Generators and defining relations for wreath products of groups are given. Under a certain condition

(conormality of generators), they are minimal.Наведено твірні та визначальні співвідношення для вінцевих добутків. За деякої умови (конормальність твірних) вони є мінімальними

Drozd, Yu.A.

Skuratovskii, R.V.

Ukrains’kyi Matematychnyi Zhurnal

English

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

UDC 512.543 + 512.542Yu. A. Drozd  (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),R. V. Skuratovskii  (Kyiv. Nat. Taras Shevchenko Univ.)GENERATORS AND RELATIONS FOR WREATH PRODUCTSTVIRNI TA SPIVVIDNOÍENNQ DLQ VINCEVYX DOBUTKIVGenerators and defining relations for wreath products of groups are given.  Under a certain condition(conormality of generators), they are minimal.Navedeno tvirni ta vyznaçal\ni spivvidnoßennq dlq vincevyx dobutkiv.  Za deqko] umovy (konor-mal\nist\ tvirnyx) vony [ minimal\nymy.Let  G,  H   be two groups.  Denote by  HG  the group of all maps  f : G →  H   withfinite support, i.e., such that  f ( x ) = 1  for all but a finite set of elements of  G.  Recallthat their (restricted regular) wreath product  W = H  G   is defined as the semidirectproduct  HG   G  with the natural action of  G  on  H G  : f ag( )  →  f ( a g )  [1, p. 175].We are going to find a set of generators and relations for  H  G   knowing those for  Gand  H.  Then we shall extend this result to the multiple wreath products    knkG=1  == ( … ( ( G1  G2 )  G3 ) … )  Gn .If  x = { x1 , x2 , … , xn }  are generators for  G  and  R  = { R1 , R2 , … , Rm  }  aredefining relations for this set of generators, we write  G  : = 〈 x1 , x2 , … , xn | R1 , R2 , …… , Rm 〉  or  G : = 〈 x | R 〉.  A presentation is called minimal if neither of the generatorsx1 , x2 , … , xn  nor of the relations  R1 , R2 , … , Rm  can be excluded.  We call the set ofgenerators  x  conormal if neither element  x ∈ x  belongs to the normal subgroup  Nxgenerated by all  y ∈ x \ { x }.  For instance, any minimal set of generators of a finite  p-group  G  is conormal since their images are linear independent in the factorgroupG / G p [ G, G ]  [1] (Theorem 5.48).Theorem 1.  Let  G  : = 〈  x | R ( x ) 〉,  H  : = 〈  y | S ( y ) 〉  be presentations of  G  andH.  Choose a subset  T  ⊆  G   such that  T  ∩  T – 1 = ∅  and   T  ∪  T – 1 = G \ { 1 },where  T – 1 = { t– 1 | t ∈  T }.  Then the wreath product  W = H  G   has a presentationof the formW  : =  〈  x, y | R ( x ), S ( y ), [ y, t– 1 z t ]  = 1    for all  y, z ∈ y,  t ∈ T〉. (1)If the given presentations of  G   and   H   are minimal and the set of generators   yis conormal, the presentation (1) is minimal as well.Theorem 2.  Let  Gi : = 〈  xi | Ri  ( xi ) 〉  be presentations of the groups  Gi  ,  1 ≤ i ≤≤ m.  For  1 < i ≤ m  choose a subset  Ti ⊆  Gi  such that  T i ∩ Ti−1 = ∅  and  Ti ∪∪ Ti−1 = Gi \ { 1 }.  