A transitive permutation group on a set $A$ is called doubly transitive if for any (hence all) $a \in A$ the subgroup $G_a$ is transitive on the set $A - \{ a \}$.
- (a) Prove that $S_n$ is doubly transitive on $\{1, 2, \dotsc, n\}$ for all $n ≥ 2$.
- (b) Prove that a doubly transitive group is primitive. Deduce that $D_8$ is not doubly transitive in its action on the $4$ vertices of a square.
My Attempt:
(a) $S_n$ is transitive on $\{1,2,\dotsc,n\}$ and for any $(i,j)\in G_a$, $(i,j)i=j$ whence $G_a$ is transitive on $\{1,2,\dotsc,n\}-\{a\}$.
(b) Without loss of generality let $|A|\ge2$.
Case I: $|A|=2$ or $3$ then $|A-\{a\}|=1$ or $2$ whence the blocks became trivial. So $G$ become primitive in this case.
Case II: $|A|\ge4$ then $|A-\{a\}|\ge3$. $A-\{a\}$ can’t have a nontrivial block $B$ since for any $i\in \{A-a\}-B$ and $j\in B$, $(i,j)\in G_a$ and $(i,j)(B)$ is neither equal or disjoint with $B$. Hence the result.
$D_8$ is not doubly transitive: Label the four vertices as $1,2,3,4$ consecutively. Consider the action of $G_4,$ the stabilizer of $4,$ on $\{1,2,3\}$. Then $\{1,3\}$ is a nontrivial block since $G_4$ contains only the reflection about the line of symmetry passing through $4$ and the identity.
My Questions:
Is my attempt correct?
Do we define ‘blocks’ (and hence ‘primitive’) only when $A$ is finite? (Even though in this exercise I never used finiteness of $A$ this question comes into my mind from their definition as given in Dummit-Foote text:
Let $G$ be a transitive permutation group on a finite set $A$. A block is a nonempty subset $B$ of $A$ such that for all $\sigma \in G$, either $\sigma(B) = B$ or $\sigma(B) \cap B = \emptyset$ (here $\sigma(B)$ is the set $\{ \sigma(b) \mid b \in B \}$.
- (a) Prove that if $B$ is a block containing the element $a$ of $A$, then the set $G_B$ defined by $G_B = \{ \sigma \in G \mid \sigma(B) = B \}$ is a subgroup of $G$ containing $G_a$.
- (b) Show that if $B$ is a block and $\sigma_1(B), \sigma_2(B), \dotsc, \sigma_n(B)$ are all the distinct images of $B$ under the elements of $G$, then these form a partition of $A$.
- (c) A (transitive) group $G$ on a set $A$ is said to be primitive if the only blocks in $A$ are the trivial ones: the sets of size $1$ and $A$ itself. Show that $S_4$ is primitive on $A = \{1, 2, 3, 4\}$. Show that $D_8$ is not primitive as a permutation group on the four vertices of a square.
