Non-solvable groups generated by involutions in which every involution is left 2 -Engel
aa r X i v : . [ m a t h . G R ] A p r NON-SOLVABLE GROUPS GENERATED BY INVOLUTIONS INWHICH EVERY INVOLUTION IS LEFT -ENGEL ALIREZA ABDOLLAHI
Abstract.
The following problem is proposed as Problem 18.57 in [The KourovkaNotebook, No. 18, 2014] by D. V. Lytkina:Let G be a finite 2-group generated by involutions in which [ x, u, u ] = 1 forevery x ∈ G and every involution u ∈ G . Is the derived length of G bounded?The question is asked of an upper bound on the solvability length of finite2-groups generated by involutions in which every involution (not only the gen-erators) is also left 2-Engel. We negatively answer the question. Introduction and Result
The following problem is proposed as Problem 18.57 of [2] by D. V. Lytkina:
Question 1.1.
Let G be a finite -group generated by involutions in which [ x, u, u ] =1 for every x ∈ G and every involution u ∈ G . Is the derived length of G bounded? Question 1.1 is asked of an upper bound on the solvability length of finite 2-groups generated by involutions in which all involutions of groups (not only thegenerators) are also left 2-Engel elements. We negatively answer the question. Inthe proof we need some well-known facts about the groups of exponent 4.It is know that groups of exponent 4 are locally finite [6] and the free Burnsidegroup B of exponent 4 with infinite countable rank is not solvable [5]. In [1], it isproved that the solvability of B is equivalent to the one of the group H defined asfollows:Let H be the freest group generated by elements { x i | i ∈ N } with respect to thefollowing relations:(1) x i = 1 for all i ∈ N ;(2) The normal closure h x i i H is abelian for all i ∈ N .(3) h = 1 for all h ∈ H ;Therefore H is a non-solvable group of exponent 4 generated by involutions x i ( i ∈ N ). Note that the relation (2) above is equivalent to say that x i is a left2-Engel element of H that is [ x, x i , x i ] = 1 for all x ∈ H . We do not know if H has the property requested in Question 1.1, that is, whether every involution u ∈ H is a left 2-Engel element of H . Instead we find a quotient of H which isstill non-solvable but it satisfies the latter property. The latter quotient of H willprovide a counterexample for Question 1.1. To introduce the quotient we need torecall some definitions and results on right 2-Engel elements. Mathematics Subject Classification.
Key words and phrases.
Groups of exponent 4; Involution; Left 2-Engel; Abelian normal clo-sure .
For any group G , R ( G ) denotes the set of all right 2-Engel elements of G , i.e. R ( G ) = { a ∈ G | [ a, x, x ] = 1 for all x ∈ G } . It is known [3] that R ( G ) is a characteristic subgroup of G . The subgroup R ( G )is a 2-Engel group that is [ x, y, y ] = 1 for all x, y ∈ R ( G ). Thus R ( G ) is nilpotentof class at most 3 [4] and so it is of solvable length at most 2. Theorem 1.2.
Let H be the freest group defined above. Then H = H/R ( H ) satisfies the following condition: (4) all involutions u ∈ H are left -Engel in H .Furthermore, H is not solvable so that there is no upper bound on solvability lengthsof finite -groups H n = h x ,...,x n i R ( h x ,...,x n i ) which satisfy all conditions (1), (2), (3) and (4) above. Proof of Theorem 1.2
The following is the key lemma of the paper.
Lemma 2.1.
Let G be any group of exponent and b ∈ G is such that b ∈ R ( G ) .Then [ a, b, b ] ∈ R ( G ) for all a ∈ G . This means that, in every group G of exponent , every involution of the quotient G/R ( G ) has an abelian normal closure.Proof. Let N be the freest group generated by elements a, b, c subject to the fol-lowing relations:(1) x = 1 for all x ∈ N ;(2) [ b , x, x ] = 1 for all x ∈ N .By [6] it is known that N is finite. Now by nq package [7] one can construct N in GAP [8] by the following commands:
LoadPackage("nq");F:=FreeGroup(4);a:=F.1;b:=F.2;c:=F.3;x:=F.4;G:=F/[x^4,LeftNormedComm([b^2,x,x])];N:=NilpotentQuotient(G,[x]);gen:=GeneratorsOfGroup(G,[x]);LeftNormedComm([gen[1],gen[2],gen[2],gen[3],gen[3]]);
Note that in above gen[1] , gen[2] and gen[3] correspond to the free generators a, b and c , respectively. The output of last command in above (which is id thetrivial element of N ) shows that [ a, b, b, c, c ] = 1. This completes the proof. (cid:3) It may be interesting in its own right that the group N defined in the proof ofLemma 2.1 is nilpotent of class 7 and order 2 . Proof of Theorem 1.2.
It follows from Lemma 2.1 that H = H/R ( H ) has theproperty (4) mentioned in the statement of Theorem 1.2. Since H is not solvableby [1] and [5], it follows that there is no upper bound on the solvable lengths offinite 2-groups H n . By construction H n is generated by involutions and by Lemma2.1 all involutions in H n are left 2-Engel. This completes the proof. (cid:3) Acknowledgements
This research was in part supported by a grant (No. 92050219) from Schoolof Mathematics, Institute for Research in Fundamental Sciences (IPM). The au-thor gratefully acknowledges the financial support of the Center of Excellence forMathematics, University of Isfahan.
References [1] N. D. Gupta and K. N. Weston, On groups of exponent four, J. Algebra, (1969) 59-66.[2] V.D. Mazurov, E.I. Khukhro (Eds.), Unsolved Problems in Group Theory, The KourovkaNotebook, No. 18, Russian Academy of Sciences, Siberian Division, Institute of Mathematics,Novosibirsk, 2014.[3] W. P. Kappe, Die A -Norm einer Gruppe, Illinois J. Math., (1961) 187-197.[4] C. Hopkins, Finite groups in which conjugate operations are commutative, American J. Math., (1929) 35-41.[5] Yu. P. Razmyslov, On a problem of Hall and Higman, Izv. Akad. Nauk SSSR Ser. Mat., (1978) 833-847[6] I. N. Sanov, Solution of Burnside’s problem for exponent 4, Ucinyi Zapiski-Leningrad, Gos.Univ. Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran; andSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O.Box 19395-5746, Tehran, Iran
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