On $n^{th}$ class preserving automorphisms of $n$-isoclinism family

0.13 MB·

by  Surjeet Kour

ON nth CLASS PRESERVING AUTOMORPHISMS OF
n-ISOCLINISM FAMILY
7
1
SURJEET KOUR
0
2
n Abstract. LetGbeafinitegroupandM,N betwonormalsubgroups
a of G. Let AutMN(G) denote the group of all automorphisms of G which
J fix N element wise and act trivially on G/M. Let n be a positive

9 integer. In this article we have shown that if G and H are two n-
1 isoclinic groups, then there exists an isomorphism from Autγn+1(G)(G)
Zn(G)
to Autγn+1(H)(H), which maps the group of nth class preserving auto-
] Zn(H)
R morphisms of G to the group of nth class preserving automorphisms of
G H. Also, for a nilpotent group of class at most (n+1),with some suit-
h. ableconditionsonγn+1(G),weprovethatAutZγnn+(G1()G)(G)isisomorphic
to the group of innerautomorphisms of a quotient group of G.
t
a
m
[

1
v 1. Introduction
8
3 Oneof themostfundamentalproblemingrouptheory is theclassification
4
of groups and the notion of isomorphism between two groups has played a
5
0 very important role in it. However, the notion of isomorphism which parti-
. tions the class of all groups into equivalence classes(isomorphism classes)is
1
0 very strong. If one would like to have all abelian groups or all finite p-
7 groups fall into one equivalence class then the notion of isomophism will not
1
work. With these thoughts in mind, Hall in 1940 defined a more general
:
v equivalence relation in the class of all groups which could give a satisfactory
i
X classification of all finite p-groups and all abelian groups. In 1940 in [2],
r Hall introduced the notion of isoclinism, which is an equivalence relation
a
in the class of all groups. The notion of isoclinism is weaker than isomor-
phism and all abelian groups form one equivalence class. Roughly speaking,
two groups are isoclinic if their central quotients are isomorphic and their
commutator subgroups are also isomorphic. More precisely, if G and H
are two groups, we say, G is isoclinic to H if there exists an isomorphism
α : G/Z(G) → H/Z(H) and an isomorphism β : γ (G) → γ (H) such that
2 2
the following diagram commutes.
2010 Mathematics Subject Classification. 20D15, 20D45.
Key words and phrases. Finite group, p- group, Class preserving automorphism, Cen-
tral automorphism.
1
2 SURJEETKOUR
G × G ) −α−×−→α H × H
Z(G) Z(G) Z(H) Z(H)
↓ γ ↓ γ
G H
β
γ (G) −→ γ (H),
2 2
where Z(G) and γ (G) denote the center of G and the commutator sub-
2
group of G respectively. The maps γ and γ are given by
G H
γ (g Z(G),g Z(G)) = [g ,g ]
G 1 2 1 2
for all g ,g ∈ G and
1 2
γ (h Z(H),h Z(H)) = [h ,h ]
H 1 2 1 2
for all h ,h ∈ H.
1 2
Notion of isoclinism has played a very important role in the classification
offinitep-groups. Isoclinismdividesp-groupsintovariousisoclinismfamilies
and it has been studied by many mathematicians (see [2, 3]). Many authors
investigated various group theoretical properties which are invariant under
the isoclinism. Later, Hall in [3], generalized the notion of isoclinism to that
of isologism, which is in fact an isoclinism with respect to certain varieties
of groups.
Heksterin1986in[4],conceptualizedandstudiedthenotionofn-isoclinism
over the variety of all nilpotent groups of class at most n. In this article, he
has also done an extensive study of group theoretical properties which are
invariant under n-isoclinism. He has shown that most of the known results
for isoclinism can carry over to n-isoclinism. In recent years, people also
started study of various subgroups of the automorphism group, which are
invariant(isomorphic) for isoclinic groups(see [7, 6]).
Let M and N be two normal subgroups of G. Let AutM(G) denote the
group of all automorphisms of G which fix M set wise and act trivially on
G/M andletAut (G)denotethegroupofallautomorphismsofGwhichfix
N
N element wise. The group AutM(G)∩Aut (G) is denoted by AutM(G).
