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**tttcomrader** Prove that the notion of group isomorphism is transitive.

My proof so far:

Suppose G, H, and K are groups.

Let the function f be the isomorphism from G onto H, then $\displaystyle f(g_{1}g_{2})=f(h_{1})f(h_{2})$ with f is onto and 1-1.

That equation should be $\displaystyle f(g_{1}g_{2})=f(g_{1})f(g_{2})$.

Now let the function t be the isomorphism from H onto K, then $\displaystyle t(h_{1}h_{2})=t(k_{1})t(k_{2})$ with t is onto and 1-1.

That equation should be $\displaystyle t(h_{1}h_{2})=t(h_{1})t(h_{2})$.

My plan is to get a function, say X, such that $\displaystyle X(g_{1}g_{2}) = X(k_{1}k_{2})$

That equation should be $\displaystyle X(g_{1}g_{2}) = X(g_{1})X(g_{2})$.

First I tried with X = f(t), but it doesn't really work out, so am I at least doing something right here?