Generators and relations for wreath products
Generators and defining relations for wreath products of groups are given. Under a certain condition (conormality of generators), they are minimal.
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2008
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509267662995456 |
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| author | Drozd, Yu. A. Skuratovskii, R. V. Дрозд, Ю. А. Скуратовський, Р. В. |
| author_facet | Drozd, Yu. A. Skuratovskii, R. V. Дрозд, Ю. А. Скуратовський, Р. В. |
| author_sort | Drozd, Yu. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:48:23Z |
| description | Generators and defining relations for wreath products of groups are given. Under a certain condition (conormality of generators), they are minimal. |
| first_indexed | 2026-03-24T02:38:23Z |
| format | Article |
| fulltext |
UDC 512.543 + 512.542
Yu. A. Drozd (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv),
R. V. Skuratovskii (Kyiv. Nat. Taras Shevchenko Univ.)
GENERATORS AND RELATIONS FOR WREATH PRODUCTS
TVIRNI TA SPIVVIDNOÍENNQ DLQ VINCEVYX DOBUTKIV
Generators 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 with
finite support, i.e., such that f ( x ) = 1 for all but a finite set of elements of G. Recall
that their (restricted regular) wreath product W = H � G is defined as the semidirect
product 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 G
and H. Then we shall extend this result to the multiple wreath products
�k
n
kG=1 =
= ( … ( ( G1 � G2 ) � G3 ) … ) � Gn .
If x = { x1 , x2 , … , xn } are generators for G and R = { R1 , R2 , … , Rm } are
defining 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 generators
x1 , x2 , … , xn nor of the relations R1 , R2 , … , Rm can be excluded. We call the set of
generators x conormal if neither element x ∈ x belongs to the normal subgroup Nx
generated 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 factorgroup
G / G
p
[ G, G ] [1] (Theorem 5.48).
Theorem 1. Let G : = 〈 x | R ( x ) 〉, H : = 〈 y | S ( y ) 〉 be presentations of G and
H. 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 presentation
of the form
W : = 〈 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 y
is 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 = �i
n
iG=1 has a presentation of the
form
W : = 〈 xi , 1 ≤ i ≤ m | Ri ( xi ), 1 ≤ i ≤ m, [ x, t– 1
y t ] = 1
for 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, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7 997
998 YU. A. DROZD, R. V. SKURATOVSKII
are 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 by
h ( 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 presentation
N � 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 G
and 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 presentation
W : = 〈 x, y ( a ) | R ( x ), S ( y ( a ) ), [ y ( a ), z ( b ) ] = 1
x– 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 have
to replace y ( a ) and z ( b ) by a– 1
y ( 1 ) a and b– 1
z ( 1 ) b. So we shall write h instead
of 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 free
product 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 the
commutators [ 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 ′ . Consider
the homomorphism ϕ of H * H′ to P such that
ϕ ( y ) =
1 if ,
if ,
y
u y u
u∈
=
y
ϕ ( z′ ) =
1 if ,
if .
z
z
∈
′ =
yv
v v
Obviously, ϕ is well defined and ϕ ( [ y, z′ ] ) = 1 if ( y, z ) ≠ ( u, v ), so it induces a
homomorphism 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 presentation
Kc, u , v : = 〈 y ( a ), a ∈ G | S ( y ( a ) ), [ y ( a ), z ( t a ) ] = 1
for all y, z ∈ y, a ∈ G, t ∈ T, ( t, y, z ) ≠ ( c, u, v )〉.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
GENERATORS AND RELATIONS FOR WREATH PRODUCTS 999
Corollary 2. Let the set of generators y be conormal. Then [ u ( 1 ), v ( c ) ] ≠ 1
in the group Kc, u , v .
Proof. There is a homomorphism ψ : Kc, u , v → Hu, v , where Hu, v is the group
from Lemma 2, mapping u ( 1 ) � u, v ( c ) � v′, y ( a ) → 1 in all other cases. Then
ψ ( [ 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 set
of generators y is conormal, the presentation (1) is minimal.
Proof. Obviously, we can omit from (1) neither of generators x , y nor of the
relations 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 Corollary
2. The group G acts on K by the rule: h ( a )
g = h ( a g ). Let Q = K � G. Then, just
as in the proof of Corollary 1, this group has a presentation
Q : = 〈 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 is
the set of generators x ∪ y of H � G.
Proof. Since G � ( H � G ) / Ĥ , where Ĥ is the normal subgroup generated by
all 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 , where
C = H / Ny for some y ∈ y; in particular, C ≠ { 1 } and is generated by the image y
of y. Since C is commutative, the map C � G → C , ( f ( x ), g ) � f x
x G
( )∈∏ is
also an epimorphism mapping y to itself. The resulting homomorphism H � G → C
maps all x ∈ x as well as all z ∈ y \ { y } to 1 and y to y ≠ 1, which accomplishes
the proof.
Example 1. The wreath product Cn � Cm , where Cn denotes the cyclic group of
order n, has a minimal presentation
Cn � 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 should
be 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 18.04.08
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 7
|
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| language | English |
| last_indexed | 2026-03-24T02:38:23Z |
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| spelling | umjimathkievua-article-32162020-03-18T19:48:23Z Generators and relations for wreath products Твірні тa співвідношення для вінцевих добутків Drozd, Yu. A. Skuratovskii, R. V. Дрозд, Ю. А. Скуратовський, Р. В. 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. Institute of Mathematics, NAS of Ukraine 2008-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3216 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 7 (2008); 997–999 Український математичний журнал; Том 60 № 7 (2008); 997–999 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3216/3178 https://umj.imath.kiev.ua/index.php/umj/article/view/3216/3179 Copyright (c) 2008 Drozd Yu. A.; Skuratovskii R. V. |
| spellingShingle | Drozd, Yu. A. Skuratovskii, R. V. Дрозд, Ю. А. Скуратовський, Р. В. Generators and relations for wreath products |
| title | Generators and relations for wreath products |
| title_alt | Твірні тa співвідношення для вінцевих добутків |
| title_full | Generators and relations for wreath products |
| title_fullStr | Generators and relations for wreath products |
| title_full_unstemmed | Generators and relations for wreath products |
| title_short | Generators and relations for wreath products |
| title_sort | generators and relations for wreath products |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3216 |
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