N N
Let n be a positive integer. An automorphism θ of G is called nth class
preserving if for each g ∈ G, there exists x ∈ γ (G) such that θ(g)= x−1gx,
n
where γ (G) denotes the nth term of the lower central series. We denote
n
the group of all nth class preserving automorphisms by Autn(G). Note that
c
Aut1(G) is denoted by Aut (G) and called the group of all class preserving
c c
automorphisms.
In [7], Yadav proved that if G and H are two finite non abelian isoclinic
groups, then Aut (G) is isomorphic to Aut (H). In [6], Rai extended Ya-
c c
dav’s result to the group Autγ2(G)(G). He proved that, if G and H are
Z(G)
two finite non abelian isoclinic groups then there exists an isomorphism
φ : Autγ2(G)(G) → Autγ2(H)(H) such that φ(Aut (G)) = Aut (H). In
Z(G) Z(H) c c
this article we study these subgroups of the automorphism group for an
n-isoclinism family. More precisely, in Theorem 3.3, we prove that if G
ON nth CLASS PRESERVING AUTOMORPHISMS OF n-ISOCLINISM FAMILY 3
and H are two finite n-isoclinic groups then there exists an isomorphism
Ψ : Autγn+1(G)(G) → Autγn+1(H)(H) such that Ψ(Autn(G)) = Autn(H).
Zn(G) Zn(H) c c
Rai [6, Theorem A ] , the extension of Yadav [7, Theorem 4.1], is obtained
as Corollary 3.4 of Theorem 3.3.
In the same article [6], Rai also proved that if G is a finite p-group of
nilpotency class two, then Autγ2(G)(G) = Inn(G) if and only if γ (G) is
Z(G) 2
cyclic. In section 3, we generalize this result in the following two ways.
(1) We consider an arbitrary nilpotent group(not just a finite p-group).
(2) WestudyAutM (G)foracentralsubgroupM ofGwhichcontains
Zn(G)
γ (G).
n+1
In Theorem 3.5, we prove that if G is a finite non abelian group with a
central subgroup M such that γ (G) ⊆ M, then
n+1
AutMZn(G)(G) ≃ Inn(G/Zn−1(G))
if and only if Mpi is cyclic for each pi ∈ π(G/Zn−1(G)), where Mpi de-
notes the p -primary component of M. We obtain Rai [6, Theorem B(2)] as
i
Corollary 3.7 of Theorem 3.5.
2. Notations and Preliminaries
In this section, we recall a few definitions and some known results. Let
G be a finite group. We denote the identity of a group by 1. Throughout
the article n denotes an integer and p denotes a prime number. The lower
central series of a group G is the series; γ (G) ≥ γ (G) ≥ ··· defined as
1 2
γ (G) = G and γ (G) = [γ (G),G] for all n ≥ 1. Note that each γ (G) is
1 n+1 n i
acharacteristic subgroupofGandγ (G) is called thecommutator subgroup
2
of G . G is a nilpotent group of class n if and only if γ (G) = {1} and
n+1
γ (G) 6= {1}.
n
The upper central series of G is a sequence of normal subgroups {1} =
Z0(G) ≤ Z1(G) ≤ Z2(G) ≤ ··· such that Zi(G)/Zi−1(G) = Z(G/Zi−1(G)).
From the definition of upper central series it follows that g ∈ Z (G) if
n
and only if [g,g ,...,g ] = 1, where g ∈ G for 1 ≤ i ≤ n. Hence
1 n i
[γ (G),Z (G)] = {1}. With this observation, it is easy to see that there
n n
exists a natural well defined map
G G
γ(n,G) : ×···× → γ (G)
n+1
Z (G) Z (G)
n n
given by
γ(n,G)(g Z (G),...,g Z (G)) = [g ,g ,...,g ].
1 n n+1 n 1 2 n+1
Now we recall the notions of n-homoclinism and n-isoclinism which are
introduced by Hall [3] and extensively investigated by Hekster [4].
