# Isomorphism

Let H be a group and tau_1 : H ---->G_1, tau_2 : H ----> G_2, ... , tau_n : H -----> G_n homomorphism with this property: whenever G is a group and g_1 : G ---->G_1, g_2 : G ----G_2, ..., g_n : G ----> G_n are homomorphism, then there exists a unique homomorphism g* : G ----> H such that (tau)_i â?¢ g* = g_i for every i. Prove that H is isomorphic to G_1 x G_2 x ... x G_n.

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#### Solution Preview

Proof:

We consider G = G_1 x G_2 x ... x G_n and g_i: G --> G_i is a natural injection.

For any x = (x_1, x_2, ..., x_n) in G, we have g_i(x) = x_i.

From the condition, we know that we have a homomorphism g*: G ...

#### Solution Summary

This solution shows the steps to proving that a given variable is isomorphic.

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