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Compute $\langle 5 \rangle$ in integers.

I thought the answer would have been $$\{5^n| \text{ for $n$ in integers }\}$$ However my teacher has marked it $$\{5*n|\text{ for $n$ in integers }\}$$ What have I done wrong?

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  • $\begingroup$ Teachers teaching on that subject are always right -- well, almost. So you've put the wrong question. It should read instead: “What have I done wrong?” Sapere aude! $\endgroup$ – Michael Hoppe Nov 7 '13 at 21:29
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The integers do not form a group under multiplication, for almost no element has an inverse under multiplication. The integers do form a group under addition, however.

So if we're viewing $\Bbb{Z}$ as a cyclic group in the usual sense, the operation is addition and

$$\langle 5 \rangle = \{..., -5-5, -5, 0, 5, 5 + 5, 5 + 5 + 5 , ...\} = \{5n : n \in \Bbb{Z}\}$$

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They do, however, form a group under addition, hence we get ⟨5⟩ = {5∗n| for n in integers }. Hope that helps!

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