The normalizers of free subgroups in free burnside groups of odd period n ≥ 1003

Atabekyan, V. S.

doi:10.1007/s10958-010-9885-1

Cited by 8 publications

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References 11 publications

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“…According to one other theorem of S.I.Adyan (see [1,Theorem 3.21]) for m > 1 and odd periods n ≥ 665 the center of B(m, n) is trivial and hence, B(m, n) is isomorphic to the inner automorphism subgroup Inn(B(m, n)) of the automorphism group Aut(B(m, n)). Other results on automorphisms and monomorphisms of the groups B(m, n) appeared relatively recently in [2]- [10]. In this paper we describe the automorphism groups of the endomorphism semigroups of B(m, n) for odd exponents n ≥ 1003.…”

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confidence: 94%

The automorphisms of endomorphism semigroups of free burnside groups

Atabekyan

2015

Int. J. Algebra Comput.

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In this paper we describe the automorphism groups of the endomorphism semigroups of free Burnside groups B(m, n) for odd exponents n ≥ 1003. We prove, that the groups Aut(End(B(m, n))) and Aut(B(m, n)) are canonically isomorphic. In particular, if the groups Aut(End(B(m, n))) and Aut(End(B(k, n))) are isomorphic, then m = k. 1The free Burnside group B(m, n) is the free group of rank m of the variety B n of all groups satisfying the identity x n = 1. The group B(m, n) is isomorphic to the quotient group of the absolutely free group F m of rank m by normal subgroup F n m generated by all n-th powers of its elements. It is well known (see [1, Theorem 2.15] that for all odd n ≥ 665 and rank m > 1 the group B(m, n) is infinite (and even has exponential growth). According to one other theorem of S.I.Adyan (see [1, Theorem 3.21]) for m > 1 and odd periods n ≥ 665 the center of B(m, n) is trivial and hence, B(m, n) is isomorphic to the inner automorphism subgroup Inn(B(m, n)) of the automorphism group Aut (B(m, n)). Other results on automorphisms and monomorphisms of the groups B(m, n) appeared relatively recently in [2]- [10]. In this paper we describe the automorphism groups of the endomorphism semigroups of B(m, n) for odd exponents n ≥ 1003. In particular, we prove, that the groups Aut(End(B(m, n))) and Aut(End(B(k, n))) are isomorphic if and only if m = k. This is a particular problem about End(A), for A a free algebra in a certain variety, was raised by B.I.Plotkin in [14]. Analogous problems for End(F ) with F a finitely generated free group or free monoid were solved by Formanek in [12] and Mashevitzky and Schein in [13] respectively.

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confidence: 94%

The automorphisms of endomorphism semigroups of free burnside groups

Atabekyan

2015

Int. J. Algebra Comput.

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“…If n ≥ 1039 is an arbitrary odd number and a and b are two non commuting elements of the group B(2, n), then for some p = 2 k , where 0 ≤ k ≤ 9, the words u(a p , b), v(a p , b) freely generate a free Burnside subgroup of the group B(2, n), where the words u(x, y) and v(x, y) are defined by equalities (1) and (2). P r o o f. The starting point for proving Theorem 2 is the following assertion, proved in [7] (see also [8,9]).…”

Section: P R O O F Let For Some Word T We Havementioning

confidence: 97%

Powers of Subsets in Free Periodic Groups

Atabekyan¹,

Aslanyan

2022

Proc. YSU A: Phys. Math. Sci.

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It is proved that for every odd $n \ge 1039$ there are two words $u(x, y), v(x,y)$ of length $\le 658n^2$ over the group alphabet $\{x,y\}$ of the free Burnside group $B(2 ,n),$ which generate a free Burnside subgroup of the group $B(2,n)$. This implies that for any finite subset $S$ of the group $B(m,n)$ the inequality $|S^t|>4\cdot 2.9^{[\frac{t}{658s^2}]}$ holds, where $s$ is the smallest odd divisor of $n$ that satisfies the inequality $s\ge1039$.

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“…In [11], it was shown that Adyan's conjecture is valid for all odd n ≥ 1003. For prime 665 < n ≤ 997, the question is still open.…”

Section: Adyan's Conjecturementioning

confidence: 99%

Automorphism Groups and Endomorphism Semigroups of Groups B(m, n)

Atabekyan

2015

Algebra Logic

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A free group of rank m in the variety B n of all groups satisfying the identical relation x n = 1 is denoted B(m, n) and is called a free periodic or free Burnside group of exponent n and rank m. The group B(m, n) is a quotient group of a free group F m of rank m with respect to a normal subgroup F n m generated by all possible nth powers of elements of F m . A well-known theorem of S. I. Adyan [1] asserts that for all odd n ≥ 665 and for m > 1, B(m, n) is an infinite group. In [1], also, it was proved that for all odd n ≥ 665, every commutative subgroup of B(m, n) is cyclic, the center of B(m, n) is trivial, and the group B(2, n) contains isomorphic copies of free periodic groups B(m, n) of any finite rank m ≥ 1.Further investigations showed that the subgroup structure of B(m, n) is in fact saturated with free Burnside subgroups (see [2]). In 1980 Adyan posed the following: Question 1 [3, Question 7.1]. It is known that free periodic groups B(m, n) of prime exponent n > 665 possess many properties that are similar to properties of absolutely free groups. Is it true that all proper normal subgroups of B(m, n) are not free periodic groups?The answer to this question appeared to be, to a certain extent, allied to the question formulated by A. Yu. Ol'shanskii in 1982.Question 2 [3, Question 8.53]. Let n be a sufficiently large odd number.

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The normalizers of free subgroups in free burnside groups of odd period n ≥ 1003

Cited by 8 publications

References 11 publications

The automorphisms of endomorphism semigroups of free burnside groups

The automorphisms of endomorphism semigroups of free burnside groups

Powers of Subsets in Free Periodic Groups

Automorphism Groups and Endomorphism Semigroups of Groups B(m, n)

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