On Commuting Automorphisms of p-Groups |
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Authors: | Fatemeh Vosooghpour |
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Affiliation: | Department of Mathematics , Alzahra University , Vanak , Tehran , Iran |
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Abstract: | Let G be a group. If the set 𝒜(G) = {α ∈Aut(G) | xα(x) = α(x)x, for all x ∈ G} forms a subgroup of Aut(G), then G is called 𝒜(G)-group. We show that the minimum order of a non-𝒜(G) p-group is p 5 for any prime p. We also find the smallest group order of a non-𝒜(G) group. This is related to a question introduced by Deaconescu, Silberberg, and Walls [4 Deaconescu , M. , Silberberg , Gh. , Walls , G. ( 2002 ). On commuting automorphisms of groups . Arch. Math 79 : 423 – 429 .[Crossref] , [Google Scholar]]. Moreover, we prove that for any prime p and for all integer n ≥ 5, there exists a non-𝒜(G) group of order p n . |
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Keywords: | Commuting automorphism Extraspecial p-group Generalized extraspecial p-group p-group |
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