The problem is to find all normal subgroups up to a given finite index n in a given finitely presented group G. Peter Dobcsanyi developed a method of doing this which essentially involved finding all subgroups of index up to n, using the standard low index subgroups algorithm, and then picking out the normal ones - of course, he used a number of tricks to avoid traversing branches of the search tree that could not possibly lead to normal subgroups.

We describe an alternative method, developed and implemented in Magma by David Firth, which runs considerably faster on the examples considered by Dobcsanyi, such as finding the normal subgroups of up to index 1500. We still use the low index subgroups algorithm, but only to find the normal subgroups of G with perfect quotients, for which we need the subgroups of G up to a much smaller index than n. This is because the normal subgroups arise as cores in G of the subgroups H of G found by low index subgroups. Normal subgroups with soluble quotients are found by using the p-quotient algorithm. The method works recursively, and for each normal subgroup H of G, we compute a presentation of H and, if necessary, apply the algorithm to H up to index n/ | G : H|. New normal subgroups are also found as intersections of existing ones.

It has been succesfully applied for n up to 50000 in some examples.