Some properties of commutative transitive of groups and Lie algebras

Some properties of commutative transitive of groups and Lie algebras Mohammad Reza R Moghaddam Department of Mathematics, Khayyam University, Mashhad, Iran, and Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, POBox 1159, Mashhad, 91775, Iran January 23-26, 2017 Mohammad Reza R Moghaddam

1 n-commutative transitive groups 2 3 4 Mohammad Reza R Moghaddam

n-commutative transitive groups In 2007 a general notion of χ-transitive groups was introduced by C Delizia, P Moravec and C Nicotera, where χ is a class of groups In 2013, L Ciobanu, B Fine and G Rosenberger studied the relationship among the notions of conjugately separated abelian, commutative transitive (CT) and fully residually χ-groups Mohammad Reza R Moghaddam

Let χ be a class of groups Then a group G is residually χ if given any non-trivial element g G there is a homomorphism ϕ : G H where H is a group in χ such that ϕ(g) 1 Mohammad Reza R Moghaddam

Definition A group G is fully residually χ if given finitely many non-trivial elements g 1,, g n in G there is a homomorphism ϕ : G H, where H is a group in χ, such that ϕ(g i ) 1 for all i = 1,, n Fully residually free groups have played a crucial role in the study of equations and first order formulas over free groups Mohammad Reza R Moghaddam

Definition A subgroup H of a group G is called conjugately separated, if H H x = 1, for all x G \ H It is clear that the intersection of a family of conjugately separated subgroups is again conjugately separated Mohammad Reza R Moghaddam

Definition Given a non-zero integer n, a group G is said to be n-central if [x, y n ] = 1, for all x, y G Hence n-central property is equivalent to G/Z (G) having finite exponent dividing n Here [x, y] = x 1 y 1 xy is the usual commutator of the elements x and y of the group G and x y = y 1 xy is the conjugate of x by y Mohammad Reza R Moghaddam

A group G is called Conjugately Separated n-central or CSC n -group, if all of its maximal n-central subgroups are conjugately separated A group G is said to be a CSA-group, if all of its maximal abelian subgroups are conjugately separated and it is commutative transitive or CT group if commutativity is transitive on the set of non-trivial elements of G L Ciobanu et al 2013 examined the relationships among the classes of non-abelian CSA, CT and fully residually χ-groups Mohammad Reza R Moghaddam

We will extend the definition of CT groups and we introduce the notion of n-commutative transitive groups, then we study its relationship with CSC n and fully residually χ-groups Definition A group G is n-commutative transitive (henceforth n-ct), if [x, y n ] = 1 and [y, z n ] = 1 imply that [x, z n ] = 1, for non-trivial elements x, y, z in G and n 1 Clearly n-ct groups are the usual CT groups, for n = 1 Also 1-central groups are abelian and CSC 1 groups are CSA Mohammad Reza R Moghaddam

We show that under some conditions the three notions n-commutative transitivity, CSC n, and fully residually χ-groups are equivalent for many important classes of groups, including those of free products of cyclics not containing the infinite dihedral group, torsion-free hyperbolic groups and one-relator groups with only odd torsion Mohammad Reza R Moghaddam

So, the main result of this section reads as follows; Theorem Let χ be a class of groups such that each of its n-noncentral group is CSC n and G be n-noncentral residually χ-group Then the following statements are equivalent: (i) G is fully residually χ; (ii) G is CSC n ; (iii) G is n-ct Mohammad Reza R Moghaddam

In 1967, B Baumslag introduced the notion of fully residually free groups and proved that a residually free group is fully residually free if and only if it is commutative transitive Here, using the previous result we show that this is true in the case of n-commutative transitive Corollary If G is a residually free group Then G is fully residually free if and only if G is n-commutative transitive Mohammad Reza R Moghaddam

A group G is called a conjugately separated 2-Engel group (henceforth CSE 2 -group), if all of its maximal 2-Engel subgroups are conjugately separated In the following, we discuss the notion of 2-Engel transitive group and then give its relationship with CSE 2 -group and fully residually χ-groups Mohammad Reza R Moghaddam

Definition (a) A group G is 2-Engel transitive (henceforth 2-ET), whenever [x, y, y] = 1 and [y, z, z] = 1 imply that [x, z, z] = 1, for every non-trivial elements x, y, z in G (b) For a given element x in G, we call EG 2 (x) = {y G : [x, y, y] = 1, [y, x, x] = 1} to be the set of 2-Engelizer of x in G The family of all 2-Engelizers in G is denoted by E 2 (G) and E 2 (G) denotes the number of distinct 2-Engelizers in G Mohammad Reza R Moghaddam

