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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 ni of a randomly chosen element gi in
G/ < g1,..., gi - 1 >. This probabilistically determines
the order of G as the product of the ni.
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.
Kapil Hari Paranjape