# Subgroup structure of symmetric group:S3

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Since this group is a complete group, every automorphism of it is inner, and in particular, this means that the classification of subgroups upto conjugacy is the same as the classification up to automorphism.

## Tables for quick information

FACTS TO CHECK AGAINST FOR SUBGROUP STRUCTURE: (finite solvable group)
Lagrange's theorem (order of subgroup times index of subgroup equals order of whole group, so both divide it), |order of quotient group divides order of group (and equals index of corresponding normal subgroup)
Sylow subgroups exist, Sylow implies order-dominating, congruence condition on Sylow numbers|congruence condition on number of subgroups of given prime power order
Hall subgroups exist in finite solvable|Hall implies order-dominating in finite solvable| normal Hall implies permutably complemented, Hall retract implies order-conjugate
MINIMAL, MAXIMAL: minimal normal implies elementary abelian in finite solvable | maximal subgroup has prime power index in finite solvable group

### Quick summary

Item Value
Number of subgroups 6
Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,6,30,156,1455,11300, 151221
Number of conjugacy classes of subgroups 4
Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,4,11,19,56,96,296,554,1593
Number of automorphism classes of subgroups 4
Compared with $S_n, n = 1,2,3,4,5,\dots$: 1,2,4,11,19,37,96,296,554,1593
Isomorphism classes of Sylow subgroups and the corresponding Sylow numbers and fusion systems 2-Sylow: cyclic group:Z2, Sylow number is 3, fusion system is the trivial one
3-Sylow: cyclic group:Z3, Sylow number is 1, fusion system is non-inner fusion system for cyclic group:Z3
Hall subgroups Given that the order has only two distinct prime factors, the Hall subgroups are the whole group, trivial subgroup, and Sylow subgroups. Interestingly, all subgroups are Hall subgroups, because the order is a square-free number
maximal subgroups maximal subgroups have order 2 (S2 in S3) and 3 (A3 in S3).
normal subgroups There are three normal subgroups: the trivial subgroup, the whole group, and A3 in S3.

### Table classifying subgroups up to automorphisms

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
trivial subgroup $\{ () \}$ trivial group 1 6 1 1 1 symmetric group:S3 trivial
S2 in S3 $\{ (), (1,2) \}, \{ (), (2,3) \}, \{ (), (1,3) \}$ cyclic group:Z2 2 3 1 3 3 -- 2-Sylow
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 (4 rows) -- -- -- -- 4 -- 6 -- --

### Table classifying isomorphism types of subgroups

Note that the first part of the GAP ID is the order.

Group name GAP ID Index Occurrences as subgroup Conjugacy classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
Trivial group $(1,1)$ 6 1 1 1 1
Cyclic group:Z2 $(2,1)$ 3 3 1 0 0
Cyclic group:Z3 $(3,1)$ 2 1 1 1 1
Symmetric group:S3 $(6,1)$ 1 1 1 1 1
Total (4 rows) -- -- 6 4 3 3

### Table listing number of subgroups by order

Note that these orders satisfy the congruence condition on number of subgroups of given prime power order: the number of subgroups of order $p^r$ is congruent to $1$ modulo $p$.

Group order Index Occurrences as subgroup Conjugacy classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
1 6 1 1 1 1
2 3 3 1 0 0
3 2 1 1 1 1
6 1 1 1 1 1
Total (4 rows) -- 6 4 3 3

### Table listing numbers of subgroups by group property

Group property Occurrences as subgroup Conjugacy classes of occurrence as subgroup Occurrences as normal subgroup Occurrences as characteristic subgroup
Cyclic group 5 3 2 2
Abelian group 5 3 2 2
Nilpotent group 5 3 2 2
Solvable group 6 4 3 3

## Subgroup structure viewed as symmetric group

### Classification based on partition given by orbit sizes

For any subgroup of $S_3$, the natural action on $\{ 1,2,3 \}$ induces a partition of the set $\{ 1,2,3 \}$ into orbits, which in turn induces an unordered integer partition of the number 3. Below, we classify this information for the subgroups.

