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A remark about central automorphisms of groups. (English) Zbl 1119.20036

An automorphism \(\alpha\) of a group \(G\) is called a central automorphism if it acts trivially on \(G/Z(G)\), where \(Z(G)\) is the centre of \(G\). Let \(\Gamma\) denote the group of central automorphisms of \(G\). In this paper the author proves the following result: Let \(Z\) be a central subgroup of finite exponent, \(n\), of the group \(G\) and suppose that \(G\) has no nontrivial direct factor contained in \(Z\). Then (i) \(\Gamma\) has finite exponent dividing \(n\); (ii) If \(k\) is the least positive integer such that \(n\) is not divisible by the \((k+1)\)-st power of any prime then \([Z,{_{2k-1}\Gamma}]=1\); (iii) \(\Gamma\) is nilpotent of class at most \(2k-1\), with \(k\) defined as in (ii). – The author gives examples to show that the bounds \(2k-1\) are actually attained in parts (ii) and (iii).

MSC:

20E36 Automorphisms of infinite groups
20F28 Automorphism groups of groups
20F50 Periodic groups; locally finite groups
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References:

[1] J.E. ADNEY - T. YEN, Automorphisms of a p-group, Illinois J. Math. 9 (1965), pp. 137-143. Zbl0125.28803 MR171845 · Zbl 0125.28803
[2] M.R. DIXON - M.J. EVANS, On groups with a central automorphism of infinite order, Proc. Amer. Math. Soc. 114 (1992), no. 2, pp. 331-336. Zbl0751.20026 MR1072334 · Zbl 0751.20026 · doi:10.2307/2159651
[3] O. MÜLLER, On p-automorphisms of finite p-groups, Arch. Math. (Basel) 32 (1979), no. 6, pp. 533-538. Zbl0417.20025 MR550318 · Zbl 0417.20025 · doi:10.1007/BF01238537
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