Algebra Solutions: Characterize the elements of cyclic subgroups

Let $G$ be a group, and let $x \in G$ be an element of finite order; say $|x| = n$ . Use the Division Algorithm to show that any integral power of $x$ is equal to one of the elements in the set $A = \{ 1, x, x^2, \ldots, x^{n-1} \}$ . Conclude that $A$ is precisely the set of distinct elements of the cyclic subgroup of $G$ generated by $x$ .

Let $k \in \mathbb{Z}$ . By the Division Algorithm, there exist unique integers $(q,r)$ such that $k = qn + r$ and $0 \leq r < |n|$ . Thus $x^k = x^{qn + r} = (x^n)^q x^r = x^r$ , where $x^r \in A$ .
Hence the cyclic subgroup of $G$ generated by $x$ is $A$ ; moreover, by a previous result, the elements of $A$ are all distinct.

Algebra Solutions

Characterize the elements of cyclic subgroups

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