Normal Automorphisms of Free Burnside Groups of Large Odd Exponents

Abstract. An automorphism α of a group G is normal if it fixes every normal subgroup of G setwise. We give an algebraic description of normal automorphisms of relatively hyperbolic groups. In particular, we show that for any such group G, Inn(G) has finite index in the subgroup Aut n (G) of normal automorphisms. If, in addition, G is non-elementary and has no finite non-trivial normal subgroups, then Aut n (G) = Inn (G). As an application, we show that Out(G) is residually finite for every finitely generated residually finite group G with infinitely many ends.

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“…Evidently there is an integer n ≥ N such that g n , h n / ∈ Ξ. Our assumption (9) implies that there is …”

Section: By Lemma 24 G Is Hyperbolic Relative To {Hmentioning

confidence: 98%

Normal automorphisms of relatively hyperbolic groups

Minasyan

Osin

2010

Trans. Amer. Math. Soc.

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“…The equality Inn (F ) = Aut N (F ) was stated by various authors for different interesting classes of groups. In [12], it was proved that for sufficiently large odd n (n > 10 78 ), an automorphism ϕ of a free Burnside group B(m, n) is outer iff there exists a normal subgroup N ¡ B(m, n) the quotient group with respect to which is a simple non-Abelian group and ϕ(N ) = N . This result was strengthened and extended to all odd exponents n ≥ 1003 in [13].…”

Section: Normal Automorphisms Of Groups B(m N)mentioning

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 question of study of automorphisms of free Burnside groups was stated by Ol'shanskii in the Kourovka Notebook [7]. The first results were obtained by Cherepanov in [4,5] and by Atabekyan in [2,3]. In paper [4] it was proved that the Fibonacci morphism is an automorphism of infinite order of free Burnside groups for all odd n > 10 10 and even n = 16k ≥ 8000.…”

Section: Introductionmentioning

confidence: 99%