Applications of Groups

We now turn to some applications of group theory. The first application makes use of the observation that computing in ℤ can be replaced by computing in ℤ_n, if n is sufficiently large; ℤ_n can be decomposed into a direct product of groups with prime power order, so we can do the computations in parallel in the smaller components. In §25, we look at permutation groups and apply these to combinatorial problems of finding the number of “essentially different” configurations, where configurations are considered as “essentially equal” if the second one can be obtained from the first one, e.g., by a rotation or reflection.