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Section4.1Cyclic groups

Often a subgroup will depend entirely on a single element of the group; that is, knowing that particular element will allow us to compute any other element in the subgroup.


If we are using the “+” notation, as in the case of the integers under addition, we write \(\langle a \rangle = \{ na : n \in {\mathbb Z} \}\text{.}\)

For \(a \in G\text{,}\) we call \(\langle a \rangle \) the cyclic subgroup generated by \(a\text{.}\) If \(G\) contains some element \(a\) such that \(G = \langle a \rangle \text{,}\) then \(G\) is a cyclic group. In this case \(a\) is a generator of \(G\text{.}\) If \(a\) is an element of a group \(G\text{,}\) we define the order of \(a\) to be the smallest positive integer \(n\) such that \(a^n= e\text{,}\) and we write \(|a| = n\text{.}\) If there is no such integer \(n\text{,}\) we say that the order of \(a\) is infinite and write \(|a| = \infty\) to denote the order of \(a\text{.}\)

The groups \({\mathbb Z}\) and \({\mathbb Z}_n\) are cyclic groups. The elements 1 and \(-1\) are generators for \({\mathbb Z}\text{.}\) We can certainly generate \({\mathbb Z}_n\) with 1 although there may be other generators of \({\mathbb Z}_n\text{,}\) as in the case of \({\mathbb Z}_6\text{.}\)

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Figure4.1.8Subgroups of \(S_3\)