  In recent work, Rosenbaum and Wagner showed that isomorphism of explicitly
listed $p$-groups of order $n$ could be tested in $n^{\frac{1}{2}\log_p n +
O(p)}$ time, roughly a square root of the classical bound. The $O(p)$ term is
entirely due to an $n^{O(p)}$ cost of testing for isomorphisms that match fixed
composition series in the two groups. We focus here on the
fixed-composition-series subproblem and exhibit a polynomial-time algorithm
that is valid for general groups. A subsequent paper will construct canonical
forms within the same time bound.
