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Automorphisms of certain p-groups (p odd).

Published online by Cambridge University Press:  17 April 2009

M.J. Curran
M.J. Curran
Affiliation:
Department of Maths and Statistics, University of Otago, P.O. Box 56, Dunedin, New Zealand.
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Abstract

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This paper shows that amongst the p-groups of order p5, where p denotes an odd prime, there is only one group whose automorphism group is again a p-group. This automorphism group has order p6 and it is shown that this is the smallest order a p-group may have when it occurs as an automorphism group. The paper also shows that all groups of order p5 have an automorphism of order 2 apart from the group above and three other related groups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 38 , Issue 2 , October 1988 , pp. 299 - 305
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Davitt, R.M., ‘On the automorphism group of a finite p-group with a small central quotient’, Canad. J. Math. (1980), 1168–1176.Google Scholar
[2]Flannery, D. and MacHale, D., ‘Some finite groups which are rarely automorphism groups I’, Proc. Roy. Irish Acad. Sect. A. 91 (1981), 209–215.Google Scholar
[3]Gorenstein, D., Finite Groups (Harper and Row, New York, 1968).Google Scholar
[4]Heineken, H. and Liebeck, H., ‘On p-groups with odd order automorphism groups’, Arch. Math. (Basel) 34 (1973), 465–471.Google Scholar
[5]James, R., ‘The groups of order p 6 (p an odd prime)’, Math. Comp. 34 (1980), 613–637.Google Scholar
[6]MacHale, D., ‘Some finite groups which are rarely automorphism groups II’, Proc. Roy. Irish. Acad. Sect. A. 83 (1983), 189–196.Google Scholar
[7]MacHale, D., ‘Characeristic elements in a group’, Proc. Roy. Irish Acad. Sect. A 86 (1986), 63–65.Google Scholar
[8]Ying, J.H., ‘On finite groups whose automorphism groups are nilpotent’, Arch. Math. (Basel) 29 (1977), 41–44.Google Scholar