Some properties on $$\mathrm{IA_Z}$$ IA Z -automorphisms of groups
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Arabian Journal of Mathematics
Hamid Taheri · Mohammd Reza R. Moghaddam · Mohammad Amin Rostamyari
Some properties on IAZ -automorphisms of groups
Received: 19 September 2018 / Accepted: 7 May 2019 © The Author(s) 2019
Abstract Let G be a group and IA(G) denote the group of all automorphisms of G, which induce identity map on the abelianized group G ab = G/G . Also the group of those IA-automorphisms which fix the centre element-wise is denoted by IAZ (G). In the present article, among other results and under some condition we prove that the derived subgroups of finite p-groups, for which IAZ -automorphisms are the same as central automorphisms, are either cyclic or elementary abelian. Mathematics Subject Classification
20F28 · 20F18 · 20D45 · 20F14
1 Introduction and preliminaries An automorphism α of a group G is called IA-automorphism if x −1 α(x) ∈ G , for all x ∈ G. This concept was introduced by Bachmuch [1] in 1965. We remark the letters I and A as to remind the reader that are those automorphisms which induce identity on the abelianized group, G/G . Also, if x −1 α(x) ∈ Z (G) for all x ∈ G, then we say that α is a central automorphism, and if α preserves all conjugacy classes of G, then it is called a class preserving automorphism. The set of all such automorphisms are denoted by IA(G), AutZ (G) and AutC (G), respectively. These concepts were introduced and studied by Curran [2] in 2001 and Yadav [15,16] in 2009 and 2013. Clearly, the set of all IA-automorphisms, which fix the centre element-wise, forms a normal subgroup of IA(G) and is denoted by IAZ (G) (see [11,12] for more information). For any element x of a group G and automorphism α of G, the autocommutator of x and α is defined as follows: [x, α] = x −1 x α = x −1 α(x). H. Taheri · M. R. R. Moghaddam Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran E-mail: [email protected] M. R. R. Moghaddam · M. A. Rostamyari Department of Mathematics, Khayyam University, Mashhad, Iran E-mail: [email protected] M. R. R. Moghaddam (B) Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.BOX 1159, Mashhad 91775, Iran E-mail: [email protected]; [email protected]
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Now, using the above notation, we have the following: AutZ (G) = {α ∈ Aut(G) | [x, α] ∈ Z (G), ∀x ∈ G}, AutC (G) = {α ∈ Aut(G) | x α ∈ x G , ∀x ∈ G}, IAZ (G) = {α ∈ Aut(G) | [x, α] ∈ G , α(z) = z, ∀x ∈ G and z ∈ Z (G)}. One can easily check that any class preserving automorphism is an IA-automorphism, which fixes the centre element-wise and hence Inn(G) ≤ AutC (G) ≤ IAZ (G) ≤ IA(G) ≤ Aut(G), Z (Inn(G)) ≤ AutZ (G) ≤ Aut(G). The following example shows that every IAZ -automorphism is not necessarily inner automorphism. Example 1.1 Consider the group G = a, b, x | [a, x] = [b, x] = 1, [a, b] = x s , s = 1. ∼ Inn(G). The IAZ -automorphism α defined by ¯ = Clearly, G = x s , Z (G) = x and G/Z (G) = a, ¯ b α(a) = ax s , α(b) = bx s
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