Let G and H be two finite groups. A pair (α,β) is called n-homoclinism
between G and H, if α : G/Z (G) → H/Z (H) is a group homomorphism
n n
4 SURJEETKOUR
and β : γ (G) → γ (H) is a group homomorphism such that the fol-
n+1 n+1
lowing diagram commutes
G ×···× G ) −α−n−+→1 H ×···× H
Zn(G) Zn(G) Zn(H) Zn(H)
↓γ(n,G) ↓ γ(n,H)
β
γ (G) −→ γ (H).
n+1 n+1
Observe that the notion of n-homoclinism generalizes the notion of ho-
momorphism. In fact 0-homoclinisms are nothing but homomorphisms. If
α is surjective then β is surjective and if β is injective then α is injective.
We say G and H are n-isoclinic if α and β are isomorphisms. In this case,
the pair (α,β) is called n-isoclinism between G and H.
Note that n-isoclinism is an equivalence relation on the class of all groups
and an equivalence class is called an n-isoclinism family.
Now we recall some well known results on n-isoclinism(n-homoclinism).
Lemma 2.1. [4, Lemma 3.8] Let (α,β) be an n-homoclinism from G to H
and let x ∈ γ (G). Then the following statements hold:
n+1
(1) α(xZ (G)) = β(x)Z (H).
n n
(2) For g ∈ G and h ∈ α(gZ (G)), β(g−1xg) = h−1β(x)h.
n
We also recall the following theorem of Hekster.
Theorem 2.2. [4, Theorem 3.12] Let (α,β) be an n-isoclinism from G to
H. Then for all i≥ 0, β(γ (G)∩Z (G)) = γ (H)∩Z (H).
n+1 i n+1 i
The following lemma is an easy observation.
Lemma 2.3. Let G be a group and n be a positive integer. Then for all
n≥ 1, Autn(G) ⊆ Autγn+1(G)(G).
c Zn(G)
Proof. Let f ∈ Autn(G). Then for each g ∈ G there exists x ∈ γ (G) such
c n
that f(g) = x−1gx. Hence g−1f(g) = [g,x] ∈ γ (G). Furthermore, if
n+1
g ∈ Z (G) then g−1f(g) = [g,x] = 1. Thus f(g) = g for all g ∈ Z (G). (cid:3)
n n
The exponent of a group G is the smallest positive integer n such that
gn = 1 for all g ∈ G. We denote exponent of a group G by exp(G). For a
finite group G, π(G) denotes the set of all prime divisors of order of G. If
G is finite abelian, then G = G , where G denotes the p-primary
Qp∈π(G) p p
component of G.
The following lemmas are well known.
Lemma2.4. LetGbeanilpotentgroupofclass(n+1). Thenexp(G/Z (G)) =
n
exp(γ (G)).
n+1
ON nth CLASS PRESERVING AUTOMORPHISMS OF n-ISOCLINISM FAMILY 5
Lemma 2.5. Let G and H be two finite groups such that exp(H) |exp(G).
Then π(H) ⊆ π(G).
A subgroup H of G is called central if H ⊆ Z(G). For an abelian group
H, Hom(G,H) denotes the abelian group of all homomorphisms from G to
H. Now we recall the following theorem from [5].
Theorem 2.6. [5, Theorem 3.3] Let G be a group and let M and N be two
normal subgroups of G. Suppose, M is a central subgroup of G. Then the
following statements are true:
(1) For each f ∈ AutM(G), the map α : G/N → M given by α (gN) =
N f f
g−1f(g) is well defined and it is a homomorphism.
(2) IfGisfiniteandM ⊆ N, thenthemapφ :AutM(G) → Hom(G/N,M)
N
defined by φ(f)= α is an isomorphism.
f
The following result is due to Azhdari and Akhavan-Malayeri [1].
Lemma 2.7. [1,Proposition1.3] Let G and H be two finite abelian p-groups
and let exp(H) |exp(G). Then Hom(G,H) ≃ G if and only if H is cyclic.
3. nth-class preserving automorphisms of n-isoclinism family
Throughout the section n denotes a positive integer. In this section we
studythesubgroupsoftheautomorphismgroupofn-isoclinicgroups. In[6],
Rai proved that, if G and H are two finite non abelian 1-isoclinic(isoclinic)
groups then there exists an isomorphism φ : Autγ2(G)(G) → Autγ2(H)(H)
Z(G) Z(H)
such that φ(Aut (G)) = Aut (H). Here, we extend these results to n-
c c
isoclinic groups. Theorem A of Rai [6] is a special case of Theorem 3.3. If G
is a nilpotent group of class at most (n+1). We also obtained a necessary
and sufficient condition on γ (G) such that Autγn+1(G)(G) is isomorphic
n+1 Zn(G)
to the group of inner automorphisms of a quotient group of G. Rai [6,
Theorem B] is obtained as a corollary to Theorem 3.5.
