Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups
We define a wreath product of a Lie algebra L with the one-dimensional Lie algebra L1 over Fp and determine some properties of this wreath product. We prove that the Lie algebra associated with the Sylow p-subgroup of finite symmetric group Spm is isomorphic to the wreath product of m copies of...
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irk-123456789-1566112019-06-19T01:28:51Z Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups Sushchansky, V.I. Netreba, N.V. We define a wreath product of a Lie algebra L with the one-dimensional Lie algebra L1 over Fp and determine some properties of this wreath product. We prove that the Lie algebra associated with the Sylow p-subgroup of finite symmetric group Spm is isomorphic to the wreath product of m copies of L1. As a corollary we describe the Lie algebra associated with Sylow p-subgroup of any symmetric group in terms of wreath product of one-dimensional Lie algebras. 2005 Article Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups / V.I. Sushchansky, N.V. Netreba // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 122–132. — Бібліогр.: 11 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 17B30, 17B60, 20F18, 20F40. http://dspace.nbuv.gov.ua/handle/123456789/156611 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We define a wreath product of a Lie algebra L
with the one-dimensional Lie algebra L1 over Fp and determine
some properties of this wreath product. We prove that the Lie
algebra associated with the Sylow p-subgroup of finite symmetric
group Spm is isomorphic to the wreath product of m copies of L1.
As a corollary we describe the Lie algebra associated with Sylow
p-subgroup of any symmetric group in terms of wreath product of
one-dimensional Lie algebras. |
| format |
Article |
| author |
Sushchansky, V.I. Netreba, N.V. |
| spellingShingle |
Sushchansky, V.I. Netreba, N.V. Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups Algebra and Discrete Mathematics |
| author_facet |
Sushchansky, V.I. Netreba, N.V. |
| author_sort |
Sushchansky, V.I. |
| title |
Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups |
| title_short |
Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups |
| title_full |
Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups |
| title_fullStr |
Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups |
| title_full_unstemmed |
Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups |
| title_sort |
wreath product of lie algebras and lie algebras associated with sylow p-subgroups of finite symmetric groups |
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Інститут прикладної математики і механіки НАН України |
| publishDate |
2005 |
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http://dspace.nbuv.gov.ua/handle/123456789/156611 |
| citation_txt |
Wreath product of Lie algebras and Lie algebras associated with Sylow p-subgroups of finite symmetric groups / V.I. Sushchansky, N.V. Netreba // Algebra and Discrete Mathematics. — 2005. — Vol. 4, № 1. — С. 122–132. — Бібліогр.: 11 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT sushchanskyvi wreathproductofliealgebrasandliealgebrasassociatedwithsylowpsubgroupsoffinitesymmetricgroups AT netrebanv wreathproductofliealgebrasandliealgebrasassociatedwithsylowpsubgroupsoffinitesymmetricgroups |
| first_indexed |
2025-07-14T09:00:16Z |
| last_indexed |
2025-07-14T09:00:16Z |
| _version_ |
1837612260773593088 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 1. (2005). pp. 122 – 132
c© Journal “Algebra and Discrete Mathematics”
Wreath product of Lie algebras and Lie algebras
associated with Sylow p-subgroups of finite
symmetric groups
Vitaly I. Sushchansky, Nataliya V. Netreba
Dedicated to Yu.A. Drozd on the occasion of his 60th birthday
Abstract. We define a wreath product of a Lie algebra L
with the one-dimensional Lie algebra L1 over Fp and determine
some properties of this wreath product. We prove that the Lie
algebra associated with the Sylow p-subgroup of finite symmetric
group Spm is isomorphic to the wreath product of m copies of L1.
As a corollary we describe the Lie algebra associated with Sylow
p-subgroup of any symmetric group in terms of wreath product of
one-dimensional Lie algebras.
1. Introduction
Lie rings associated to a group are already the classical objects of modern
algebra. One can find their usefulness in a variety of applications, includ-
ing the restricted Burnside problem, the study of some group identities,
the theory of fixed point of automorphism, the coclass theory for p-groups
and pro-p groups, the investigation of just-infinite pro-p groups, and the
recent study of Hausdorff dimension and the spectrum of pro-p groups.
Lie ring methods provide a recipe for translating some group-theoretic
questions to Lie-theoretic ones.
A classical operation in group theory is the wreath product of groups.
