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International Journal of Group Theory
ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 3 No. 3 (2014), pp. 35-38.
c
2014 University of Isfahan
www.ui.ac.ir
ON n-KAPPE GROUPS
A. FARAMARZI SALLES∗ AND H. KHOSRAVI
Communicated by Patrizia Longobardi
Abstract. LetGbe an infinite group and n∈ {3,6} ∪ {2k|
k∈N}. In this paper, we prove thatGis ann-Kappe group if and only if for any two infinite subsetsX andY ofG, there existx∈X andy∈Y
such that [xn, y, y] = 1.
1. Introduction
Let w be a word in a free group of rank n > 0. Let V =V(w) be the variety of groups defined by the law w(x1, . . . , xn) = 1. Define V∗ =V(w∗) to be the class of all groups Gsuch that given infinite
subsets X1, X2, . . . , Xn of G, there exist xi ∈ Xi, 1 ≤i ≤ n, satisfying w(x1, . . . , xn) = 1. In [7], P.
Longobardi et al. posed the question of when the equalityF ∪ V =V∗ holds, where F is the class of
all finite groups (see Problem 15.1 in [10]).
The origin of this question is a problem of Erd¨os (see [8]). There is no example, so far, of an infinite group in V∗ \ V. In considering this question, many authors have obtained the equality for certain
words (see [1, 2, 3, 4, 8]).
For a group G, we denote the set of right 2-Engel elements of G by R2(G) = {a ∈ G|[a, x, x] =
1 for every x∈ G}. A well-known result of W. Kappe [5] states that R2(G) is always a characteristic
subgroup of G. Let n be an integer, a group G is called an n-Kappe group if [yn, x, x] = 1 for all x and y in G. Clearly, G is an n-Kappe group if and only if the factor group G/R2(G) has finite
exponent dividingn. In this paper, we answer the above mentioned problem affirmatively for the word w= [yn, x, x], wheren∈ {3,6}∪{2k|k∈N}. More precisely, we shall prove that every infiniteK∗n-group is aKn-group, where Kn=V(w) andK∗n=V(w∗).
MSC(2010): Primary: 20F99; Secondary: 20E10, 05E15. Keywords: Kappe groups, Variety of groups, Erd¨os’ Problem. Received: 10 November 2013, Accepted: 29 January 2014. ∗Corresponding author.
36 Int. J. Group Theory 3 no. 3 (2014) 35-38 A. Faramarzi Salles and H. Khosravi
2. Results
In this section we present our main result. First, we state the following lemma, which will be used frequently.
Lemma 2.1 ([1]). LetG be an infiniteV(w∗)-group, wherewis a word in the free group of rank 2. Let
A be an infinite abelian subgroup ofGand y1, . . . , yn∈G. Then there exists an infinite subsetT of the set B ={a∈A|w(a, yi) =w(yi, a) = 1, ∀i= 1, . . . , n} such that t1t−12 ∈B for all distinct elements t1
and t2 in T. Also,A\B is finite.
LetG be a group andx an element of G. As usual, CG(x) denotes the centralizer ofx inG.
Lemma 2.2. Let G be an infinite K∗
n-group andg an element of G. Then CG(gn) is infinite.
Proof. Assume thatCG(xn) is finite for somex∈G. First we construct by induction onm, a sequence (xm)m∈N of elements ofGsatisfying the following properties:
(i) xix−1j ∈/CG(xn), ifi6=j,
(ii) xix−1j xkx−1h ∈/ CG(xn), if i, j, h andk are pairwise different.
For the induction step, assume that x1, . . . , xm satisfy (i) and (ii). Let
D=
m
[
i=1
CG(xn)xi∪ m
[
i,j,h=1
CG(xn)xix−1j xh∪ m
[
i,j,h=1
xix−1j xhCG(xn)xh.
Since D is finite, there exists xm+1 ∈G which is not in D. It is easy to verify that x1, . . . , xm, xm+1
satisfy (i) and (ii).
