I need to prove proposition 1, and this is what I have.
Definition 1: If there exists a nonzero integer such that $a^m=e$, then the order of the element $'a'$ is defined to be the least positive integer $'n'$ such that $a^n=e$
If there does not exists a nonzero integer $m$ such that $a^m=e$, then the order is infinity
Definition 2: A subset $H$ of a group $G$ is a subgroup if
- $e\in H$
- if $x,y\in H$, then $xy\in H$.
- if $x\in H$, then $x^{-1}\in H$
Proposition 1: A noneempty subset $H$ of a finite goup $G$ is a subgroup if and only if $H$ is closed
proof: $\Rightarrow )$ Let H be a noneempty set of a finite grout $G$, then we know by definition 2(2) that $H$ is closed under group multiplication.
$\Leftarrow $): $H$ is finite therefore if $a\in H$, then $\text{ord}(a)=n$. Thus $e=a^n\in H$. We have shown that $e\in H$. If we can show that for every $h\in H$ that $h^{-1}\in H$, then we can conclude that $H$ is a subgroup of $G$. So let $h\in H$, then we know that there exists a element $k\in \Bbb{N}$ such that $h^k=e$. But then $h^{k-1}=h^{-1}$ and $h^{k-1} h=h h^{k-1}=e$. So $h^{-1}\in H$.
Therefore $H$ is a subgroup of $G$
My question is:
- Is this a correct proof, because I don't know if $\text{ord}(a)=n$ when $H$ is finite?