The wreath product of Lie algebras was defined by A. L. Shmelkin [6]
2000 Mathematics Subject Classification: 17B30, 17B60, 20F18, 20F40.
Key words and phrases: Lie algebra, wreath product, semidirect product, Lie
algebra associated with the lower central series of the group, Sylow p-subgroup, sym-
metric group.
V. I. Sushchansky, N. V. Netreba 123
already in 1973. In spite of this the notion is almost non-investigated by
now.
We define another notion of a wreath product of a Lie algebra with
the one-dimensional Lie algebra over the finite field Fp. Our idea of the
construction comes from the study of some class of Lie algebras associated
with p-groups, namely the Sylow p-subgroups of finite symmetric groups.
The Sylow p-subgroup Pm of the symmetric group Spm is isomorphic to
a wreath product of cyclic groups of order p [7]. The structure of the
Lie algebra associated with Pm was investigated in [9]. Our definition
of wreath product allows us to prove the main result of the article: Lie
algebra associated with the Sylow p-subgroup of finite symmetric group
Spm is isomorphic to the wreath product of one-dimensional Lie algebra,
i.e.
L(Cp ≀ . . . ≀ Cp) = L(Cp) ≀ . . . ≀ L(Cp).
Using this theorem we describe the Lie algebra associated with the Sylow
p-subgroup of any finite symmetric group Sn in terms of wreath product
of one-dimensional Lie algebra. Also we investigate some basic properties
of our definition of wreath product.
2. The wreath product of Lie algebras and its properties
Recall the definition of the semidirect product of Lie algebras (see [1]).
Let M and N be Lie algebras over K and a 7→ ϕa be a homomorphism
from M to the Lie algebra of differentiations of the algebra N . Define a
Lie bracket on the direct sum L of K-modules M and N by the equality:
([a, b], [a′, b′]) = [(a, a′), (b, b′) + ϕa(b
′) − ϕa′(b)],
where a, a′ ∈M and b, b′ ∈ N .
Definition 1. Lie algebra L is called the semidirect product of algebraM
and algebra N which corresponds to the homomorphism ϕ : M → D(N),
and we denote it as L = M ⋌ϕ N .
Let L be a Lie Algebra over the field Fp and L1 be the one-dimensional
Lie algebra over Fp.
Let L[x]/〈xp〉 be the Lie algebra of polynomials over L of degree at
most p − 1. The Lie bracket of the monomials in this algebra is defined
in the following way:
(lxn, l′xm) =
{
(l, l′)xn+m, if n+m < p;
0, if n+m ≥ p.
(1)
By linearity the Lie bracket is determined for all polynomials.
124 Wreath product of Lie algebras...
The following proposition determines the one-to-one correspondence
between the set L[x]/〈xp〉 and the set of all maps from L1 to L.
Proposition 1. Every map f : L1 → L corresponds to the unique poly-
nomial q(x) over L of degree at most p − 1 such that f(α) = q(ε(α)),
where ε : L1 → Fp is the some isomorphism of vector spaces.
Proof. Let f(α0), . . ., f(αp−1) be the images of the elements of Lie algebra
L1 under the map f : L1 → L. Consider the linear system of equalities
with respect to l0, . . . , lp−1 ∈ L:
lp−1ε(α0)
p−1 + . . .+ l1ε(α0) + l0 = f(α0)
lp−1ε(α1)
p−1 + . . .+ l1ε(α1) + l0 = f(α1)
...
...
lp−1ε(αp−1)
p−1 + . . .+ l1ε(αp−1) + l0 = f(αp−1),
where {ε(αi)} are all elements of the field Fp.
Or we may write down it as:
ε(α0)
p−1 . . . ε(α0) 1
ε(α1)
p−1 . . . ε(α1) 1
...
...
...
...
ε(αp−1)
p−1 . . . ε(αp−1) 1
lp−1
...
l0
=
f(α0)
...
f(αp−1)
Determinant of the matrix det(A) is the Vandermond determinant
and thus is nonzero. Hence, there is only one set of elements lp−1, . . . , l0
for arbitrary f(α0), . . . f(αp−1). That is
(lp−1, . . . , l0)
T = A−1(f(α0), . . . f(αp−1))
T .
Thus, to every map f : L1 → L corresponds the unique polynomial
q(x) = lp−1x
p−1 + . . . + l1x + l0 over L and by construction f(α) =
q(ε(α)).
