Why isomorphisms

I often wondered why isomorphisms were important. And if you haven’t done the same, maybe you have studied Abstract Algebra rather passively. We shall explore this question today.

We know (or assume for the moment) that $\Bbb{Z}_6$ is isomorphic to $\Bbb{Z}_2\times\Bbb{Z}_3$ . The mappings are $n\to(n\mod 2,n\mod 3)$ . You can verify all this for yourself.

Now let us suppose we don’t know what $4+5$ is, where $4,5\in\Bbb{Z_6}$ . We can determine $4+5$ by studying the manipulations of $(4\mod 2,4\mod 3)=(0,1)$ and $(5\mod 2,5\mod 3)=(1,2)$ .

We have $(0,1)+(1,2)=(1,0)$ . We also know that the element which had mapped to $(1,0)$ had been $3$ . Hence, $4+5=3$ .

Isomorphisms help us study the properties of one algebraic structure, provided we know the properties of the algebraic structure isomorphic to it. It is a tremendously useful concept in the fuzzy world of mathematics, where studying even a fraction of the concepts we ourselves have invented often proves to be an arduous task.

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