= Solution
A <group> has a closed <binary operation>, associativity, an identity, and an inverse for each element. For a finite group, <Lagrange's theorem> states $|G|=[G:H]|H|$ for each subgroup $H$, so $|H|$ divides $|G|$.
The <order of a group element> $g$ is the least positive $n$ with $g^n=e$, or infinity if none exists. In a finite group the powers form the <cyclic subgroup> $\langle g\rangle$ of that order; Lagrange's theorem proves $\operatorname{ord}(g)\mid|G|$. If $|G|=p$ is prime and $g\ne e$, its order must be $p$. Thus $g$ generates the whole group, proving $G\cong C_p$.
If every nonidentity element has order two, every element is its own inverse. Then $ab=(ab)^{-1}=b^{-1}a^{-1}=ba$. This proves that a <group of exponent two is abelian>.
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