External Direct Product isomorphism

**tttcomrader** · September 30th 2007, 02:12 PM

Suppose that $G_{1} \approx G_{2}, H_{1} \approx H_{2}$ , prove that $G_{1} \oplus H_{1} \approx G_{2} \oplus H_{2}$

For this problem, I really don't have much clue, I feel I'm not too comfortable with the whole direct product thingy.

Thanks.

**Plato** · September 30th 2007, 02:42 PM

Did you understand this problem?
http://www.mathhelpforum.com/math-he...-products.html

**tttcomrader** · September 30th 2007, 06:25 PM

No, I do not yet know how to solve that other problem as well. But I was hoping that by understanding how to solve this one, I might be able to solve the other one.

**ThePerfectHacker** · October 1st 2007, 11:10 AM

Originally Posted by tttcomrader

Suppose that $G_{1} \approx G_{2}, H_{1} \approx H_{2}$ , prove that $G_{1} \oplus H_{1} \approx G_{2} \oplus H_{2}$

For this problem, I really don't have much clue, I feel I'm not too comfortable with the whole direct product thingy.

Thanks.

There exists a bijective homomorphism $\phi: G_1\mapsto G_2$ and $\psi: H_1\mapsto H_2$ by definition.

Now create $G_1\times H_1 = \{(g_1,h_1)|g_1\in G_1 \mbox{ and }h_1\in H_1$ and similarly $G_2\times H_2$ .

For $x\in G_1\times H_1$ , i.e. $x = (g_1,h_1)$ for some $g_1\in G_1$ and $h_1\in H_1$ . Define $\tau (x) = (\phi (g_1),\psi (h_1))$ .

Show that $\tau$ is a mapping from $G_1\times H_1$ into $G_2\times H_2$ . Now show it is one-to-one and onto. Finally conclude by showing it is a homomorphism between these groups. It is a straightforward computation.

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