Quiz:Symmetric group:S3

See symmetric group:S3. We take the symmetric group on the set $\{ 1,2,3 \}$ of size three.

Elements

See element structure of symmetric group:S3 for full details.

Element orders and conjugacy class structure

Review the conjugacy class structure: [SHOW MORE]

Partition	Partition in grouped form	Verbal description of cycle type	Elements with the cycle type in cycle decomposition notation	Elements with the cycle type in one-line notation	Size of conjugacy class	Formula for size	Even or odd? If even, splits? If splits, real in alternating group?	Element order	Formula calculating element order
1 + 1 + 1	1 (3 times)	three fixed points	$()$ -- the identity element	123	1	$\! \frac{3!}{(1)^3(3!)}$	even; no	1	$\operatorname{lcm} \{ 1, 1, 1 \}$
2 + 1	2 (1 time), 1 (1 time)	transposition in symmetric group:S3: one 2-cycle, one fixed point	$(1,2)$, $(1,3)$, $(2,3)$	213, 321, 132	3	$\! \frac{3!}{(2)(1)}$	odd	2	$\operatorname{lcm} \{ 2,1 \}$
3	3 (1 time)	3-cycle in symmetric group:S3: one 3-cycle	$(1,2,3)$, $(1,3,2)$	231, 312	2	$\! \frac{3!}{3}$	even; yes; no	3	$\operatorname{lcm} \{ 3 \}$
Total (3 rows -- 3 being the number of unordered integer partitions of 3)	--	--	--	--	6 (equals 3!, the size of the symmetric group)	--	odd: 3 even;no: 1 even; yes; no: 2	order 1: 1, order 2: 3, order 3: 2	--

1 What is the number of non-identity elements of the symmetric group of degree three?

	2
	3
	4
	5
	6

2 How many elements are there of order exactly three in the symmetric group of degree three?

	2
	3
	4
	5
	6

3 Which of the following is a correct description of the conjugacy class structure of the symmetric group of degree three?

	Conjugacy class of size 1 and order (of elements in the conjugacy class) 1, conjugacy class of size 2 and order (of elements in the conjugacy class) 2, conjugacy class of size 3 and order (of elements in the conjugacy class) 3
	Conjugacy class of size 1 and order (of elements in the conjugacy class) 1, conjugacy class of size 2 and order (of elements in the conjugacy class) 3, conjugacy class of size 3 and order (of elements in the conjugacy class) 2
	Conjugacy class of size 1 and order (of elements in the conjugacy class) 2, conjugacy class of size 2 and order (of elements in the conjugacy class) 1, conjugacy class of size 3 and order (of elements in the conjugacy class) 3
	Conjugacy class of size 1 and order (of elements in the conjugacy class) 2, conjugacy class of size 2 and order (of elements in the conjugacy class) 3, conjugacy class of size 3 and order (of elements in the conjugacy class) 1
	Conjugacy class of size 1 and order (of elements in the conjugacy class) 3, conjugacy class of size 2 and order (of elements in the conjugacy class) 1, conjugacy class of size 3 and order (of elements in the conjugacy class) 2
	Conjugacy class of size 1 and order (of elements in the conjugacy class) 3, conjugacy class of size 2 and order (of elements in the conjugacy class) 2, conjugacy class of size 3 and order (of elements in the conjugacy class) 1

Multiplication, conjugacy and generating sets

Review the multiplication table in cycle decomposition notation: [SHOW MORE]

Element	$()$	$(1, 2)$	$(2, 3)$	$(1, 3)$	$(1, 2, 3)$	$(1, 3, 2)$
$()$	$()$	$(1, 2)$	$(2, 3)$	$(1, 3)$	$(1, 2, 3)$	$(1, 3, 2)$
$(1, 2)$	$(1, 2)$	$()$	$(1, 2, 3)$	$(1, 3, 2)$	$(2, 3)$	$(1, 3)$
$(2, 3)$	$(2, 3)$	$(1, 3, 2)$	$()$	$(1, 2, 3)$	$(1, 3)$	$(1, 2)$
$(1, 3)$	$(1, 3)$	$(1, 2, 3)$	$(1, 3, 2)$	$()$	$(1, 2)$	$(2, 3)$
$(1, 2, 3)$	$(1, 2, 3)$	$(1, 3)$	$(1, 2)$	$(2, 3)$	$(1, 3, 2)$	$()$
$(1, 3, 2)$	$(1, 3, 2)$	$(2, 3)$	$(1, 3)$	$(1, 2)$	$()$	$(1, 2, 3)$