Note that the subgroups of the automorphism group, which we study in
this section, are trivial for an abelian group. Thus we may further assume
that all groups are non-abelian.
Lemma 3.1. Let G and H be two finite groups and (α,β) be an n-isoclinism
from G to H. Then the following statements are true:
(1) For each f ∈ Autγn+1(G)(G), the map θ :H → H given by θ (h) =
Zn(G) f f
hβ(g−1f(g)), where g ∈ G such that α(gZ (G)) = hZ (H), is well
n n
defined.
(2) For all h ∈ H, h−1θ (h) ∈ γ (H).
f n+1
(3) If f ,f ∈ Autγn+1(G)(G) such that f 6= f , then θ 6= θ .
1 2 Zn(G) 1 2 f1 f2
Proof. (1) Suppose g Z (G) = g Z (G) for some g ,g ∈ G. Then
1 n 2 n 1 2
g−1g ∈ Z (G). As f fixes Z (G) element wise, f(g−1g ) = g−1g .
1 2 n n 1 2 1 2
This implies that g−1f(g ) = g−1f(g ). Hence hβ(g−1f(g )) =
1 1 2 2 1 1
6 SURJEETKOUR
hβ(g−1f(g )).
2 2
(2) Trivial.
(3) Let f ,f ∈ Autγn+1(G)(G) and let f 6= f . Then there exists 1 6=
1 2 Zn(G) 1 2
g ∈ G such that f (g) 6= f (g). Therefore, g−1f (g) 6= g−1f (g).
1 2 1 2
Since β is injective, β(g−1f (g)) 6= β(g−1f (g)). Hence θ (h) 6=
1 2 f1
θ (h), where h ∈ H such that α(gZ (G)) = hZ (H).
f2 n n
(cid:3)
Theorem 3.2. Let G and H be two finite groups and (α,β) be an n-
isoclinism from G to H. Then the following statements are true:
(1) For each f ∈ Autγn+1(G)(G), the map θ defined in Lemma 3.1(1),
Zn(G) f
is a group homomorphism.
(2) θ is an isomorphism fixing Z (H) element wise.
f n
Proof. (1) From Lemma3.1, θ is welldefinedandh−1θ (h) ∈ γ (H).
f f n+1
Let h ,h ∈ H and let g ,g ∈ G such that α(g Z (G)) = h Z (H)
1 2 1 2 1 n 1 n
andα(g Z (G)) = h Z (H). Thenα(g g Z (G)) = h h Z (H)and
2 n 2 n 1 2 n 1 2 n
θ (h h ) = h h β(g−1g−1f(g g ))
f 1 2 1 2 2 1 1 2
= h (h β(g−1g−1f(g )g g−1f(g )))
1 2 2 1 1 2 2 2
= h (h β(g−1g−1f(g )g )h−1)h β(g−1f(g )).
1 2 2 1 1 2 2 2 2 2
Notethatg−1f(g ) ∈ γ (G). ThereforebyusingLemma2.1(2),
1 1 n+1
β(g−1g−1f(g )g ) = h−1β(g−1f(g ))h .
2 1 1 2 2 1 1 2
Hence θ (h h )= h β(g−1f(g ))h β(g−1f(g )) = θ (h )θ (h ).
f 1 2 1 1 1 2 2 2 f 1 f 2
(2) It is enough to prove that θ is injective. Let θ (h) = 1 for some
f f
h ∈ H. Then hβ(g−1f(g)) =1, where g ∈ G such that α(gZ (G)) =
n
hZ (H). Hence h = (β(g−1f(g)))−1 ∈ γ (H). Without loss of
n n+1
generality, we may assume that g ∈ γ (G). If not, then there
n+1
′ ′ ′
exists g ∈ γ (G) such that β(g ) = h and g Z (G) = gZ (G).
n+1 n n
′
Hence we can replace g by g .