Therefore exists the bijection between the set of all maps f : L1 → L
and the set of all polynomials over L of degree at most p − 1. The
structure of Lie algebra L[x]/〈xp〉 defines the structure of Lie algebra
on the set of all maps f : L1 → L. We will denote this Lie algebra as
Fun(L1, L) ≃ L[x]/〈xp〉.
The identification ε between L1 and Fp gives us the structure of L1-
module on the algebra Fun(L1, L). Thus, we also consider the Lie algebra
Fun(L1, L) as L1–module.
V. I. Sushchansky, N. V. Netreba 125
Further we will not distinguish the notations of the elements of one-
dimension Lie algebra L1 and the field Fp. From the context it is clear
from which structures the elements are considered.
Let f ∈ Fun(L1, L). Denote by f ′ ∈ Fun(L1, L) the derivative of the
polynomial f .
Proposition 2. For every α ∈ L1 the mapDα : Fun(L1, L) → Fun(L1, L)
which is defined by the rule Dα(f) = αf ′ is the differentiation.
Proof. The linearity of the mapDα follows from the linearity of derivative
of the polynomials. So the fact that Dα is differentiation is enough to
verify for monomials.
Dα(lxn, l′xm) =
{
α(n+m)(l, l′)xn+m−1, if n+m < p;
0, if n+m ≥ p.
(Dα(lxn), l′xm) + (lxn, D(l′xm)) = αn(lxn−1, l′xm) +
+ mα(lxn, l′xm−1) =
{
α(n+m)(l, l′)xn+m−1, if n+m− 1 < p;
0, if n+m− 1 ≥ p.
Notice that if the degree n + m = p, then by definition (1) of the Lie
bracket in Lie algebra Fun(L1, L) holds n + m = 0. Thus the upper
equality coincides with the lower one and Dα is a differentiation.
Therefore we can define a map ϕ from Lie algebra L1 to the alge-
bra of differentiations D(Fun(L1, L)) given by the rule α 7→ Dα, where
Dα(f) = αf ′. The map ϕ is a homomorphism. Really, ϕ((α, β)) = 0 and
DαDβ(f) −DβDα(f) = αβf ′′ − βαf ′′ = 0.
Definition 2. The semidirect product of Lie algebra L1 with Lie alge-
bra Fun(L1, L), which corresponds to the homomorphism ϕ, we call the
wreath product of Lie algebra L with L1 and denote by L ≀ L1.
Thus, L ≀ L1 := L1 ⋌ϕ Fun(L1, L) = {[a, f ]| a ∈ L1, f ∈ Fun(L1, L)}
with Lie bracket
([a1, f1], [a2, f2]) = [0, a1
∂f2
∂x
− a2
∂f1
∂x
+ (f1, f2)]. (2)
Remark 1. Definition 2 allows us to consider the wreath product
L ≀ L1 ≀ . . . ≀ L1 for an arbitrary Lie algebra L.
The subset of elements [a, e] of L ≀ L1 forms the subalgebra P , which
is isomorphic to L1. The subset H of elements [0, f ] is a subalgebra of
L ≀ L1 which is isomorphic to Fun(L1, L).
126 Wreath product of Lie algebras...
Proposition 3. Let L be a solvable Lie algebra of the derived length n.
Then L ≀ L1 is solvable of the derived length n+ 1.
Proof. By the definition of the Lie bracket in Lie algebra Fun(L1, L)
the coefficients of a polynomial (f, g), f, g ∈ Fun(L1, L), belong to the
algebra L(1) = (L,L). Thus the inclusion
(
Fun(L1, L), Fun(L1, L)
)
⊆
Fun(L1, L
(1)) holds.
The following inclusion (L ≀ L1)
(1) ⊂ [0, Fun(L1, L)] is also correct.
Thus we have
(
[0, Fun(L1, L)], [0, Fun(L1, L)]
)
= [0,
(
Fun(L1, L), Fun(L1, L)
)
] ⊆
⊆ [0, Fun(L1, L
(1))].
Thus, (L ≀ L1)
(2) ⊆ [0, Fun(L1, L
(1))]. If we continue this process we
obtain that
(L ≀ L1)
(n+1) ⊆ [0, Fun(L1, L
(n))].
Thus, if L is solvable of derived length n then L ≀L1 is solvable of derived
length at most n+ 1.
Notice that [0, L] is contained in (L≀L1)
(1), where we consider elements
of L as constant polynomials. Thus
[0, L(n−1)] ⊆ (L ≀ L1)
(n).