As an example consider Q 16 = a, b : a 8 = 1, a 4 = b 4, b 1 ab = a 1, the Quaternion group of order 16 and take the element b in Q 16 Then one can easily check that the 2-Engelizer set of b is as follows: E 2 Q 16 (b) = {1, a 2, a 4, a 6, b, a 2 b, a 4 b, a 6 b} Mohammad Reza R Moghaddam

Clearly in general, the 2-Engelizer of each non-trivial element of an arbitrary group G does not form a subgroup The following example shows our claim Example Let G be a finitely presented group of the following form: G = a 1, a 2, a 3, a 4 a 3 3 = a3 4 = 1, [a 1, a 2 ] = 1, [a 1, a 3 ] = a 4, [a 1, a 4 ] = 1, [a 2, a 3 ] = 1, [a 2, a 4 ] = a 2, [a 3, a 4 ] = 1 It is clear that G is an infinite group One can easily check that G is not 2-ET, as [a 2, a 1, a 1 ] = 1 and [a 1, a 4, a 4 ] = 1, while [a 2, a 4, a 4 ] = a 2 Moreover, EG 2 (a 1) is not a subgroup of G, since it is easily calculated that a 2, a 3 EG 2 (a 1) but a 2 a 3 EG 2 (a 1) Mohammad Reza R Moghaddam

Now, we give a condition under which the 2-Engelizer of each non-trivial element of a group G forms a subgroup Theorem Let G be an arbitrary group, then the set of each 2-Engelizer of a non-trivial element in G forms a subgroup if and only if the group x E G 2 (x) is abelian for all non-trivial elements x in G Mohammad Reza R Moghaddam

Now we study the relationship between the non 2-Engel CSE 2, 2-ET and fully residually χ-groups Theorem Let χ be a class of groups such that each non 2-Engel χ-group is CSE 2 and G be a non 2-Engel and residually χ-group Then (i) G is a CSE 2 (ii) If G is a 2-Engel transitive, then G is fully residually χ-group Mohammad Reza R Moghaddam

Finally, we show that Baumslag s Theorem is also true in the case of Theorem Let G be a residually free group Then G is fully residually free if and only if G is 2-Engel transitive Mohammad Reza R Moghaddam

Definition A Lie algebra L is commutative transitive (henceforth CT), if [x, y] = 0 and [y, z] = 0 imply that [x, z] = 0, for any non-zero elements x, y, z in L The property of CT is clearly subalgebra closed, yet it is not quotient closed, as every free Lie algebra is CT (see I Klep and P Moravec (2010), Example 44 for more detail) Mohammad Reza R Moghaddam

The Frattini subalgebra Φ(L) of a Lie algebra L, is the intersection of all maximal subalgebras of L or it is L itself, when there are no maximal subalgebras we prove the following useful result Theorem Every non-abelian Lie algebra L with non-zero Frattini subalgebra is CT Mohammad Reza R Moghaddam

The following lemma is needed to prove the main theorem of this section Lemma ( Bokut and Kukin (1994), Lemma 4162 ) A Lie algebra L is fully residually free if and only if, for every two linearly independent elements x 1 and x 2 in L, there exists a homomorphism ϕ from the Lie algebra L into a free Lie algebra F such that the elements ϕ(x 1 ) and ϕ(x 2 ) are linearly independent in F Mohammad Reza R Moghaddam

Now, using the previous lemma, it can be shown that Baumslag s Theorem is also held for free Lie algebras Theorem Let L be a residually free Lie algebra Then L is fully residually free, if and only if L is CT Mohammad Reza R Moghaddam

Thanks For Your Attention Mohammad Reza R Moghaddam

[1] Bokut, L A and Kukin, G P, Algorithmic and Combinatorial Algebra, Kluwer Academic Publisher, 1994 [2] Ciobanu, L, Fine, B, and Rosenberger, G, Classes of groups generalizing a theorem of Benjamin Baumslag, To Appear-Comm in Alg [3] Klep, I and Moravec, P, Lie algebras with abelian centralizers, Algebra Colloq 17 4, (2010), 629-636 Mohammad Reza R Moghaddam

[4] Moghaddam, M R R, Rosenberger, G and Rostamyari, M A, Some properties of n-commutative transitive groups, Submitted [5] Moghaddam, M R R and Rostamyari, M A, 2-Engelizer subgroup of a, Bull Korean Math Soc 53 no 3, (2016), 657 665 [6] Weisner, L, Groups in which the normaliser of every element except identity is abelian, Bull Amer Math Soc 31 (1925), 413-416 Mohammad Reza R Moghaddam