Transitive subgroup of symmetric group

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Definition

Let $S_n$ be the symmetric group on $n$ letters. Let it act naturally on $X=\{1,2,\dots ,n\}$. A subgroup $G\le S_n$ is said to be a transitive subgroup of the symmetric group on n letters if its group action on $X$ made from the restriction of the action of $S_n$ on $X$ is a transitive group action.

Examples

General examples

• Any $G\le S_n$ that contains an $n$-cycle is certainly transitive - successive applications of that cycle can send any $x\in X=\{1,2,\dots ,n\}$ to any other element of $X$ under the natural group action.
• A partial converse of the above exists. For $p$ prime, if $G\le S_p$ is transitive then it must contain a $p$-cycle.

Small symmetric groups

n=1, n=2, symmetric groups of order 1, 2

In these cases, the symmetric group only has itself as a transitive subgroup. The symmetric group is isomorphic to the trivial group and cyclic group:Z2 respectively.

n=3, symmetric group of order 6

See also: subgroup structure of symmetric group:S3

The transitive subgroups of symmetric group:S3 are, up to automorphism classes of subgroups:

Automorphism class of subgroups	List of all subgroups	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Total number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Note
A3 in S3	$\{(),(1,2,3),(1,3,2)\}$	cyclic group:Z3	3	2	1	1	1	cyclic group:Z2	3-Sylow
whole group	$\{(),(1,2,3),(1,3,2),$ $(1,2),(1,3),(2,3)\}$	symmetric group:S3	6	1	1	1	1	trivial group
Total (2 rows)	--	--	--	--	2	--	2	--	--

n=4, symmetric group of order 24

See also: subgroup structure of symmetric group:S4

The transitive subgroups of symmetric group:S4 are, up to automorphism classes of subgroups:

Automorphism class of subgroups	Representative	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (=1 iff automorph-conjugate subgroup)	Size of each conjugacy class (=1 iff normal subgroup)	Number of subgroups (=1 iff characteristic subgroup)	Isomorphism class of quotient (if exists)	Subnormal depth (if subnormal)	Note
Z4 in S4	$⟨(1,2,3,4)⟩$	cyclic group:Z4	4	6	1	3	3	--	--
normal Klein four-subgroup of S4	$\{(),(1,2)(3,4),$ $(1,3)(2,4),(1,4)(2,3)\}$	Klein four-group	4	6	1	1	1	symmetric group:S3	1	2-core
D8 in S4	$⟨(1,2,3,4),(1,3)⟩$	dihedral group:D8	8	3	1	3	3	--	--	2-Sylow, fusion system is non-inner non-simple fusion system for dihedral group:D8
A4 in S4	$⟨(1,2,3),(1,2)(3,4)⟩$	alternating group:A4	12	2	1	1	1	cyclic group:Z2	1
whole group	$⟨(1,2,3,4),(1,2)⟩$	symmetric group:S4	24	1	1	1	1	trivial group	0
Total (5 rows)	--	--	--	--	5	--	9	--	--	--

n=5, symmetric group of order 120

See also: subgroup structure of symmetric group:S5

The transitive subgroups of symmetric group:S5 are, up to automorphism classes of subgroups:

Automorphism class of subgroups	Representative subgroup	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes (= 1 iff automorph-conjugate subgroup)	Size of each conjugacy class (= 1 iff normal subgroup)	Total number of subgroups (= 1 iff characteristic subgroup)
Z5 in S5	$⟨(1,2,3,4,5)⟩$	cyclic group:Z5	5	24	1	6	6	5-Sylow
D10 in S5	$⟨(1,2,3,4,5),(2,5)(3,4)⟩$	dihedral group:D10	10	12	1	6	6
GA(1,5) in S5	$⟨(1,2,3,4,5),(2,3,5,4)⟩$	general affine group:GA(1,5)	20	6	1	6	6
A5 in S5	$⟨(1,2,3,4,5),(1,2,3)⟩$	alternating group:A5	60	2	1	1	1	only proper nontrivial normal subgroup, quotient is cyclic group:Z2
whole group	$⟨(1,2,3,4,5),(1,2)⟩$	symmetric group:S5	120	1	1	1	1
Total (5 rows)	--	--	--	--	5	--	20	--

Facts

Galois theory

The Galois group of a polynomial of degree $n$ can be viewed as a subgroup of $S_n$. This group is transitive if and only if the polynomial is irreducible. For example, $X^5-2$ is irreducible over $\mathbb{Q}$, so it's Galois group must be a transitive subgroup of $S_5$. Indeed, it turns out that the Galois group of this polynomial is isomorphic to general affine group:GA(1,5).