\left\{{3}{,}{4}\right\}{,}\left\{{5}{,}{6}\right\}{,}\left\{{7}{,}{8}\right\}{,}\left\{{9}{,}{10}\right\}{,}\left\{{11}{,}{12}\right\}{,}\left\{{13}{,}{14}\right\}{,}\left\{{15}{,}{16}\right\}{,}\left\{{17}{,}{18}\right\}{,}\left\{{19}{,}{20}\right\}{,}\left\{{21}{,}{22}\right\}{,}\left\{{23}{,}{24}\right\}{,}\left\{{25}{,}{26}\right\}{,}\left\{{27}{,}{28}\right\}{,}\left\{{29}{,}{30}\right\}{,}\left\{{31}{,}{32}\right\}{,}\left\{{33}{,}{34}\right\}{,}\left\{{35}{,}{36}\right\}{,}\left\{{37}{,}{38}\right\}{,}\left\{{39}{,}{40}\right\}{,}\left\{{41}{,}{42}\right\}{,}\left\{{43}{,}{44}\right\}{,}\left\{{45}{,}{46}\right\}{,}\left\{{47}{,}{48}\right\}{,}\left\{{49}{,}{50}\right\}{,}\left\{{51}{,}{52}\right\}{,}\left\{{53}{,}{54}\right\}{,}\left\{{55}{,}{56}\right\}{,}\left\{{57}{,}{58}\right\}{,}\left\{{59}{,}{60}\right\}{,}\left\{{61}{,}{62}\right\}{,}\left\{{63}{,}{64}\right\}{,}\left\{{65}{,}{66}\right\}{,}\left\{{67}{,}{68}\right\}{,}\left\{{69}{,}{70}\right\}{,}\left\{{71}{,}{72}\right\}{,}\left\{{73}{,}{74}\right\}{,}\left\{{75}{,}{76}\right\}{,}\left\{{77}{,}{78}\right\}{,}\left\{{79}{,}{80}\right\}{,}\left\{{81}{,}{82}\right\}{,}\left\{{83}{,}{84}\right\}{,}\left\{{85}{,}{86}\right\}{,}\left\{{87}{,}{88}\right\}{,}\left\{{89}{,}{90}\right\}{,}\left\{{91}{,}{92}\right\}{,}\left\{{93}{,}{94}\right\}{,}\left\{{95}{,}{96}\right\}{,}\left\{{97}{,}{98}\right\}{,}\left\{{99}{,}{100}\right\}{,}\left\{{101}{,}{102}\right\}{,}\left\{{103}{,}{104}\right\}{,}\left\{{105}{,}{106}\right\}{,}\left\{{107}{,}{108}\right\}{,}\left\{{109}{,}{110}\right\}{,}\left\{{111}{,}{112}\right\}{,}\left\{{113}{,}{114}\right\}{,}\left\{{115}{,}{116}\right\}{,}\left\{{117}{,}{118}\right\}{,}\left\{{119}{,}{120}\right\}\right\}$ (22)
 > $S≔\mathrm{Stabilizer}\left(109,W\right):$
 > $\mathrm{Index}\left(S,W\right)$
 ${120}$ (23)
 > $\mathrm{GroupOrder}\left(S\right)$
 ${576460752303423488}$ (24)
 > $\mathrm{IsElementary}\left(S\right)$
 ${\mathrm{true}}$ (25)
 > $\mathrm{DirectFactors}\left(S\right)$
 ${\mathrm{< a permutation group on 120 letters with 31 generators >}}{,}{\mathrm{< a permutation group on 118 letters with 28 generators >}}{,}{\mathrm{< a permutation group on 116 letters with 27 generators >}}{,}{\mathrm{< a permutation group on 114 letters with 23 generators >}}{,}{\mathrm{< a permutation group on 112 letters with 25 generators >}}{,}{\mathrm{< a permutation group on 108 letters with 28 generators >}}{,}{\mathrm{< a permutation group on 106 letters with 31 generators >}}{,}{\mathrm{< a permutation group on 104 letters with 37 generators >}}{,}{\mathrm{< a permutation group on 102 letters with 31 generators >}}{,}{\mathrm{< a permutation group on 100 letters with 30 generators >}}{,}{\mathrm{< a permutation group on 98 letters with 27 generators >}}{,}{\mathrm{< a permutation group on 96 letters with 32 generators >}}{,}{\mathrm{< a permutation group on 94 letters with 29 generators >}}{,}{\mathrm{< a permutation group on 92 letters with 30 generators >}}{,}{\mathrm{< a permutation group on 90 letters with 23 generators >}}{,}{\mathrm{< a permutation group on 88 letters with 30 generators >}}{,}{\mathrm{< a permutation group on 86 letters with 27 generators >}}{,}{\mathrm{< a permutation group on 84 letters with 25 generators >}}{,}{\mathrm{< a permutation group on 82 letters with 28 generators >}}{,}{\mathrm{< a permutation