The below-given problem is in black box setting means input is given by set of generators.

Given an abelian $p$-group $A$ and two matrices $U_1$ and $U_2$ in $R(A)$ such that the order of $U_1$ and $U_2$ are coprime with $p$, output an element $U \in R(A)$ such that $UU_1=U_2U$ if such an element exists.

$R(A)$ denotes the set of all automorphisms of $A$.

The brute force way seems to give a $\Theta(|A|^{\log |A|})$ bound.

Question: Is the above problem solvable in the polynomial time?


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