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According to the introductory abstract algebra, it says that $g^n =e$ where $g$ is an element of some finite group. Then, it talks about the subgroup $\langle g^d \rangle$. What is it exactly, and what would be the elements of this group? (where $d$ is some integer.)

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    @William modified $n$ to $d$.2012-08-18

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The notation $\langle g^d \rangle$ denotes the cyclic group generated by the element $g^d$. More explicitly, we begin with the element $g^d$. Then we consider $g^d \cdot g^d = g^{2d}$. We continue generating more elements of the subgroup and have the set $\{g^d, g^{2d}, g^{3d}, \cdots\}$ with the group operation coming from the finite group.

Of course it requires proof that this set generates a group. I will leave this as a fun exercise for you, but please ask if you need more help.