Then the wreath product  W = in iG=1   has a presentation of theformW  : =  〈  xi , 1 ≤ i ≤ m | Ri  ( xi ), 1 ≤ i ≤ m, [ x, t– 1 y t ]  = 1for all  x, y ∈  xii j<∪ ,  t ∈ Tj〉. (2)If all given presentations of  Gi  are minimal and the sets of generators  xi , 1 ≤ i < n,©  YU. A. DROZD, R. V. SKURATOVSKII, 2008ISSN  1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7 997998 YU. A. DROZD, R. V. SKURATOVSKIIare conormal, the presentation (2) is minimal as well.In what follows, we keep the notations of Theorem 1.  Note that  HG  =   a G H a∈ ( ) ,where  H  ( a )  is a copy of the group  H;  the elements of  H ( a )  will be denoted byh ( a ),  where  h  runs through  H.  Then  h ( a ) g = h ( a g )  and  HG  = 〈  y ( a ) | S ( y ( a ) ),[ y ( a ), z ( b ) ]  = 1〉,  where  a, b ∈ G,  a ≠ b.The following lemma is quite evident.Lemma 1.  Suppose a group  G  acting on a group  N .  Let  G  = 〈  x | R ( x ) 〉,  N  == 〈  y | S ( y ) 〉  be presentations of  G   and  N,  and  yx = wxy ( y )  for each  x ∈  x,  y ∈∈ y.   Then their semidirect product  N  G  has a presentationN  G  : =  〈  x, y | R ( x ), S ( y ), x– 1 y x  = wxy ( y )    for all  x ∈ x,  y ∈ y〉.Note that this presentation may not be minimal even if both presentations for  Gand  N  were so, since some elements of  y  may become superfluous.Corollary 1.  The wreath product  W = H  G  has indeed a presentation (1).Proof.  Lemma 1 gives a presentationW  : =  〈  x, y ( a ) | R ( x ), S ( y ( a ) ), [ y ( a ), z ( b ) ]  = 1x– 1 y ( a ) x  = y ( a x )   for  x ∈ x,  y, z ∈ y,  a, b ∈ G,  a ≠ b〉.Using the last relations, we can exclude all generators  y ( a )  for  a ≠ 1;  we only haveto replace  y ( a )  and  z ( b )  by  a– 1 y ( 1 ) a  and  b– 1 z ( 1 ) b.  So we shall write  h  insteadof  h ( 1 )  for  h ∈  H;  especially, the relations for  y ( a )  and  z  ( b )  are rewritten as[ a– 1 y a, b– 1 z b ]  = 1.  The latter is equivalent to  [ y, t– 1 z t ]  = 1,  where  t = b a– 1 ≠ 1.Moreover, the relations  [ y, t– 1 z t ]  = 1  and  [  z, t  y  t– 1 ]  = 1  are also equivalent;therefore we only need such relations for  t ∈ T.The corollary is proved.Lemma 2.  Suppose that  y  is a conormal set of generators of the group  H,  u ,v ∈ y,  and consider the group  H u, v = ( H *  H ′ ) / Nu , v ,  where  *  denotes the freeproduct of groups,  H ′   is a copy of the group  H  whose elements are denoted by  h′( h ∈  H ),   a n d    N u, v   is the normal subgroup of   H H* ′    generated by thecommutators  [ y, z′ ]  with  y, z ∈ y,  ( y, z ) ≠ ( u, v ).  Then  [ u, v′ ] ≠ 1  in  Hu, v .Proof.  Let  C = H / Nu ,  C ′ = H′ / Nv′ ,  P = C * C ′,  u  = u  Nu ,  v = v′ Nv ′ .  Considerthe homomorphism  ϕ  of  H * H′  to  P  such thatϕ ( y )  =  1 if ,if ,yu y uu∈=yϕ ( z′ )  =    1 if ,if .zz∈′ =yvv vObviously,  ϕ  is well defined and  ϕ  ( [ y, z′ ] ) = 1  if  ( y, z ) ≠ ( u, v  ),  so it induces ahomomorphism  Hu, v → P.  Since  ϕ ( [ u, v′ ] ) =  [ ′]u, v  ≠ 1,  it accomplishes the proof.Now fix elements  c ∈ T,  u, v ∈ y,  and let  Kc, u , v  be the group with a presentationKc, u , v  : =  〈  y ( a ),  a ∈ G | S ( y ( a ) ), [ y ( a ), z ( t a ) ]  = 1for all  y, z ∈ y,  a ∈ G,  t ∈ T,  ( t, y, z ) ≠ ( c, u, v )〉.ISSN  1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7GENERATORS AND RELATIONS FOR WREATH PRODUCTS 999Corollary 2.  