On the Complexity of Group Isomorphism

The group isomorphism problem consists in deciding whether two groups G and H given by their multiplication tables are isomorphic. An algorithm for group isomorphism attributed to Tarjan runs in time n, c.f. [Mil78]. Miller and Monk showed in [Mil79] that group isomorphism can be many-one reduced to isomorphism testing for directed graphs. For groups with n elements, the graphs have valence at least n. We many-one reduce group isomorphism onto graph isomorphism, such that the valence of the graphs is at most d+ 1 if d is the size of the largest factor group in a composition series of the groups. Combining this with the fact that isomorphism testing for graphs of valence d can be solved in time n, p-group isomorphism is in time n for a constant c. We extend this algorithm to work for general groups, improving the exponential runtime behavior if factor groups of large size exist. Then we also consider the following simple group isomorphism algorithm, namely we compute a composition series and then guess coset representatives as generators. This algorithm has the same worst-case behavior as Tarjans algorithm. But if we combine both algorithms then we can show that group isomorphism is in time n logn/ log logn for a constant c.