group on 80 letters with 28 generators >}}{,}{\mathrm{< a permutation group on 78 letters with 35 generators >}}{,}{\mathrm{< a permutation group on 76 letters with 28 generators >}}{,}{\mathrm{< a permutation group on 74 letters with 28 generators >}}{,}{\mathrm{< a permutation group on 72 letters with 27 generators >}}{,}{\mathrm{< a permutation group on 70 letters with 27 generators >}}{,}{\mathrm{< a permutation group on 68 letters with 30 generators >}}{,}{\mathrm{< a permutation group on 66 letters with 20 generators >}}{,}{\mathrm{< a permutation group on 64 letters with 26 generators >}}{,}{\mathrm{< a permutation group on 62 letters with 25 generators >}}{,}{\mathrm{< a permutation group on 60 letters with 29 generators >}}{,}{\mathrm{< a permutation group on 30 letters with 31 generators >}}{,}{\mathrm{< a permutation group on 28 letters with 28 generators >}}{,}{\mathrm{< a permutation group on 26 letters with 34 generators >}}{,}{\mathrm{< a permutation group on 24 letters with 29 generators >}}{,}{\mathrm{< a permutation group on 22 letters with 32 generators >}}{,}{\mathrm{< a permutation group on 20 letters with 32 generators >}}{,}{\mathrm{< a permutation group on 18 letters with 29 generators >}}{,}{\mathrm{< a permutation group on 16 letters with 35 generators >}}{,}{\mathrm{< a permutation group on 14 letters with 30 generators >}}{,}{\mathrm{< a permutation group on 2 letters with 27 generators >}}{,}{\mathrm{< a permutation group on 4 letters with 36 generators >}}{,}{\mathrm{< a permutation group on 6 letters with 29 generators >}}{,}{\mathrm{< a permutation group on 8 letters with 26 generators >}}{,}{\mathrm{< a permutation group on 10 letters with 29 generators >}}{,}{\mathrm{< a permutation group on 12 letters with 32 generators >}}{,}{\mathrm{< a permutation group on 32 letters with 24 generators >}}{,}{\mathrm{< a permutation group on 34 letters with 25 generators >}}{,}{\mathrm{< a permutation group on 36 letters with 25 generators >}}{,}{\mathrm{< a permutation group on 38 letters with 24 generators >}}{,}{\mathrm{< a permutation group on 40 letters with 35 generators >}}{,}{\mathrm{< a permutation group on 42 letters with 29 generators >}}{,}{\mathrm{< a permutation group on 44 letters with 31 generators >}}{,}{\mathrm{< a permutation group on 46 letters with 31 generators >}}{,}{\mathrm{< a permutation group on 48 letters with 28 generators >}}{,}{\mathrm{< a permutation group on 50 letters with 26 generators >}}{,}{\mathrm{< a permutation group on 52 letters with 30 generators >}}{,}{\mathrm{< a permutation group on 54 letters with 28 generators >}}{,}{\mathrm{< a permutation group on 56 letters with 25 generators >}}{,}{\mathrm{< a permutation group on 58 letters with 32 generators >}}$ (26)

Notice that, if a group is already regular, then its regular wreath product is isomorphic to the ordinary wreath product.

 > $R≔\mathrm{TransitiveGroup}\left(6,2\right)$
 ${R}{≔}⟨\left({1}{,}{3}{,}{5}\right)\left({2}{,}{4}{,}{6}\right){,}\left({1}{,}{4}\right)\left({2}{,}{3}\right)\left({5}{,}{6}\right)⟩$ (27)
 > $\mathrm{IsRegular}\left(R\right)$
 ${\mathrm{true}}$ (28)
 > $\mathrm{AreIsomorphic}\left(\mathrm{WreathProduct}\left(\mathrm{CyclicGroup}\left(2\right),R\right),\mathrm{RegularWreathProduct}\left(\mathrm{CyclicGroup}\left(2\right),R\right)\right)$
 ${\mathrm{true}}$ (29)