= Solution
A <discrete subgroup> of the <Euclidean isometry> group has the identity isolated in the group topology. Equivalently, no sequence of distinct group elements converges to the identity; translating neighbourhoods then isolates every element. Such a <subgroup> is closed, and its intersection with any compact subset is finite: otherwise two elements of a convergent sequence would have quotient tending to the identity.
Write each <Euclidean isometry> uniquely as $x\mapsto Qx+b$, $Q\in O(2)$. Its linear part gives the <group homomorphism> $G\to O(2)$, since the linear part of a composite is the product of the linear parts. Its kernel is exactly the <translation subgroup> $G_T$. The translation <vectors> form a discrete additive <subgroup> $L\subset\mathbb R^2$.
If $L\ne\{0\}$, choose a shortest nonzero <vector> $a$, possible by discreteness and finiteness in compact balls. On its line, Euclidean division of a scalar coefficient by one proves $L\cap\mathbb Ra=\mathbb Za$: a nonzero remainder would be shorter than $a$. If all <vectors> are collinear, this finishes the proof. Otherwise orient the perpendicular coordinate so some <vector> has positive height. Reduce every <vector>'s component along $a$ modulo $a$ into a bounded interval. Among <vectors> with positive height at most that of a chosen <vector>, these reduced representatives lie in a compact rectangle and are finite. Choose $b$ with the smallest such positive height. Subtracting an integer multiple of $b$ from any <vector> of $L$ reduces its height into $[0,b_\perp)$; minimality makes the remainder's height zero, so it is an integer multiple of $a$. Thus
$$
\boxed{L=\{0\},\quad\mathbb Za,\quad\text{or }\mathbb Za+\mathbb Zb\text{ with }a,b\text{ linearly independent}.}
$$
This derives the rank-at-most-two <translation lattice> rather than assuming every <discrete subgroup> of the plane is a full lattice.
Two examples with $G_T\cong\mathbb Z$ are the group generated by $T(x,y)=(x+1,y)$ alone, and the <infinite dihedral group> generated by $T$ together with $R(x,y)=(-x,y)$. The latter has $RTR=T^{-1}$; its elements are translations $T^k$ and reflections $T^kR$, whose mirror lines are $x=k/2$. The first group preserves the orientation of every translated motif; the second also reverses it across these mirrors.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-1-isometries.png]
{title=Translation-only and dihedral plane-isometry groups with a rank-one translation subgroup}
Back to article page
