Exercise 13.3.10

Proof.

(a)

An orderer pair $(u,v)\in {𝔽}_2^2$ is a base of a vectorial plane if $u\ne 0$, and $v\notin ⟨u⟩=\{0,u\}$, so there are $7\times 6$ (ordered) bases of planes in ${𝔽}_3^2$.

A vectorial plane $P$ of ${𝔽}_3^2$, which contains 4 vectors, has $3\times 2$ bases, with the same reasoning: 3 choices for the first non null vector, and 2 choices for the second vector.

Therefore the number of two-dimensional subspaces of ${𝔽}_2^3$ is $\frac{7\times 6}{3\times 2}=7$.

(b)

Let $\{e_1,e_2,e_3\}$ be the standard basis of $F^3$, where $e_1=(1,0,0),e_2=(0,1,0),e_3=(0,0,1)$. Any element of $\mathrm{GL}(3,F)$ applies a base on a base, thus the orbit of $\{e_1,e_2,e_3\}$ is included in $B$.

Conversely, let $\{{\nu }_1,{\nu }_2,{\nu }_3\}$ be any element of $B$. Define $g=[{\nu }_1,{\nu }_2,{\nu }_2]$ the matrix whose columns are ${\nu }_1,{\nu }_2,{\nu }_3$. Since ${\nu }_1,{\nu }_2,{\nu }_3$ are linearly independent, $g\in \mathrm{GL}(3,{𝔽}_2)$, and $g\cdot e_1={\nu }_1,g\cdot e_2={\nu }_2,g\cdot e_3={\nu }_3$, so that $\{{\nu }_1,{\nu }_2,{\nu }_3\}=g\cdot \{e_1,e_2,e_3\}$ is in the orbit of $\{e_1,e_2,e_3\}$. Therefore the orbit of $\{e_1,e_2,e_3\}$ is $B$, thus $\mathrm{GL}(3,F)$ acts transitively over $B$.

(c)

Let $P,Q$ be two-dimensional subspaces of $F^3$. Take $\{{\nu }_1,{\nu }_2\}$ a basis of $P$, and $\{{\nu }_1^{\prime },{\nu }_2^{\prime }\}$ a basis of $Q$. We can complete them into bases of $F^3$, say $b=\{{\nu }_1,{\nu }_2,{\nu }_3\}$ and $b^{\prime }=\{{\nu }_1^{\prime },{\nu }_2^{\prime },{\nu }_3^{\prime }\}$. By part $b$, there exists $g\in \mathrm{GL}(3,F)$ such that $g\cdot b=b^{\prime }$. Since $g$ maps a basis of $P$ on a basis of $Q$, $g\cdot P=Q$. Therefore $\mathrm{GL}(3,F)$ acts transitively on the set of two-dimensional subspaces of $F^3$

Note: Since there are exactly 3 nonzero vectors in a two-dimensional subspace of ${𝔽}_2^3$, each triple of distinct nonzero linearly dependant vectors in ${𝔽}_2^3$ determines a unique two-dimensional space. Therefore part (c) explains the statement in the proof of 13.3.9, that $\mathrm{GL}(3,{𝔽}_2^3)$ acts transitively on this set of triples. □