From this follows that L ≀ L1 is solvable of derived length at least n+ 1.
Thus L ≀ L1 is solvable of derived length n.
Proposition 4. Let L be a nilpotent Lie algebra of nilpotent class n.
Then L ≀ L1 is nilpotent of nilpotent class np.
Proof. Consider the lower central series of the Lie algebra L ≀ L1. Let
γ0 = L ≀ L1, γk = (γk−1, L ≀ L1) be the k-th term of the lower central
series.
Denote Fk = {f | [0, f ] ∈ γk} ⊂ Fun(L1, L). Then γk = [0, Fk]. From
formula (2) follows that every polynomial f ∈ Fk has monomials of degree
≤ p− 1 − k with coefficients from L and f has also monomials of degree
≤ p− 1 with coefficients from γ1(L).
Hence, Fp ⊂ Fun(L1, γ1(L)). Notice, that polynomials of Fp have
monomials of degree ≤ p − 1 with coefficients from γ1(L), γ2(L), . . . ,
γp(L).
In a similar we obtain Fp+p ⊂ Fun(L1, γ2(L)) and so on. Thus,
γp·n = [0, Fp·n] ⊂ [0, Fun(L1, γn(L))] = [0, 0].
V. I. Sushchansky, N. V. Netreba 127
Thus, if L is nilpotent of nilpotent class n then L ≀ L1 is nilpotent of
nilpotent class at most np.
Notice that [0, L] ⊆ γk(L ≀ L1), 1 ≤ k ≤ p − 1. In a similar way we
obtain [0, γ1(L)] ⊆ γl(L ≀ L1), p ≤ l ≤ 2p− 1.
Thus, [0, γ(n−1)(L)] ⊆ γs(L ≀L1), (n−1)p ≤ s ≤ np−1. Consequently,
Lie algebra L ≀ L1 is nilpotent of nilpotent class at least np.
Thus L ≀ L1 is nilpotent of nilpotent class np.
3. Lie algebras associated with the Sylow p-subgroups of
symmetric groups
We will consider the notion of "tableau" introduced by L. Kaloujnine in
[4]. On the set of all tableaux of the length m over Fp we introduce the
structure of Lie algebra in the following way. Define the addition, Lie
bracket ( , ) and the multiplication on the elements of Fp for tableaux
u = [u1, u2(x1), u3(x1, x2), . . .], v = [v1, v2(x1), v3(x1, x2), . . .]
by the following equalities (1 ≤ k ≤ m):
(i) {u+ v}k = uk + vk;
(ii) {(u, v)}k =
k−1∑
i=1
(
∂vk
∂xi
· ui − vi ·
∂uk
∂xi
);
(iii) {α · u}k = α · uk, α ∈ Fp.
where u1 = a1 ∈ Fp,
uk = ak(x1, x2, . . . , xk−1) = ak(xk−1) ∈ Fp[x1 . . . , xk−1]/Ik−1,
where Ik−1 is an ideal, generated by polynomials xp
1, x
p
2, . . . , x
p
k−1.
According to [9] the set of all tableaux over Fp with operations (i) −
(iii) forms the Lie algebra denoted by Lm.
Denote by L(Pm) the Lie algebra associated with the lower central
series of the Sylow p-subgroup Pm of the symmetric group Spm . The
structure of Lie algebra L(Pm) was investigated in [9]. In particular, the
following theorem was proved:
Theorem 5. Lie algebra L(Pm) is isomorphic to the algebra Lm.
The following theorem holds:
Theorem 6. Lm ≃ L1 ≀ L1 ≀ . . . ≀ L1.
128 Wreath product of Lie algebras...
Proof. Note that since Pm ≃ Cp ≀Cp ≀. . .≀Cp, and Lie algebra Lm ≃ L(Pm),
then we can replace the assertion of the theorem by L(Cp ≀Cp ≀ . . . ≀Cp) ≃
L1 ≀ L1 ≀ . . . ≀ L1.
We will prove the theorem by induction on the number of the com-
ponents of the wreath product. Define
Pn = Cp ≀ . . . ≀ Cp
︸ ︷︷ ︸
n
and Ln = L1 ≀ . . . ≀ L1
︸ ︷︷ ︸
n
.
If n = 1 then L(Cp) ≃ L1 and the assertion is correct. Assume
that the assertion is true for n, that is L(Pn) ≃ Ln. We will show that
Ln ≀ L1 ≃ L(Pn ≀ Cp).
