How to show an isomorphism between groups?

[blahblah8724]

Is the only way to show an isomorphism between groups is to just define a map which has the isomorphism properties?

So for example for a group $G$ with order 15 to show that $G \cong C_3 \times C_5$ would I just have to define all the possible transformations to define the isomorphism?


Thanks!

 

[morphism]

Is the only way to show an isomorphism between groups is to just define a map which has the isomorphism properties?

In a sense, yes, but you don't have to define the map explicitly. You can use theorems about the structure theory of groups to help you out. These are theorems that say something along the lines of "if a group satisfies property P, then it must also satisfy property Q". So, for instance, a group of order 15 must have subgroups H and K of orders 3 and 5 (by Cauchy's theorem), and these subgroups must be normal (by Sylow's theorem), and their intersection must be trivial (by Lagrange's theorem), so G must be isomorphic to the internal direct product HxK (this follows from another structure theorem - which one?).

Of course, after making all these deductions, you can explicitly write down an isomorphism $G \to C_3 \times C_5$, but the point is you didn't have to start out by looking for such a map.