4-Engel Groups Research Articles

On the exponent of Bogomolov multipliers

Abstract We prove that if G is a finite group, then the exponent of its Bogomolov multiplier divides the exponent of G in the following four cases: (i) G is metabelian, (ii) exp ⁡ G = 4 {\exp G=4} , (iii) G is nilpotent of class ≤ 5 {\leq 5} , or (iv) G is a 4-Engel group.

Results on Engel Fuzzy Subgroups

‎In the classical group theory there is‎ an open question‎: ‎Is every torsion free n-Engel group (for n ≥ 4)‎, nilpotent?‎. ‎To answer the question‎, ‎Traustason‎ [11] showed that with some additional conditions all‎ ‎4-Engel groups are locally nilpotent‎. ‎Here‎, ‎we gave some partial‎ answer to this question on Engel fuzzy subgroups‎. ‎We show that if μ is a normal 4-Engel fuzzy‎ subgroup of group G‎, ‎x,y in G and a =yx‎, ‎then μ| is a generalized nilpotent of class at‎ most 2‎. ‎Also we define a torsion free fuzzy subgroup and show‎ ‎that if μ is a 4-Engel torsion free fuzzy subgroup of G‎, ‎then μ| is a generalized nilpotent of class at most 4‎, ‎for conjugate elements a,y in G‎.

The local nilpotence theorem for 4-Engel groups revisited

‎The proof of the local nilpotence theorem for $4$-Engel groups was‎ ‎completed by G‎. ‎Havas and M‎. ‎Vaughan-Lee in 2005‎. ‎The complete proof on the other hand is spread over several articles and the aim of this paper is to give a‎ ‎complete coherent linear version‎. ‎In the process we are also‎ ‎able to make a few simplifications and in particular we are able to merge two of‎ ‎the key steps into one‎.

SCHUR MULTIPLIERS OF n-ENGEL GROUPS

We find a bound for the exponent of the Schur multiplier of a finite p-group in terms of the exponent and Engel length of the given group. It is also proved that if G is a 3-Engel group of finite exponent, then the exponent of H2(G) divides exp G. When G is a 4-Engel group of exponent e, the exponent of H2(G) divides 10e.

On 4-Engel Groups

In this note, the author proves that a group G is a 4-Engel group if and only if the normal closure of every element g ∈ G is a 3-Engel group

A NOTE ON THE LOCAL NILPOTENCE OF 4-ENGEL GROUPS

Recently Havas and Vaughan-Lee proved that 4-Engel groups are locally nilpotent. Their proof relies on the fact that a certain 4-Engel group T is nilpotent and this they prove using a computer and the Knuth–Bendix algorithm. In this paper we give a short handproof of the nilpotency of T.

4-ENGEL GROUPS ARE LOCALLY NILPOTENT

Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

TWO GENERATOR 4-ENGEL GROUPS

Using known results on 4-Engel groups one can see that a 4-Engel group is locally nilpotent if and only if all its 3-generator subgroups are nilpotent. As a step towards settling the question whether all 4-Engel groups are locally nilpotent we show that all 2-generator 4-Engel groups are nilpotent.

Locally nilpotent 4-Engel groups are Fitting groups

Let G be a locally nilpotent 4-Engel group. We show that the normal closure of any element from G is nilpotent of class at most 4. When G has no element of order 2 or 5, the normal closure has class at most 3. These bounds are sharp.

SOME COMMUTATOR GROUP LAWS EQUIVALENT TO THE COMMUTATIVE LAW

ABSTRACT Let G be a group, n an integer greater than 2 and . Let be a commutator of weight n with entries from the set . The question, originally raised by N. D. Gupta, is: Under what conditions is a group satisfying the law abelian? Some sufficient conditions for an affirmative answer are the finiteness or the solvability of the group. It is known that in the case the answer is positive for an arbitrary group. We prove that every group satisfying the law is abelian. We establish some group laws of the form which imply the commutative law without any other restriction. We also prove that every 4-Engel group satisfying the law is abelian.

Semigroup identities in 4-Engel groups

We say that a group G or a variety of groups V satisfies a semigroup law, if it satisfies a nontrivial law of the form u(x1, . . . , xn) = v(x1, . . . xn), where u and v are words in the free semigroup freely generated by x1, . . . , xn. It follows from a result of J. Lewin and T. Lewin [2] that a variety V of groups which satisfies a semigroup law can be characterised by its semigroups laws. Furthermore, we have then a sufficient and necessary condition for a semigroup to be embeddable in some group in V . A semigroup S is embeddable in some group in V if and only if it is cancellative and it satisfies all the semigroup laws that hold in V . In other words we have that S is embeddable in some group in V if and only if S is a cancellative semigroup in the corresponding semigroup variety. In [4] B. H. Neumann and T. Taylor show that nilpotent groups satisfy semigroup laws. We will be using their work later on so we will now describe it in more details. Let F be a free group that is freely generated by the variables x, y, z1, z2, . . .. We define a sequence of words q1, q2, . . . in the variables x, y, z1, z2, . . . by induction as follows.

A Classification of Some Insoluble Varieties of Groups of Exponent Four

We classify the subvarieties of the variety B4∩E4of all 4-Engel groups of exponent 4 which contain the proper subvariety defined by7, and from this we deduce a classification of all varieties of groups of exponent 4 which contain this subvariety. We also use the first classification to obtain a reduction for the finite basis problem for varieties of groups of exponent 4.