Now let N = A∪B with A and B infinite and disjoint. Then also the sets {xxi : i ∈ A} and
{xxj :j∈ B} are infinite. Since G∈ K∗
n, there exist s∈A and t ∈B such that [(xxs)n, xxt, xxt] = 1. Thus [(xn)xsx−
1
t , x, x] = 1, it follows thatx(xn)xsx −1 t
∈CG(x). LetA1 =A\ {s}andB1 =B\ {t}. Then,
arguing as before, we find v∈A1 and w∈B1 such that x(x
n)xv x−1 w
∈CG(x). Hence, by induction, we
have infinite subsets I ⊆A and J ⊆B such that x(xn)xsx−t1
∈ CG(x), for an infinite number of pairs (s, t) (with all s’s different and all t’s different). But CG(x) is finite, so there exist i0 ∈ I and j0 ∈J
such that, for infinitely many i∈I and j ∈J we have x(xn)xi0x− 1 j0
=x(xn)xix− 1 j
. From this we conclude that (xn)xi0x−
1
j0 ((xn)xix− 1
j )−1∈CG(x), is a finite set, and hence there existi
1 6=i2 ∈I and j16=j2 ∈J,
such that (xn)xi0x− 1
j0 ((xn)xi1x −1
j1 )−1 = (xn)xi0x −1
j0 ((xn)xi2x −1
j2 )−1, and so (xn)xi1x −1
j1 = (xn)xi2x −1
j2 . Thus
we get xi1x
−1
j1 xj2x
−1
i2 ∈CG(x
n), which contradicts (ii). Therefore, CG(xn) is infinite, as required.
In the next results,nis an element of{3,6}∪{2k|k∈N}. The following corollary will be used repeatedly later.
Corollary 2.3. Let G be an infinite group in Kn∗. Then G has an infinite abelian subgroup. Proof. We show that in any infinite groupG∈ K∗
Int. J. Group Theory 3 no. 3 (2014) 35-38 A. Faramarzi Salles and H. Khosravi 37
n. If n ∈ {3,6} then G is locally finite (see page 425 of [9]) and if n ∈ {2k|k ∈ N} then G is an infinite 2-group, therefore by Corollary 2.5 of [6],Ghas an infinite abelian subgroup,Asay. Now every
non-trivial element of A has the required property.
Let G be a group, g an element of G and X a subset of G. We denote the subsets {gx :x ∈ X},
{xg :x∈X}and {gx:x∈X}by gX,Xg and gX respectively.
Lemma 2.4. Let G be an infinite group in K∗
n and x and y elements ofG such that CG(x) is infinite and CG(y) is finite. Then [yn, x, x] = 1.
Proof. We may assume thatyn6= 1. ThenCG(yn) is infinite by Lemma 2.2, then there exists an infinite abelian subgroup Ayn ≤ CG(yn) by Corollary 2.3. By applying Lemma 2.1, there exists an infinite subsetTyn ⊆Byn ⊆Ayn, whereByn ={a∈Ayn|[an, x, x] = [xn, a, a] = 1}such that t1t−1
2 ∈Byn for all
distinct elements t1, t2 ∈Tyn. Now we claim that yTyn is an infinite subset of G. If yTyn is finite then there exists an elementt0 ∈Tyn and an infinite subsetT0 ofTyn such thatyt0 =yt, for allt∈T0. Thus CG(y) is infinite, that gives a contradiction. Therefore yTyn is infinite. Now we consider two cases for xTyn. In the first case, we assume that xTyn is finite. Then CG(x)∩CG(yn) is infinite. Hence there exists an infinite abelian subgroupA of CG(x)∩CG(yn). It is clear thatyA andxAare infinite. Thus there aret1 and t2 inA such that [(yn)t1, xt2, xt2] = [yn, x, x] = 1.
In the second case, we suppose that xTyn is infinite. Then there are elements t
1 and t2 of Tyn such that 1 = [(yn)t1, xt2, xt2] = [(yn)t1t−
1
2 , x, x]t2 = [yn, x, x]. This completes the proof.
Lemma 2.5. Let Gbe an infinite group in K∗
n and x, y∈Gsuch that CG(x)∩CG(y) is infinite. Then [yn, x, x] = 1.
Proof. Let A be an infinite abelian subgroup of CG(x)∩CG(y), by Corollary 2.3. It is clear that Ax and Ay are infinite subsets of G. Thus, there exist elementsaand bof Asuch that [(by)n, ax, ax] = 1.
Then [bnyn, ax, ax] = 1 and therefore [yn, x, x] = 1.
Lemma 2.6. LetGbe an infinite group inK∗
nandy an element ofGsuch thatCG(y)is infinite. Then [yn, x, x] = 1, for all elementsx of G.
