Prove that in an infinite cyclic group order of every element ($\ne e$) is infinite.
I have tried proving this way : If there exists an element of finite order, then it must generate a finite subgroup of $G.$ But this is not possible as by Lagrange's theorem, order of a subgroup must divide order of a group, which is not true here as order of $G$ is infinity.
Is this correct and complete ?
On
It doesn't quite work, as there are infinite groups with elements of (any possible) finite order.
Instead, think that every element of your infinite cyclic group $\langle g \rangle$ is of the form $x = g^k$ for some $k$. What happens if $1 = x^ n = (g^k)^n$ for some $n > 0$?
On
That is not correct because Lagrange's theorem presumes the group to be finite, and you can't use it that way, specifically because you're not using the fact that is cyclic.
For example $$G=\mathbb{R}-\lbrace 0 \rbrace$$
$G$ is a group under multiplication and $H=\lbrace 1,-1\rbrace$ is a finite subgroup.
You're correct trying to use negation: if a element generates a finite subgroup, the it means that $g^k=1$ for some $k$. Is that possible if the group is cyclic?
On
If you mean the infinite cyclic group $G := \{x^n | n\in \mathbb{Z}\}$. Then the proof would be by contradiction. Suppose there exists an element $y(\neq 1)\in G$ such that $y^n = 1$. But $y = x^m$ for some $m\in \mathbb{Z}$. Hence $1= y^n = (x^m)^n = x^{mn}$. But this implies G is finite, as then $G = \{x^{-mn+1},\ldots, x^{mn-1}\}$. Contradiction!
This proof is invalid, since some infinite groups have finite subgroups. E.g. $\mathbb{Z}_2 \times \mathbb{Z}$ has the subgroup $\mathbb{Z}_2 \times \{0\}$.
In an infinite cyclic group the elements are $$\{\ldots,g^{-3},g^{-2},g^{-1},1,g,g^2,g^3,\ldots,\}$$ for some generator $g$.
If $h \neq 1$ is in this group, then $h=g^k$ for some $k \in \mathbb{Z} \setminus \{0\}$. So, the subgroup generated by $h$ has the underlying set $$\{\ldots,g^{-3k},g^{-2k},g^{-k},1,g,g^{2k},g^{3k},\ldots,\}.$$