Nowobservethat,1 = hβ(g−1f(g)) = hβ(g−1)β(f(g)) ∈ γ (H).
n+1
Hence β(f(g)) = h−1β(g) ∈ γ (H). Furthermore, α(gZ (G)) =
n+1 n
β(g)Z (H) = hZ (H), implies that, h−1β(g) ∈ Z (H). There-
n n n
fore, β(f(g)) ∈ γ (H)∩Z (H). By using Theorem 2.2, we have
n+1 n
f(g) ∈ Z (G)∩γ (G). As f fixes Z (G) element wise, f(g) = g
n n+1 n
and h = 1. Hence θ is an isomorphism. Also, if h ∈ Z (H), then
f n
g ∈ Z (G), whereg ∈ Gsuchthatα(gZ (G)) = hZ (H). Therefore,
n n n
θ (h) = h. Hence θ ∈ Autγn+1(H)(H).
f f Zn(H)
(cid:3)
ON nth CLASS PRESERVING AUTOMORPHISMS OF n-ISOCLINISM FAMILY 7
Theorem 3.3. Let G and H be two finite groups and let (α,β) be an n-
isoclinismfromGtoH. Thenthereexistsanisomorphism Ψ : Autγn+1(G)(G) →
Zn(G)
Autγn+1(H)(H) such that Ψ(Autn(G)) = Autn(H).
Zn(H) c c
Proof. Define Ψ : Autγn+1(G)(G) → Autγn+1(H)(H) such that Ψ(f) = θ ,
Zn(G) Zn(H) f
where θ is the map defined in Lemma 3.1(1). By Theorem 3.2, θ ∈
f f
Autγn+1(H)(H).
Zn(H)
FirstweshowthatΨisagrouphomomorphism. Letf ,f ∈ Autγn+1(G)(G).
1 2 Zn(G)
Then we need to show that θf1◦f2 = θf1 ◦θf2. Consider h ∈ H, then
(θ ◦θ )(h) = θ (hβ(g−1f (g))),
f1 f2 f1 2
where g ∈ G such that α(gZ (G)) = hZ (H).
n n
Note that
α(f (g)Z (G)) =α(gZ (G))α(g−1f (g)Z (G)) = hZ (H)α(g−1f (g)Z (G)).
2 n n 2 n n 2 n
Since g−1f (g) ∈ γ (G), by Lemma 2.1 we have
2 n+1
α(g−1f (g)Z (G)) = β(g−1f (g))Z (H)
2 n 2 n
Therefore, α(f (g)Z (G)) = hβ(g−1f (g))Z (H). Thus
2 n 2 n
(θ ◦θ )(h) = θ (hβ(g−1f (g)))
f1 f2 f1 2
= hβ(g−1f (g))β(f (g−1)f (f (g))
2 2 1 2
= hβ(g−1(f ◦f )(g))
1 2
= θf1◦f2(h).
Hence Ψ is a group homomorphism.
InordertoprovethatΨisanisomorphism,defineamapΦfromAutγn+1(H)(H)
Zn(H)
to Autγn+1(G)(G) such that Φ(θ) = f , where f : G → G is given by
Zn(G) θ θ
f (g) = gβ−1(h−1θ(h)), where h ∈ H such that α−1(hZ (H)) = gZ (G).
θ n n
Using the fact that α and β are isomorphisms, one can prove that Φ is a
group homomorphism and Φ(θ ) = f, for f ∈ Autγn+1(G)(G). Therefore,
f Zn(G)
(Φ◦Ψ)(f) = Φ(θ ) = f. Hence Φ◦Ψ = I. Similarly, one can show that
f
Ψ◦Φ = I. Thus Ψ is an isomorphism.
Next we show that Ψ(Autn(G)) = Autn(H). Consider f ∈ Autn(G) ⊆
c c c
Autγn+1(G)(G). Clearly, θ ∈ Autγn+1(H)(H) and θ (h) = hβ(g−1f(g))
Zn(G) f Zn(H) f
where g ∈ G such that α(gZ (G)) = hZ (H). Also, for g ∈ G, there ex-
n n
istsx ∈ γ (G)suchthatf(g) = x−1gx. Therefore,θ (h) = hβ(g−1x−1gx) =
n f
hβ[g,x]. Observethat,β[g,x] = [h,y],wherey ∈γ (H)suchthatα(xZ (G)) =
n n
yZ (H). Hence θ (h) = y−1hy and Ψ(Autn(G)) ⊆ Autn(H). Similarly, we
n f c c
can show that Φ(Autn(H)) ⊆ Autn(G). Therefore, we have Ψ(Autn(G)) =
c c c
Autn(H). (cid:3)
c
8 SURJEETKOUR
Corollary 3.4. [6, Theorem A] Let G and H be two finite isoclinic groups.
