## Homology, Homotopy and Applications

### Group extensions and automorphism group rings

#### Abstract

We use extensions to study the semi-simple quotient of the group ring $\mathbf{F}_pAut(P)$ of a finite $p$-group $P$. For an extension $E: N \to P \to Q$, our results involve relations between $Aut(N)$, $Aut(P)$, $Aut(Q)$ and the extension class $[E]\in H^2(Q, ZN)$. One novel feature is the use of the intersection orbit group $\Omega([E])$, defined as the intersection of the orbits $Aut(N)\cdot[E]$ and $Aut(Q)\cdot [E]$ in $H^2(Q,ZN)$. This group is useful in computing $|Aut(P)|$. In case $N$, $Q$ are elementary Abelian $2$-groups our results involve the theory of quadratic forms and the Arf invariant.

#### Article information

Source
Homology Homotopy Appl., Volume 5, Number 1 (2003), 53-70.

Dates
First available in Project Euclid: 13 February 2006

https://projecteuclid.org/euclid.hha/1139839926

Mathematical Reviews number (MathSciNet)
MR1989613

Zentralblatt MATH identifier
1033.20047

Subjects
Primary: 20J06: Cohomology of groups
Secondary: 20D45: Automorphisms 55P42: Stable homotopy theory, spectra

#### Citation

Martino, John; Priddy, Stewart. Group extensions and automorphism group rings. Homology Homotopy Appl. 5 (2003), no. 1, 53--70. https://projecteuclid.org/euclid.hha/1139839926