I was looking at this question on the odd torsion points of elliptic curves. The accepted answer states that the "coprime-to-$p$" points form a subgroup (so, in particular, the $C_{oddtors}(\mathbb{Q})$ would form a subgroup). However, I don't understand why this is a subgroup.
More generally, given an abelian group, do the elements of odd order form a subgroup? I cannot convince myself that the sum of two elements with odd order will necessarily have odd order.
Say we have an abelian group with two elements $a, b$ of order $o_a, o_b$ respectively, and let $d = \operatorname{lcm}(o_a, o_b)$. Then $d\cdot(a+b) = 0$, so the order of $a+b$ must divide $d$. If $o_a$ and $o_b$ are both odd, then $d$ is odd, and thus the order of $a + b$ must also be odd.
There is nothing special about "odd" here, we can use "coprime to $n$" for any natural number $n$ in place of "odd".
In a finite abelian group $G$, the order of the product $ab$ (additively written the sum) is a divisor of $$ \operatorname{lcm}(\operatorname{ord}(a),\operatorname{ord}(b)). $$ The lcm of two odd numbers is odd. Note that the torsion group of an elliptic curve over $\Bbb Q$ is a finite abelian group.
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