Definition:Transitive Subgroup

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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 $S_n$.


$n$-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.