Polynomial-time Isomorphism Test for Groups with Abelian Sylow Towers

Abstract

We consider the problem of testing isomorphism of groups of order n
given by Cayley tables. The trivial n^{log n} bound on the time
complexity for the general case has not been improved over the past
four decades. Recently, Babai et al. (following Babai  et al. in SODA
2011) presented a polynomial-time algorithm for groups without abelian
normal subgroups, which suggests solvable groups as the hard case for
group isomorphism problem. Extending recent work by Le Gall (STACS
2009) and  Qiao et  al.  (STACS  2011), in  this  paper we design  a
polynomial-time algorithm to test isomorphism for the largest class of
solvable groups yet, namely groups with abelian Sylow towers, defined
as  follows. A group G is said to possess a Sylow tower, if there
exists a normal series where each quotient is isomorphic to Sylow
subgroup of G. A group has an abelian Sylow tower if it has a Sylow
tower and all its Sylow subgroups are abelian. In fact, we are able
to compute the coset of isomorphisms of groups formed as coprime
extensions of an abelian group, by a group whose automorphism group is
known.

The  mathematical   tools  required  include   representation  theory,
Wedderburn's theorem  on semisimple  algebras, and M.E.  Harris's 1980
work on  p'-automorphisms of abelian  p-groups. We use tools  from the
theory of permutation group algorithms, and develop an algorithm for a
parameterized versin of  the graph-isomorphism-hard setwise stabilizer
problem, which may be of independent interest.