Conjugacy class of subgroups Size of conjugacy class Induced partition of 3 Direct product of transitive subgroups on each orbit? Illustration with representative
trivial subgroup 1 1 + 1 + 1 Yes Under the action of the trivial subgroup, the orbits are singleton subsets.
S2 in S3 3 2 + 1 Yes Under the action of $\{ (), (1,2) \}$, the orbits are $\{ 1,2 \}$ and $\{ 3 \}$.
A3 in S3 1 3 Yes The action is a transitive group action, so $\{ 1,2,3 \}$ is a single orbit. Each of the elements sends the point $1$ to a different element of $\{ 1,2,3 \}$, covering everything.
whole group 1 3 Yes The action is a transitive group action, so $\{ 1,2,3 \}$ is a single orbit.

## Defining functions

### Subgroup-defining functions and associated quotient-defining functions

Subgroup-defining function What it means Value as subgroup Value as group Order Associated quotient-defining function Value as group Order (= index of subgroup)
center elements that commute with every group element trivial subgroup trivial group 1 inner automorphism group symmetric group:S3 6
derived subgroup subgroup generated by all commutators A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3 abelianization cyclic group:Z2 2
Frattini subgroup intersection of all maximal subgroups trivial subgroup trivial group 1 Frattini quotient symmetric group:S3 6
Jacobson radical intersection of all maximal normal subgroups A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
socle join of all minimal normal subgroups A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
Fitting subgroup join of all nilpotent normal subgroups A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
solvable radical join of all solvable normal subgroups whole group symmetric group:S3 6  ? trivial group 1
Brauer core largest odd-order normal subgroup A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
3-Sylow core or 3-core normal core of any 3-Sylow subgroup A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2
3-Sylow closure normal closure of any 3-Sylow subgroup A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3  ? cyclic group:Z2 2

### Subgroup series-defining functions

Series-defining function Type Zeroth member First member Second member Third member Stable member
upper central series ascending trivial center: trivial second center: trivial trivial trivial
lower central series descending -- whole group derived subgroup: $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3 $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3 $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3
derived series descending whole group derived subgroup: $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3 second derived subgroup: trivial trivial trivial
Frattini series descending whole group Frattini subgroup: trivial trivial trivial trivial
Fitting series ascending trivial Fitting subgroup: $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3 whole group whole group whole group
socle series ascending trivial socle: $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3 whole group whole group whole group

### Conjugacy class-defining functions

Conjugacy class-defining function What it means Value as subgroup Value as group Order Index of subgroup Number of subgroups
2-Sylow subgroup subgroup whose order is a power of 2, index relatively prime to 2. Sylow subgroups exist, and Sylow implies order-conjugate S2 in S3: $\{ (), (1,2) \}, \{ (), (2,3) \},$, $\{ (), (1,3) \}$ cyclic group:Z2 2 3 3
3-Sylow subgroup subgroup whose order is a power of 3, index relatively prime to 3. Sylow subgroups exist, and Sylow implies order-conjugate A3 in S3: $\{ (), (1,2,3), (1,3,2) \}$ cyclic group:Z3 3 2 1

## Lattice of subgroups

### The entire lattice

The lattice of subgroups of the symmetric group of degree three has the following interesting features:

• Every non-identity automorphism of the whole group acts nontrivially on the lattice. Note that since the symmetric group of degree three is a complete group, all the automorphisms are inner.
• In fact, the non-identity automorphisms give rise to all possible permutations of the three non-abelian subgroups of order two. More specifically, a permutation of the letters $1,2,3$ gives rise to an inner automorphism that permutes the two-element subgroups fixing these elements the same way. For instance, the $3$-cycle $(1,2,3)$, acting by conjugation, sends the subgroup stabilizing $1$ (namely $\{ (), (2,3) \}$) to the subgroup stabilizing $2$ (namely $\{ (), (1,3) \}$).

### The sublattice of normal subgroups

The lattice of normal subgroups, which is in this case also the lattice of characteristic subgroups, is a totally ordered sublattice comprising the trivial subgroup, the subgroup of order three, and the whole group. This sublattice is preserved by all automorphisms.

## Subgroup operations

### Intersection

For all pairs of subgroups, either one is contained in the other, or the intersection is trivial. This makes the intersection table easy to construct.