Then there exists an isomorphism Ψ : Autγ2(G)(G) → Autγ2(H)(H) such
Z(G) Z(H)
that Ψ(Aut (G)) = Aut (H).
c c
Proof. Put n= 1 in Theorem 3.3. (cid:3)
Let Gbeafinitenonabelian groupand letM beacentral subgroupof G.
Let M denote the p-primary component of M, wherep is a primedivisor of
p
the order of M. In [6], Rai proved that, if G is a finite p-group of nilpotency
class two then Autγ2(G)(G) = Inn(G) if and only if γ (G) is cyclic. We
Z(G) 2
generalize this result to a nilpotent group of class at most (n+1). Also we
have shown that the result is true for any central subgroup M of G which
contains γ (G).
n+1
Theorem 3.5. Let G be a finite non abelian group and let M be a central
subgroup of G such that γ (G) ⊆ M. Then
n+1
AutMZn(G)(G) ≃ Inn(G/Zn−1(G))
if and only if Mpi is cyclic for each pi ∈π(G/Zn−1(G)).
Proof. Since γ (G) ⊆ M ⊆ Z(G), G is nilpotent of class at most (n+1).
n+1
If nilpotency class of G is less than (n + 1), then both groups are trivial
and result is true. Now we may assume that G is nilpotent of class (n+1).
Therefore by Lemma 2.4, exp(G/Z (G)) = exp(γ (G)) | exp(M). Hence,
n n+1
by using Lemma 2.5, we have π(G/Z (G)) ⊆π(M).
n
Let π(G/Z (G)) = {p ,...,p } and let
n 1 r
G/Z (G) ≃ H ×H ×···×H ,
n p1 p2 pr
where H denotes the p -primary component of G/Z (G).
pi i n
Similarly, let π(M) = {p ,...,p ,q ,...,q } and
1 r 1 s
M ≃ M ×···×M ×M ×···×M ,
p1 pr q1 qs
where M and M denote the p -primary and q -primary components of M
pi qi i i
respectively.
Clearly exp(H ) |exp(M ) for all 1 ≤ i≤ r. Now by Theorem 2.6, we have
pi pi
AutM (G) ≃ Hom(G/Z (G),M)
Zn(G) n
= Hom( r H , r M × s M )
Qi=1 pi Qi=1 pi Qj=1 qj
r r
≃ Hom( H , M )
Qi=1 pi Qi=1 pi
r
≃ Hom(H ,M ).
Qi=1 pi pi
By Lemma 2.7, Hom(H ,M ) ≃ H if and only if M is cyclic. Hence
pi pi pi pi
r
AutMZn(G)(G)) ≃ YHpi ≃ G/Zn(G) ≃ Inn(G/Zn−1(G))
i=1
if and only if M is cyclic for all 1 ≤ i≤ r. (cid:3)
pi
ON nth CLASS PRESERVING AUTOMORPHISMS OF n-ISOCLINISM FAMILY 9
Corollary 3.6. Let G be a finite non abelian group of class (n+1). Then
AutZγnn+(G1()G)(G) ≃ Inn(G/Zn−1(G)) if and only if (γn+1(G))p is cyclic for all
p-primary components of γ (G).
n+1
Proof. Take M = γ (G). (cid:3)
n+1
Corollary 3.7. Let G be a finite non abelian p-group of class (n+1). Then
AutZγnn+(G1()G)(G) ≃ Inn(G/Zn−1(G)) if and only if γn+1(G) is cyclic.
Rai’s [6, Theorem B(2)] follows from corollary 3.7 by taking n = 1.
Acknowledgement
This research is supported by SERB-DST grant YSS/2015/001567.
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Discipline of Mathematics, Indian Institute of Technology, Gandhinagar
382355, India.
E-mail address: [email protected]