Let the set of generators  y  be conormal.  Then  [ u ( 1 ), v  ( c ) ]  ≠ 1in the group  Kc, u , v .Proof.  There is a homomorphism  ψ : Kc, u , v →  Hu, v ,  where  Hu, v  is the groupfrom Lemma 2, mapping   u ( 1 )    u,  v ( c )     v′,  y ( a ) → 1  in all other cases.  Thenψ ( [ u ( 1 ), v ( c ) ] ) = [ u, v′ ] ≠ 1,  so  [ u ( 1 ), v ( c ) ] ≠ 1  as well.Corollary 3.  If the given presentations of    G    and   H    are minimal and the setof generators  y  is conormal, the presentation (1) is minimal.Proof.  Obviously, we can omit from (1) neither of generators  x ,  y   nor of therelations  R  ( x ),  S  ( y ).  So we have to prove that neither relation  [ u, c– 1 v c ] = 1( u, v ∈ y, c ∈ T )  can be omitted as well.  Consider the group  K = Kc, u , v  of Corollary2.  The group  G  acts on  K  by the rule:  h ( a ) g = h ( a g ).  Let  Q = K    G.  Then, justas in the proof of Corollary 1, this group has a presentationQ : = 〈  x, y | R ( x ), S ( y ), [ y, t– 1 z t ]  = 1   for all    y, z ∈ y,  t ∈ T,  ( t, y, z ) ≠ ( c, u, v )〉,where  y = y ( 1 )  for all  y ∈ y,  but  [ u, c– 1 v c ] = [ u ( 1 ), v ( c ) ] ≠ 1.The corollary is proved.Now for an inductive proof of Theorem 2 we only need the following simple result.Lemma 3.  If the sets of generators  x  of  G  and  y  of  H   are conormal, so isthe set of generators  x ∪ y  of  H  G.Proof.  Since  G  (  H  G  ) / ˆH ,  where  ˆH   is the normal subgroup generated byall  y ∈ y,  it is clear that neither  x ∈ x  belongs to the normal subgroup generated by( x \ { x } ) ∪ y.  On the other hand, there is an epimorphism  H  G  →  C  G ,  whereC = H / Ny  for some  y ∈ y;  in particular,  C  ≠ { 1 }  and is generated by the image  yof  y.  Since  C  is commutative, the map  C  G  →  C ,  (  f ( x ), g )   f xx G ( )∈∏   isalso an epimorphism mapping  y   to itself.  The resulting homomorphism  H  G  →  Cmaps all  x ∈ x  as well as all  z ∈ y \ { y }  to  1  and  y  to  y  ≠ 1,  which accomplishesthe proof.Example 1.  The wreath product  Cn  Cm  ,  where  Cn  denotes the cyclic group oforder  n,  has a minimal presentationCn  Cm  : =  〈  x, y | xm  = 1,  yn  = 1,  [ y, x– k y xk ]  = 1    for  1 ≤ k ≤ m / 2〉.(Possibly,  m = ∞  or  n = ∞,  then the relation  xm  = 1  or, respectively,  yn  = 1  shouldbe omitted.)1. Rotman J. J.  An introduction to the theory of groups. – New York: Springer, 1995. – XV + 513 p.2. Kaloujnine L. A.  Sur les  p-group de Sylow du groupe symétrique du degre  pm // C. r. Acad. sci.Paris. – 1945. – 221. – P. 222 – 224.Received 18.04.08ISSN  1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7

Generators and relations for wreath products

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Generators and relations for wreath products

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)