## Proceedings of the Japan Academy, Series A, Mathematical Sciences

### On automorphisms of some $p$-groups

#### Abstract

A group $G$ is said to enjoy Hasse principle'' if every local coboundary of $G$ is a global coboundary. Let $G$ be a non-Abelian finite $p$-group of order $p^m$, $p$ prime and $m > 4$ having a normal cyclic subgroup of order $p^{m-2}$ but having no element of order $p^{m-1}$. We prove that $G$ enjoys Hasse principle'' if $p$ is odd but in the case $p = 2$, there are fourteen such groups twelve of which enjoy Hasse principle'' but the remaining two do not satisfy Hasse principle''. We also find all the conjugacy preserving outer automorphisms for these two groups.

#### Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 4 (2002), 46-50.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148392731

Digital Object Identifier
doi:10.3792/pjaa.78.46

Mathematical Reviews number (MathSciNet)
MR1900023

Zentralblatt MATH identifier
1062.20021

Subjects
Primary: 20D45: Automorphisms 20D15: Nilpotent groups, $p$-groups

#### Citation

Kumar, Manoj; Vermani, Lekh Raj. On automorphisms of some $p$-groups. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 4, 46--50. doi:10.3792/pjaa.78.46. https://projecteuclid.org/euclid.pja/1148392731

#### References

• Burnside, W.: On the outer automorphisms of a group. Proc. London Math. Soc. (2), 11, 40–42 (1913).
• Burnside, W.: Theory of Groups of Finite Order. Dover Publication, Inc., Mineola, New York (1955).
• Kumar, M., and Vermani, L. R.: “Hasse principle” for extraspecial $p$-groups. Proc. Japan Acad., 76A, 123–125 (2000).
• Kumar, M., and Vermani, L. R.: “Hasse principle” for groups of order $p^4$. Proc. Japan Acad., 77A, 95–98 (2001).
• Ono, T.: “Shafarevich-Tate” sets for profinite groups. Proc. Japan Acad., 75A, 96–97 (1999).
• Ono, T., and Wada, H.: “Hasse principle” for free groups. Proc. Japan Acad., 75A, 1–2 (1999).
• Ono, T., and Wada, H.: “Hasse principle” for symmetric and alternating groups. Proc. Japan Acad., 75A, 61–62 (1999).
• Wall, G. E.: Finite groups with class-preserving outer automorphisms. J. London Math. Soc., 22, 315–320 (1947).