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# Homology, Homotopy and Applications

## Volume 5 (2003)

### Number 1

### Group extensions and automorphism group rings

Pages: 53 – 70

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n1.a3

#### Authors

#### 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.

#### Keywords

automorphism group, extension class, semi-simple quotient, stable splittings

#### 2010 Mathematics Subject Classification

Primary 20J06. Secondary 55P42.