Subgroup/subgroup trivial $\{ (), (1,2) \}$ $\{ (), (2,3) \}$ $\{ (), (1,3) \}$ $\{ (), (1,2,3), (1,3,2) \}$ whole group
trivial trivial trivial trivial trivial trivial trivial
$\{ (), (1,2) \}$ trivial $\{ (), (1,2) \}$ trivial trivial trivial $\{ (), (1,2) \}$
$\{ (), (2,3) \}$ trivial trivial $\{ (), (2,3) \}$ trivial trivial $\{ (), (2,3) \}$
$\{ (), (1,3) \}$ trivial trivial trivial $\{ (), (1,3) \}$ trivial $\{ (), (1,3) \}$
$\{ (), (1,2,3), (1,3,2) \}$ trivial trivial trivial trivial $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$
whole group trivial $\{ (), (1,2) \}$ $\{ (), (2,3) \}$ $\{ (), (1,3) \}$ $\{ (), (1,2,3), (1,3,2) \}$ whole group

### Join of subgroups

For all pairs of subgroups, either one is contained in the other, or the join is the whole group. This makes the table for joins of subgroups easy to construct.

Subgroup/subgroup trivial $\{ (), (1,2) \}$ $\{ (), (2,3) \}$ $\{ (), (1,3) \}$ $\{ (), (1,2,3), (1,3,2) \}$ whole group
trivial trivial $\{ (), (1,2) \}$ $\{ (), (2,3) \}$ $\{ (), (1,3) \}$ $\{ (), (1,2,3), (1,3,2) \}$ whole group
$\{ (), (1,2) \}$ $\{ (), (1,2) \}$ $\{ (), (1,2) \}$ whole group whole group whole group whole group
$\{ (), (2,3) \}$ $\{ (), (2,3) \}$ whole group $\{ (), (2,3) \}$ whole group whole group whole group
$\{ (), (1,3) \}$ $\{ (), (1,3) \}$ whole group whole group $\{ (), (1,3)\}$ whole group whole group
$\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ whole group whole group whole group $\{ (), (1,2,3), (1,3,2) \}$ whole group
whole group whole group whole group whole group whole group whole group whole group

### Commutators of pairs of subgroups

For any pair of subgroups, their commutator is trivial if one of them is trivial or they are both equal and proper, and is $\{ (), (1,2,3), (1,3,2) \}$ otherwise.

Subgroup/subgroup trivial $\{ (), (1,2) \}$ $\{ (), (2,3) \}$ $\{ (), (1,3) \}$ $\{ (), (1,2,3), (1,3,2) \}$ whole group
trivial trivial trivial trivial trivial trivial trivial
$\{ (), (1,2) \}$ trivial trivial $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$
$\{ (), (1,3) \}$ trivial $\{ (), (1,2,3), (1,3,2) \}$ trivial $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$
$\{ (), (2,3) \}$ trivial $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ trivial $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$
$\{ (), (1,2,3), (1,3,2) \}$ trivial $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ trivial $\{ (), (1,2,3), (1,3,2) \}$
whole group trivial $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$

### Unary operations on subgroups

Subgroup Normalizer Centralizer Normal closure Normal core
trivial whole group whole group trivial trivial
$\{ (), (1,2) \}$ $\{ (), (1,2) \}$ $\{ (), (1,2) \}$ whole group trivial
$\{ (), (2,3) \}$ $\{ (), (2,3) \}$ $\{ (), (2,3) \}$ whole group trivial
$\{ (), (1,3) \}$ $\{ (), (1,3) \}$ $\{ (), (1,3) \}$ whole group trivial
$\{ (), (1,2,3), (1,3,2) \}$ whole group $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$
whole group whole group trivial whole group whole group

## Subgroup series

The max-length of the group is 2 (it cannot be more, based on the prime factorization of 6) and there are four subgroup series of this maximal length, one series for each proper nontrivial subgroup.

Series Intermediate subgroup Normal series? Subnormal series? Chief series? Composition series?
$\{ () \} \le \{ (), (1,2) \} \le \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$ $\{ (), (1,2) \}$ -- S2 in S3 No No No No
$\{ () \} \le \{ (), (2,3) \} \le \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$ $\{ (), (2,3) \}$ -- S2 in S3 No No No No
$\{ () \} \le \{ (), (1,3) \} \le \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$ $\{ (), (1,3) \}$ -- S2 in S3 No No No No
$\{ () \} \le \{ (), (1,2,3), (1,3,2) \} \le \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}$ $\{ (), (1,2,3), (1,3,2) \}$ -- A3 in S3 Yes Yes Yes Yes

In particular, there is a unique composition series which is also a unique chief series for the group.