What is the cardinality of the centre of $O_2(\mathbb{R})$? (Definition: The centre of a group $G$ is $\{g \in G \mid gh = hg \text{ for every } h \in G\}$. Hint: Reflection matrices and permutation matrices are orthogonal.)
(A) 1;
(B) 2;
(C) The cardinality of $\mathbb{N}$;
(D) The cardinality of $\mathbb{R}$.

Let $\mathbb{k}$ be a field, $n$ a positive integer and $G$ a finite subgroup of $\mathrm{GL}_n(\mathbb{k})$ such that $|G| > 1$. Further assume that every $g \in G$ is upper-triangular and all the diagonal entries of $g$ are 1.
(A) Show that $\operatorname{char}\,\mathbb{k} > 0$. (Hint: consider the minimal polynomials of elements of $G$.)
(B) Show that the order of $g$ is a power of $\operatorname{char}\,\mathbb{k}$, for every $g \in G$.
(C) Show that the centre of $G$ has at least two elements.