The statement of the second exercise is incorrect. It should be the following:
Let G be a given subgroup of the group of all isometries of R^{2} for which the subgroup T(G) consisting of all translations in G is the vector lattice of all translations of a lattice Lambda in R^{2} containing the origin.
Let
be a given isometry in the group G. Show that the matrix U has the property that the linear isometry x -> U x leaves Lambda invariant. Hint: Consult exercise 3 on the assignment due April 14.
Show that given two isometries f in the group G sharing the same matrix U,
the difference b_{2} - b_{1} of their translational components is in the lattice Lambda.
The third exercise, a continuation of the second was correctly stated as follows:
Let H be the set of orthogonal matrices U that leave Lambda invariant. Show that H is a finite group; i.e., show that H is a subgroup of the group of all orthogonal matrices and that H is a finite set.