## Homology, Homotopy and Applications

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

### Group extensions and automorphism group rings

John Martino and Stewart Priddy

#### 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

**Permanent link to this document**

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