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I came across something while looking up abstract algebra which said "Let G be a group and suppose there is an element $a$ in G which generates a cyclic subgroup of order 2 and is the unique such element." Could some please give me an example of such a group and element, and/or elaborate on this topic?

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The easiest one is in $\mathbb{Z}_4$. The element 2 generates the subgroup $\{2,0\}$, and it's the only one, since all the other elements have order 4 or 1.

More generally, every $\mathbb{Z}_{2d}$, has this property, since the element $d$ generates $\{0,d\}$

Given $G$ finite group with odd order, then $\mathbb{Z}_{2}\times G$ is also ok.

switching to theory, we have that $a$ must be the only element in $G$ with order 2, so it is invariant by conjugation. It means that $H=\{0,a\}$ is a normal subgroup of $G$. If $G$ is a finite group, and $H$ is a $2$- Sylow, then $G=H\times G'$.

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Consider $\mathbb{Z}/4\mathbb{Z}$. $2$ is the unique element that generates the cyclic subgroup $\{0,2\}$.

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