Let n be a positive integer. We say that a group G is an -Engel group if it satisfies the law . The variety of -Engel groups lies between the varieties of n-Engel groups and -Engel groups. In this paper, we study these groups, and in particular, we prove that all -Engel -groups are locally nilpotent. We also show that if G is a -Engel p-group, where is a prime, then is locally nilpotent.
Let G be a group and . The commutator of g and h is the element
We recursively define , where n is a positive integer, as and for . A subset is an Engel set of G if, for every , there is a positive integer such that . If k is bounded above by some positive integer n, we say that S is an n-Engel subset, and if furthermore , then G is an n-Engel group. Recall that every 2-Engel group is nilpotent of class at most 3. By a classic result of Heineken , every 3-Engel group is locally nilpotent, and this result was later generalized to include all 4-Engel groups  (see also ).
Recall that an element is said to be left n-Engel if for all and right n-Engel if for all . We denote the subset of left n-Engel elements by and that of right n-Engel elements by .
Let G be a group and n a positive integer.
We say that is a left -Engel element if for all .
We say that is a right -Engel element if for all .
We say that G is an -Engel group if it satisfies the law .
We denote the subset of left -Engel elements by and the right -Engel elements by . Thus G is an -Engel group if and only if or equivalently . We denote the variety of m-Engel groups by .
It is not difficult to prove that
Thus, in particular, .
Let G be a group and n a positive integer. We have
In particular, .
That is obvious. To see that , let . Then, for any , we have
Thus and . ∎
Our main results on -Engel groups are the following.
Let G be a -Engel -group. Then G is locally nilpotent.
Let G be a -Engel p-group, where p is a prime and . Then is locally nilpotent.
A major ingredient to the proofs is a result on Engel sets that is also of independent interest. Let be the n-Engel word .
Let be the largest 2-generator group satisfying the relations . Then R is nilpotent of class 4.
We will see later that these relations imply that is a 3-Engel subset of R.
2 Proof of Theorem A
Consider the n-Engel word . As we will focus in particular on the 3-Engel word, we will often use instead of .
Suppose G is a group with elements , where
Then is nilpotent of class at most 2.
From the equations
we see that and . Thus we have that is nilpotent of class at most 2. ∎
Let G be a group with elements , where
Then for all .
By symmetry, it suffices to deal with the case when . As
we can also assume that . Then, from
we can also assume, without loss of generality, that . We are thus only left with showing that , but this follows from Lemma 2.1 and the fact that . ∎
It follows from Lemma 2.2 that if
then is a 3-Engel subset.
Let G be a group with elements satisfying
Then commutes with .
From the Hall–Witt identity and Lemma 2.1, we have
It follows that commutes with , and thus, using Lemma 2.1 again, commutes with as well. ∎
Proof of Theorem A.
Let be the largest group satisfying the relations
By the remark after Lemma 2.2, we know that is a 3-Engel subset of R.
In order to show that R is nilpotent of class at most 4, we need to show that . This is equivalent to showing that . As and are nilpotent of class at most 2, we see that and . In order to show that , we need to show that and are in . As , it suffices to show that . In the following calculations, we again use the fact that and are nilpotent of class at most 2. We have
From (2.1), we thus see that it suffices to show that , and in fact, it suffices to show that commutes with b as then, by symmetry, the RHS of (2.1) commutes with a, and thus commutes then with a as well.
From Lemma 2.2, we know that
In particular, equation (2.1) holds if we replace a by or b by . Calculating in the group that is nilpotent of class at most 2, we see that . From this and (2.1), it follows that is invariant under replacing a by . Notice also that
it follows that commutes with b.
As R is nilpotent of class at most 4, it follows that R is metabelian, and using the nilpotent quotient algorithm nq of Nickel  (which is implemented in GAP ), one can see that the class is exactly 4. It turns out that R is torsion-free with . ∎
An interesting related result [1, Proposition 3.1] states that if with , then is nilpotent of class at most 3.
