This is an exercise from Hungerford chapter 2 section 3. Here $G $ is said to satisfy the ACC and DCC. Then how should I apply the Krull Schmidt theorem to prove the proposition in the title? I tried to show that $G×G $ itself satisfies the ACC and DCC but to no avail. Could anyone help me?
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https://math.stackexchange.com/questions/1122839/when-are-two-direct-products-of-groups-isomorphic – Alex Youcis Apr 01 '19 at 08:51
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No it is different. I solved the case shown in the link. – Keith Apr 01 '19 at 08:57
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1Hey, sorry. I meant to link this: https://math.stackexchange.com/questions/779798/g-times-g-cong-h-times-h-longrightarrow-g-cong-h-for-g-acc-and-dcc?rq=1 – Alex Youcis Apr 01 '19 at 08:58