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I'm feeling quite stupid at the moment. The direct product $\,\Bbb Z/5\Bbb Z\times\Bbb Z/5\Bbb Z\,$ should have order 25, right?

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ofcourse 5x5 25 elements. – Ram Jan 30 '13 at 12:46
up vote 2 down vote accepted

If by Z/Z5 you mean $\Bbb Z/5\Bbb Z$, then yes, you are correct. Any time we use a direct product $A\times B$, there will be $|A|\cdot |B|$ elements.

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I wanted to edit the title, but the OP wrote it as Z5. So I stopped. – Babak S. Jan 30 '13 at 12:34
@BabakSorouh: I agree, I wasn't sure if this answered his question or not, but I thought it might be worth putting anyway. – Clayton Jan 30 '13 at 12:37
Ah yes sorry, that's what I meant. Thanks for the quick answer. – Horstenson Jan 30 '13 at 12:38
@Horstenson: You're welcome. – Clayton Jan 30 '13 at 12:40

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