One way is to assume that there is an isomorphism between them and use the
fact that an isomorphism is one-to-one, onto and operation preserving to
produce a contradiction. (You cannot any property except these three. Thus
you cannot choose a particular mapping between the two groups). See
Examples 5 and 6 in Chapter 6 for instances of this approach. A second way
is to show that one of the groups has some group-theoretic property that
the other group does not have. One of the most commons ways to do this is
to show that the two groups have a different number of elements of some
specific order (order 2 is often a good choice). For example, 
S<sub>4</sub> and D<sub>12</sub>
are both non-Abelian and have order 24 but D<sub>12</sub> has an element 
of order 12
whereas S<sub>4</sub> does not.