Gratuitously easy? Show that every isometry of n-dimensional Euclidean space has the form tau \circ phi where tau is a translation and phi has a fixed point.
Give an example of an isometry of the Euclidean plane with a fixed point but with no invariant line.
Show that if an isometry of Euclidean 3-space has a fixed point, then it has an invariant line through that fixed point.
Show that if f is an isometry of Euclidean 3-space with a fixed point c, then
The orthogonal complement through c of an invariant line through c is an invariant plane.
The orthogonal complement through c of an invariant plane through c is an invariant line.
Let G be the smallest group of isometries of R^{2} that contains the three isometries
tau_{h} : (x, y) ---> (x + 2, y).
tau_{v} : (x, y) ---> (x, y + 1).
gamma : (x, y) ---> (x + 1, 1 - y).
What type of isometry is gamma ?
Find the translation subgroup of G, i.e., the set of all translations that are in G.
Explain why the translation subgroup of G is a vector lattice.
What is the largest lattice Lambda in R^{2} containing (0, 0) that is invariant under each of the translations in the translation subgroup of G ?
Prove that G contains no reflection.