The following examples show that the hypotheses of Theorem A cannot be weakened.
Example 1 ([1, Example 4.2]).
Let x and y be elements of defined by
Then , and has order , so, in particular, G is not nilpotent.
Let x and y be elements of defined by
and has order , so, in particular, G is not nilpotent.
Let be a group, . Then .
That commutes with x is a direct consequence of . Then shows that commutes also with y. ∎
Let G be a group, and let . Suppose that, for some , we have that is an n-Engel subset of G. Then
and therefore . Replacing b by , we see that implies . Next use , which implies and thus, after conjugation by a, that . Replacing b by , we see that implies . ∎
Let G be a -Engel 2-group. Then G is locally nilpotent.
Taking the quotient of G by the Hirsch–Plotkin radical, we can assume that , and we want to show that . We argue by contradiction and suppose . As groups of exponent 4 are locally finite, there must be an element of order 8. We get a contradiction by showing that is abelian and thus .
Let , and consider the subgroup , where . Let
where and . By Lemma 3.2, we know that
By Theorem A, we then know that and therefore H is finite. Using GAP or MAGMA, one can then check that , and thus we have shown that is abelian. ∎
Let G be a -Engel 3-group. Then G is locally nilpotent.
As before, we can assume that , and the aim is then to show that . We argue by contradiction and suppose that . As groups of exponent 3 are locally finite, there must be an element of order 9. Let and . As in the proof of Proposition 3.3, one sees that H is finite and then, with the help of GAP or MAGMA, that . Thus for all , and thus is a left 3-Engel element of G. By the main result of , we then know that , which contradicts the fact that . ∎
Let G be a group, and let be two elements of finite order such that is a 4-Engel set. Then every prime divisor of is a divisor of and . In particular, if a and b are of coprime order, then .
are 3-Engel subsets of G. By Theorem A, we know that
are nilpotent. As these groups are nilpotent, we know that every prime divisor of divides and every prime divisor of divides . Now we have , and thus divides and and thus also and from the discussion above. ∎
Proof of Theorem B.
Let G be a -group that is -Engel. Let be the set consisting of all elements in G whose order is a power of 2 and that of those elements whose order is a power of 3. In view of Propositions 3.3 and 3.4, it suffices to show that and are subgroups and that G is a direct product of and . Now take any two elements of coprime orders, and let . By Lemma 3.1, we know that for all . By Lemma 3.5, it follows that . Thus T is nilpotent, and as a and b are of coprime order, it follows that . Now let , and let be an element that has odd order. By the argument above, we know that , and as was arbitrary, we see that . Thus is a 2-group, and by Proposition 3.3, it is locally nilpotent and thus so is . As is generated by 2-elements, it is then a 2-group, and thus , and thus is a subgroup. The proof that is a subgroup is similar, using Proposition 3.4. Now let and . Then and is thus trivial. Hence G is a direct product of and and thus locally nilpotent. ∎
Let be a prime, and consider the group , where and is a 3-Engel set. Then G has exponent , and is abelian.
By Theorem A, we know that G is nilpotent of class at most 4. Then G is regular, and it follows easily that and then that (for definition and properties of regular p-groups, see [2, Section 12.4], in particular Theorem 12.4.3). ∎
Proof of Theorem C.
Let be a prime, and let G be a -Engel p-group. Consider , where is the Hirsch–Plotkin radical of G. The aim is to show that H is of exponent p. Passing from G to H, we can thus without loss of generality assume that the Hirsch–Plotkin radical of G is trivial, and the aim is to show that G is then of exponent p. We argue by contradiction and suppose that G has an element g of order . Let , and consider the subgroup , where . Let , where and . By Lemma 3.2, we know that
By Theorem A, we then know that is finite, and thus also H is finite. By Lemma 3.6, we know that , that is, , and thus, in particular, . Thus is a left 3-Engel element of odd order in G. By the main result of , it follows that , which contradicts our assumption that . ∎
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