We define a cyclic group to be one all of whose elements can be written as "powers" of a single element, so G is cyclic if ##G= \{a^n ~|~ n \in \mathbb{Z} \}## for some ##a \in G##. Is it true that in this case, ##G = \{ a^0, a^1, a^2, ... , a^{n-1} \}##? If so, why? And why do we write a cyclic group as ##\{a^n ~|~ n \in \mathbb{Z} \}##, where n is allowed to be any integer, except just those 0 through n - 1?(adsbygoogle = window.adsbygoogle || []).push({});

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# I Nature of cyclic groups

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