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On The Orders of Automorphism Groups of Finite Groups
Author(s) -
Bray John N.,
Wilson Robert A.
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s002460930400400x
Subject(s) - euler's totient function , mathematics , mathematics subject classification , automorphism , automorphism group , euler's formula , combinatorics , pure mathematics , discrete mathematics , mathematical analysis
In the Kourovka notebook , Deaconescu asks whether |Aut G |⩾Φ(| G |) for all finite groups G , where Φ denotes the Euler totient function, and whether G is cyclic whenever |Aut G |. Both questions are answered in the negative in this paper. Moreover, |Aut G |/Φ(| G |) can be made arbitrarily small. 2000 Mathematics Subject Classification 20E36 (primary), 20F28 (secondary).