11.4 Other problems

Once sufficiently many relations between a collection of generators of the group G has been found, we have seen above that one can ``read off'' many of the properties of G such as its isomorphism class. The above algorithms for finding relations can be applied directly to solving other problems or questions regarding the group G.

To find the order of the group G we use the above techniques to find (in succession) the order n_i of a randomly chosen element g_i in the group G/ < g₁,..., g_{i - 1} >. This probabilistically determines the order of G as the product of the n_i.

Given h and g in G and the fact that h is a power of g we determine this power (the Discrete Log problem) by finding a minimal relation between g and h.