Definition:Transitive Subgroup

Definition

Let $S_n$ denote the symmetric group on $n$ letters for $n \in \N$.

Let $H$ be a subgroup of $S_n$.

Let $H$ be such that:

for every pair of elements $i, j \in \N_n$ there exists $\pi \in H$ such that $\map \pi i = j$.

Then $H$ is called a transitive subgroup of $S_n$.

Examples

$n$-Cycle in $S_n$

Consider the subgroup $H$ of $S_n$ generated by the cyclic permutation $\tuple {1, 2, \ldots, n}$.

Then $H$ is a transitive subgroup .

Also see

• Results about transitive subgroups can be found here.

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 86$