Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-25T07:51:36.207Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Automorphism Group of a Finite p-Group with a Small Central Quotient

Published online by Cambridge University Press:  20 November 2018

Richard M. Davitt
Richard M. Davitt*
Affiliation:
University of Louisville, Louisville, Kentucky
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In recent years there has been considerable interest in the conjecture that |G| divides |Aut G| for all finite non-cyclic p-groups G of order greater than p2. In particular, the conjecture has been established for a considerable number of (not necessarily distinct) classes of finite p-groups ([6], [7], [8], [9], [15], [16]); additionally, results have been obtained, often using homological methods, which permit reductions in any attempt to establish the overall conjecture ([5], [10], [13], [15]). In the former case, the p-groups G have generally been regular p-groups (see, for example, [6]) and the prime p = 2 has either been excluded (see, for example, [8]) or treated as a special case (as in [9]).It is the purpose of this paper to establish the conjecture for the class of all p-groups G where |G: Z(G)| ≦ p4 with no restrictions on the prime p.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 5 , 01 October 1980 , pp. 1168 - 1176
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Adney, J. E. and Yen, T., Automorphisms of a p-group, Illinois J. Math. 9 (1965), 137–143.Google Scholar
2. Arganbright, D. E., The power-commutator structure of finite p-groups, Pacific J. Math. 29 (1969), 11–17.Google Scholar
3. Blackburn, N., On prime power groups with two generators, Proc. Cambridge Phil. Soc 54 (1958), 327–337.Google Scholar
4. Brisley, W. and Macdonald, I. D., Two classes of metabelian p-groups, Math. Z. 112 (1969), 5–12.Google Scholar
5. Buckley, J., Automorphism groups of'isoclinic p-groups, J. Lond. Math. Soc. 12 (1975), 37–44.Google Scholar
6. Davitt, R. M., The automorphism group of finite p-abelian p-groups, Illinois J. Math. 16 (1972), 76–85.Google Scholar
7. Davitt, R. M. and Otto, A. D., On the automorphism group of a finite modular p-group, Proc. Amer. Math. Soc. 35 (1972), 399–404.Google Scholar
8. Davitt, R. M. and Otto, A. D., On the automorphism group of a finite p-group with the central quotient metacyclic, Proc. Amer. Math. Soc. 80 (1971), 467–472.Google Scholar
9. Faudree, R., A note on the automorphism group of a p-group, Proc. Amer. Math. Soc. 19 (1968), 1379–1382.Google Scholar
10. Gaschütz, W., Nichtabelsche p-Gruppen besitzen äussere p-Automorphismen, J. Alg. 4 (1966), 1–2.Google Scholar
11. Hall, M. Jr. and Senior, J. K., The groups of order 2n (n ≧ 6) (The Macmillan Company, New York, 1964).Google Scholar
12. Hall, P., A contribution to the theory of groups of prime-power orders, Proc. London Math. Soc. 36 (1933), 29–95.Google Scholar
13. Hummel, K., The order of the automorphism group of a central product, Proc. Amer. Math Soc. 47 (1975), 37–40.Google Scholar
14. Huppert, B., EndlicheGruppen, I, Die Grundlehren der math. Wissenschaften, Band 134 (Springer-Verlag, Berlin and New York, 1967).Google Scholar
15. Otto, A. D., Central automorphisms of a finite p-group, Trans. Amer. Math. Soc. T25 (1966), 280–287.Google Scholar
16. Ree, R., The existence of outer automorphisms of some groups, II, Proc. Amer. Math. Soc. 9 (1958), 105–109.Google Scholar
17. Scott, W. R., Group theory (Prentice-Hall, Inc., Englewood Cliffs, NJ, 1964).Google Scholar