Proof. Let x be an arbitrary element of G. Since CG(y) is infinite, there exists an infinite abelian subgroupA≤CG(y) which contains y, by Corollary 2.3. We break up the proof into two cases:
Case (i): Let CG(x) be finite. Then we conclude from Lemma 2.1 that A has a cofinite subset B ={a∈ A: [an, x, x] = [xn, a, a] = 1}. Now if A0 ={a∈A :an= 1} is infinite, then yA0 and xA0
are infinite. By the property K∗
n, there exista and b of A0 such that [(yb)n, xa, xa] = [yn, xa, xa] = 1
and so [yn, x, x] = 1. Now suppose that A
0 is finite. Since xb
n
∈CG(x) for all elements b of B, there exists an infinite subset X of B and an element a0 of B such that xa
n 0 =xa
n
for all elements aof X. This implies that ana−n
0 ∈CG(x) for all elementsa ofX. Consequently, there exists an infinite subset
X0 ⊆X and an element a1 ∈X such that an =an1 for all elements aof X0. It follows that aa−11 ∈A0
38 Int. J. Group Theory 3 no. 3 (2014) 35-38 A. Faramarzi Salles and H. Khosravi
Case (ii): LetCG(x) be infinite. Then we may assume thatCG(x)∩CG(y) is finite, by Lemma 2.5. Let B0 = {a ∈ A : [an, x, x] = 1 = [an, yx, yx]}, which is cofinite by Lemma 2.1. It is easy to check
that B0 ={a∈ A : [an, x, x] = 1 = [an, x, y]}. By applying the property K∗n for infinite subsets yB0
and xB0, we get [(b
1y)n, xb2, xb2] = 1 for some elements b1 and b2 of B0. Then [bn1yn, x, x] = 1 and so
[[bn
1, x][yn, x], x] = 1. Now we have [yn, x, x] = 1, which completes the proof. Main Theorem. Every infinite K∗
n-group is aKn-group.
Proof. Let G be an infinite K∗
n-group and x and y two elements of G such thatyn 6= 1. By Lemmas [2.4, 2.5 and 2.6], we may assume that CG(y) andCG(x) are finite. By Corollary 2.3, there exists an infinite abelian subgroupAofCG(yn). SinceCG(y) andCG(x) are finite,yAandxAare infinite subsets of G. Thus there exist elementsaand binA such that [(ya)n, xb, xb] = 1. It follows that [yn, x, x] = 1,
therefore Gis aKn-group.
Acknowledgments
The research of the second author was in part supported by a grant from Gonbad-e Qabus University (No. 6/624).
References
[1] A. Abdollahi and B. Taeri, Some conditions on infinite subsets of infinite groups,Bull. Malaysian Math. Soc. (2),22 (1999) 87-93.
[2] C. Delizia and C. Nicotera, Groups with conditions on infinite subsets, Ischia Group Theory 2006, Proc. Conf., in
honor of Akbar Rhemtulla, World Scientific Publishing, (2007) 46-55.
[3] C. Delizia and A. Tortora, Locally graded groups with a Bell condition on infinite subsets, J. Group Theory, 12
(2009) 753-759.
[4] G. Endimioni, On a combinatorial problem in varieties of groups,Comm. Algebra,23(1995) 5297–5307. [5] W. P. Kappe, Die A-Norm einer Gruppe,Illinois J. Math.,5(1961) 187-197.
[6] O. H. Kegel and B. A. F. Wehrfritz,Locally Finite Groups, North-Holland Mathematical Library,3, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.
[7] P. Longobardi, M. Maj and A. H. Rhemtulla, Infinite groups in a given variety and Ramsy’s theorem,Comm. Algebra,
20(1992) 127-139.
[8] B. H. Neumann, A problem of Paul Erd¨os on groups,J. Austral. Math. Soc. Ser. A,21(1976) 467-472. [9] D. J. S. Robinson,A course in the theory of groups, 2nd edition,80, Springer-Verlag, New York, 1996.
[10] Unsolved problems in group theory, The Kourovka notebook, Sixteenth edition, Including archive of solved problems, Edited by V. D. Mazurov and E. I. Khukhro, Russian Academy of Sciences Siberian Division, Institute of Mathematics,
Novosibirsk, 2006.
Asadollah Faramarzi Salles
Department of Mathematics, Damghan University, P. O. Box 3671641167, Damghan, Iran
Email: [email protected]
Hassan Khosravi
Department of Mathematics, Gonbad-e Qabus University, Gonbad-e Qabus, Iran
