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Group Extensions And Automorphism Group Rings

John Martino and Stewart Priddy

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 {\it 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.

Homology, Homotopy and Applications, Vol. 5(2003), No. 1, pp. 53-70

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