= Solution
A <group> is a set $G$ equipped with a binary operation satisfying the <group axioms>: closure; <associativity>, $(gh)k=g(hk)$; an <identity element> $e$ with $eg=ge=g$; and an <inverse element> $g^{-1}$ for every $g$, with $gg^{-1}=g^{-1}g=e$.
A <subgroup> $H$ is a subset that is a <group> under the restricted operation. Equivalently, it is nonempty and contains $xy^{-1}$ whenever $x,y\in H$. A <generating set of a group> is a set whose elements and inverses can be multiplied in finite words to produce every group element.
The <group order> is its number of elements, $|G|$. The <order of a group element> $g$ is the least positive integer $r$ with $g^r=e$, or infinity if there is none. <Lagrange's theorem> states that for a finite <group> and <subgroup> $H$, $|H|$ divides $|G|$ and $|G|=[G:H]|H|$. In particular, every finite element order divides the <group order>.
A <cyclic group> is generated by one element, so every element is a power of that generator. A <group isomorphism> is a bijective <group homomorphism>: a bijection $\phi:G\to G'$ satisfying $\phi(gh)=\phi(g)\phi(h)$ for all $g,h$.
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