Every function f : L1 → Ln can be uniquely represented by the
tableau
[a1(x1), a2(x1, x2), . . . , an(x1, . . . , xn)], (3)
where ak(x1, . . . , xk) ∈ Fp[x1, . . . , xk]/Ik. Really, f(x1) = lp−1x
p−1
1 +
. . .+ l0, where li ∈ Ln and according to the assumption of induction and
theorem 5 li = [bi0, b
i
1(x2), . . . , b
i
n−1(x2, . . . , xn)]. Then f(x) is uniquely
represented in the form [a1(x1), a2(x1, x2), . . . , an(x1, . . . , xn)], where
ai+1(x1, . . . , xi+1) = bp−1
i (x2, . . . , xi+1)x
p−1
1 + . . .+ b0i (x2, . . . , xi+1),
i = 0, . . . , n− 1.
Then f ′ is represented in the form
f ′ = (p− 1)lp−1x
p−2
1 + . . .+ l1 =
= (p− 1)[bp−1
0 , . . . , bp−1
n−1(x2, . . . , xn)]xp−2
1 + . . .
. . .+ [b10, . . . , b
1
n−1(x2, . . . , xn)] =
= [(p− 1)bp−1
0 xp−2
1 + . . .+ b10, . . . , (p− 1)bp−1
n−1x
p−2
1 + . . .+ b1n−1] =
= [
∂
∂x1
a1(x1),
∂
∂x1
a2(x1, x2), . . . ,
∂
∂x1
an(x1 . . . , xn)]. (4)
Moreover, for every functions f = [a1(x1), a2(x1, x2), . . . , an(xn)],
g = [b1(x1), b2(x1, x2), . . . , bn(xn)] the function (f, g) is of the form
[0, c2(x1, x2), . . . , cn(x1, . . . , xn)], where
ck(x1, . . . , xk) =
k−1∑
i=1
(ai
∂
∂xi+1
bk − bi
∂
∂xi+1
ak). (5)
Indeed, from the linearity of representation (3) follows that it is enough
to verify (5) only for monomials. Let f = lxm
1 and g = hxk
1, where l =
V. I. Sushchansky, N. V. Netreba 129
[l1, l2(x2), . . . , ln(x2, . . . , xn)], h = [h1, h2(x2), . . . , hn(x2, . . . , xn)] ∈ Ln.
Then
f = [l0x
m
1 , l1(x2)x
m
1 , . . . , ln(x2, . . . , xn)xm
1 ],
g = [h0x
k
1, h1(x2)x
k
1, . . . , hn(x2, . . . , xn)xk
1]
Then the coefficients from (5) look like:
cj(x1, . . . , xj) =
j−1
∑
i=1
(lix
m
1
∂
∂xi+1
hjx
k
1 − hix
k
1
∂
∂xi+1
ljx
m
1 ) =
=
{ ∑j−1
i=1 (li
∂
∂xi+1
hj − hi
∂
∂xi+1
lj)x
m+k
1 , if m+ k < p.;
0, if m+ k ≥ p.
Let us write down how (f, g) is represented by the tableau (3):
(f, g) =
{
(l, h)xm+k
1 , if m+ k < p;
0, if m+ k ≥ p.
=
=
{
[0, d2(x2), . . . , dn(x2, . . . , xn)]xm+k
1 , if m+ k < p;
0, if m+ k ≥ p.
=
=
{
[0, d2(x2)x
m+k
1 , . . . , dn(x2, . . . , xn)xm+k
1 ], if m+ k < p;
0, if m+ k ≥ p,
where dj(x2, . . . , xj) =
j−1
∑
i=1
(li
∂
∂xi+1
hj − hi
∂
∂xi+1
lj).
Thus the function (f, g) is of the form [0, c2(x1, x2), . . . , cn(x1, . . . , xn)].
Let us construct the map ψ : Ln ≀ L1 → L(Pn ≀ Cp) by the rule
ψ([a0, f ]) = [a0, a1(x1), . . . , an(x1, . . . , xn)]. According to proposition 1
and theorem 5 the map ψ is a bijection. Let us show that ψ is linear.
Really:
ψ(α[a0, f ] + β[b0, g]) = ψ([αa0 + βb0, αf + βg]) =
= [αa0 + βb0, αa1(x1) + βb1(x1), . . . , αan(xn) + βbn(xn)] =
= α[a0, . . . , an(xn)] + β[b0, . . . , bn(xn)] = αψ([a0, f ]) + βψ([b0, g]).
