Polynomial-Time Isomorphism Test for Groups with No Abelian Normal Subgroups [chapter]

László Babai, Paolo Codenotti, Youming Qiao
2012 Lecture Notes in Computer Science  
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 upon over the past four decades. We demonstrate that the obstacle to efficient algorithms is the presence of abelian normal subgroups; we show this by giving a polynomial-time isomorphism test for groups without nontrivial abelian normal subgroups. This concludes a project started by the authors and J. A. Grochow
more » ... SODA 2011). Two key new ingredient are: (a) an algorithm to test permutational isomorphism of permutation groups in time, polynomial in the order and simply exponential in the degree; (b) the introduction of the "twisted code equivalence problem," a generalization of the classical code equivalence problem by admitting a group action on the alphabet. Both of these problems are of independent interest.
doi:10.1007/978-3-642-31594-7_5 fatcat:qkmh5ql4mraoxaaird72svte54