Alladi Ramakrishnan Hall

Orbit Of Pairs in Finite Modules Over Discrete Valuation Rings and Permutation Representations

C. P. Anilkumar

IMSc

We study the action of the automorphism group of a finite abelian group on pairs of elements in the group (and more generally, finite torsion modules over discrete valuation rings). When the abelian group is $p$-primary, it is determined (up to isomorphism) by a partition $\lambda$. If $\lambda$ is fixed, and $p$ is allowed to vary, we show that the number of orbits, and the cardinality of each orbit of pairs, is a polynomial in $p$ with integer coefficients.

We use our description of orbits of pairs to study the permutation representation of the automorphism group on an orbit in the abelian group.
We prove that the permutation representation on certain orbits is multiplicity free.