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AUTOMORPHISMS OF METABELIAN PRIME POWER ORDER GROUPS OF MAXIMAL CLASS

Part of: Representation theory of groups

Published online by Cambridge University Press:  01 April 2008

S. FOULADI  and
R. ORFI
S. FOULADI*
Affiliation:
Faculty of Mathematics, University for Teacher Education, 599 Taleghani Avenue, Tehran 15618, Iran (email: [email protected])
R. ORFI
Affiliation:
Faculty of Mathematics, University for Teacher Education, 599 Taleghani Avenue, Tehran 15618, Iran (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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Let G be a p-group of maximal class of order pn. It is shown that the order of the group of all automorphisms of G centralizing the Frattini quotient takes the maximum value p2n−4 if and only if G is metabelian. A structure theorem is proved for the Sylow p-subgroup, Autp(G), of the automorphism group of G when G is metabelian. For p=2, Aut2(G) is the full automorphism group of G. For p=3, we prove a structure theorem for the full automorphism group of G.

Keywords

finite p-groupmaximal classmetabelian groupautomorphism group

MSC classification

Secondary: 20D45: Automorphisms 20D15: Nilpotent groups, $p$-groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 2 , April 2008 , pp. 261 - 276
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Adney, J. E. and Yen, T., ‘Automorphisms of a p-group’, Illinois J. Math. 9 (1965), 137–143.CrossRefGoogle Scholar
[2]Baartmans, A. H. and Woeppel, J. J., ‘The automorphism group of a p-group of maximal class with an Abelian maximal subgroup’, Fund. Math. 93(1) (1976), 41–46.CrossRefGoogle Scholar
[3]Blackburn, N., ‘On a special class of p-groups’, Acta Math. 100 (1958), 45–92.CrossRefGoogle Scholar
[4]Caranti, A. and Mattarei, S., ‘Automorphisms of p-groups of maximal class’, Rend. Sem. Mat. Univ. Padova 115 (2006), 189–198.Google Scholar
[5]Caranti, A. and Scoppola, C. M., ‘A remark on the orders of p-groups that are automorphism groups’, Boll. Unione Mat. Ital. A (7) 4(2) (1990), 201–207.Google Scholar
[6]Caranti, A. and Scoppola, C. M., ‘Endomorphisms of two-generated metabelian groups that induce the identity modulo the derived subgroup’, Arch. Math. 56(3) (1991), 218–227.CrossRefGoogle Scholar
[7]Huppert, B., Endliche gruppen, Vol. 1 (Springer, Berlin, 1967).CrossRefGoogle Scholar
[8]Juhász, A., ‘The group of automorphisms of a class of finite p-groups’, Trans. Amer. Math. Soc. 270(2) (1982), 469–481.Google Scholar
[9]Leedham-Green, C. R. and McKay, S., The structure of groups of prime power order (Oxford University Press, Oxford, 2002).CrossRefGoogle Scholar
[10]Malinowska, I., ‘Finite p-groups with few p-automorphisms’, J. Group Theory 4 (2001), 395–400.CrossRefGoogle Scholar