It remains to prove that ψ(([a0, f ], [b0, g])) = (ψ([a0, f ]), ψ([b0, g])).
From (4) and (5) follows:
ψ(([a0, f ], [b0, g])) = ψ([0, a0g
′ − b0f
′ + (f, g)]) =
= [0, d1(x1), . . . , dn(x1, . . . , xn)], where
130 Wreath product of Lie algebras...
dk = a0
∂
∂x1
bk − b0
∂
∂x1
ak +
k−1∑
i=1
(ai
∂
∂xi+1
bk − bi
∂
∂xi+1
ak) =
=
k−1∑
i=0
(ai
∂
∂xi+1
bk − bi
∂
∂xi+1
ak).
Thus,
ψ(([a0, f ], [b0, g])) = ([a0, a1(x1), . . . , an(xn)], [b0, b1(x1), . . . , bn(xn)]) =
= (ψ([a0, f ]), ψ([b0, g])).
Let Sn be the group of all permutations of the set of n elements, where
n = a0 + a1p+ a2p
2 + . . .+ akp
k.
We describe the Lie algebra L(Sylp(Sn)) associated with the Sylow p-
subgroup of any symmetric group Sn in terms of wreath product of one-
dimensional Lie algebras. It is well known (see [7]), that the Sylow p-
subgroup of the symmetric group Sn is isomorphic to
Sylp(Sn) ≃
k⊕
l=0
Sylp(Spl) × . . .× Sylp(Spl)
︸ ︷︷ ︸
al
(6)
Proposition 7. Let G = H ×K and γi(H), γi(K) be the i-th terms of
the lower central series of the groups H and K correspondingly. Then
γi(G) = γi(H) × γi(K).
Proof. We will prove this assertion by induction. If n = 0 we have
γ0(H) = H, γ0(K) = K and γ0(G) = G = H × K = γ0(H) × γ0(K).
Assume that the assertion is true for i, that is γi(G) = γi(H) × γi(K).
Then
γi+1(G) = [γi(G), G] = [γi(H) × γi(K), H ×K] =
= [γi(H), H] × [γi(K),K] = γi+1(H) × γi+1(K).
Hence, we obtain γi(G) = γi(H) × γi(K) by induction on i, as required.
Corollary 8. L(G) = L(H) ⊕ L(K).
V. I. Sushchansky, N. V. Netreba 131
Proof. Recall, that Lie algebra associated with the lower central series of
the group G (see [10])) is L(G) =
∞⊕
i=1
γi(G)/γi+1(G), where γi(G) is i-th
term of the lower central series of group G. Thus, we have
L(G) = L(H ×K) = ⊕i≥0γi(G)/γi+1(G) =
= ⊕i≥0(γi(H) × γi(K))/(γi+1(H) × γi+1(K)) =
= ⊕i≥0γi(H)/γi+1(H) ⊕i≥0 γi(K)/γi+1(K) = L(H) ⊕ L(K)
Theorem 9. Lie algebra associated with the Sylow p-subgroup of the
group Sn is isomorphic to
L(Sylp(Sn)) ≃ ⊕k
r=0 L(Sylp(Spr)) ⊕ . . .⊕ L(Sylp(Spr))
︸ ︷︷ ︸
ar
Proof. The assertion of the theorem directly follows from (6) and corol-
lary (8).
Remark 2. According to the theorem 5 we can write down the assertion
of the theorem in the form
L(Sylp(Sn)) ≃ ⊕k
r=0
(
⊕ar
i=1 ≀
r
j=1 L1
)
References
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Scientific Press, Utrecht, 1987.
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132 Wreath product of Lie algebras...
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IsZp, // IV All-Union school of Lie algebra and their applications in mathematics
and physics, Abstract, Kazan, 1990., p.44. (in Russia)
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Contact information
V. Sushchansky Silesian University of Technology, Gliwice,
Poland and Kyiv Taras Shevchenko Univer-
sity, Kyiv, Ukraine
E-Mail: Wital.Sushchanski@polsl.pl,
wsusz@univ.kiev.ua
N. Netreba Kyiv Taras Shevchenko University, Ukraine
E-Mail: netr@univ.kiev.ua
Received by the editors: 27.03.2005
and in final form 05.04.2005.
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