Review the multiplication table in one-line notation: [SHOW MORE]

Element	123	213	132	321	231	312
123	123	213	132	321	231	312
213	213	123	231	312	132	321
132	132	312	123	231	321	213
321	321	231	312	123	213	132
231	231	321	213	132	312	123
312	312	132	321	213	123	231

1 What can we say about the order of the product of two distinct elements, each of order exactly two, in the symmetric group of degree three?

	The product must be the identity element
	The product must have order two
	The product can have order either 1 or 2
	The product must have order three
	The product can have order either 1 or 3

2 What can we say about the order of the product of two distinct elements, each of order exactly three, in the symmetric group of degree three?

	The product must be the identity element
	The product must have order two
	The product can have order either 1 or 2
	The product must have order three
	The product can have order either 1 or 3

3 Which of the following is false in the symmetric group of degree three?

	Any two elements of the same order are conjugate
	Every element is conjugate to its inverse
	Any two elements generating the same cyclic subgroup are conjugate
	Any two elements that together generate the whole group are conjugate
	None of the above, i.e., they are all true

Conjugation and commutator operations

Review the conjugation operation: [SHOW MORE]

Review the commutator operation: [SHOW MORE]

Here, the two inputs are group elements $g,h$, and the output is the commutator. We first give the table assuming the left definition of commutator: $[g,h] = ghg^{-1}h^{-1}$. Here, the row element is $g$ and the column element is $h$. Note that $[g,h] = [h,g]^{-1}$:

	$()$	$(1,2)$	$(2,3)$	$(1,3)$	$(1,2,3)$	$(1,3,2)$
$()$	$()$	$()$	$()$	$()$	$()$	$()$
$(1,2)$	$()$	$()$	$(1,3,2)$	$(1,2,3)$	$(1,2,3)$	$(1,3,2)$
$(2,3)$	$()$	$(1,2,3)$	$()$	$(1,3,2)$	$(1,2,3)$	$(1,3,2)$
$(1,3)$	$()$	$(1,3,2)$	$(1,2,3)$	$()$	$(1,2,3)$	$(1,3,2)$
$(1,2,3)$	$()$	$(1,3,2)$	$(1,3,2)$	$(1,3,2)$	$()$	$()$
$(1,3,2)$	$()$	$(1,2,3)$	$(1,2,3)$	$(1,2,3)$	$()$	$()$

The corresponding table with the right definition: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Here is the information on the number of times each element occurs as a commutator:

Conjugacy class (indexing partition)	Elements	Number of occurrences of each as commutator	Probability of each occurring as the commutator of elements picked uniformly at random	Total number of occurrences as commutator	Total probability	Explanation
1 + 1 + 1	$()$	18	1/2	18	1/2	See commuting fraction and its relationship with the number of conjugacy classes.
2 + 1	$(1,2), (2,3), (1,3)$	0	0	0	0	Not in the derived subgroup.
3	$(1,2,3), (1,3,2)$	9	1/4	18	1/2

1 Suppose $g$ and $h$ are distinct elements of order two in the symmetric group of order three. What can we say about $ghg^{-1}$ (this is a conjugate of $h$ by $g$)?

	It equals $g$
	It equals $h$
	It equals an element of order two that is neither $g$ nor $h$
	It is an element of order three
	It is the identity element

2 Suppose $g$ and $h$ are distinct elements of order two in the symmetric group of order three. What can we say about the commutator $ghg^{-1}h^{-1}$?

	It equals $g$
	It equals $h$
	It equals an element of order two that is neither $g$ nor $h$
	It is an element of order three
	It is the identity element

3 Suppose $g$ and $h$ are distinct elements of order three in the symmetric group of order three. What can we say about $ghg^{-1}$ (this is a conjugate of $h$ by $g$)?

	It equals $g$
	It equals $h$
	It equals an element of order two
	It is the identity element

4 Suppose $g$ and $h$ are distinct elements of order three in the symmetric group of order three. What can we say about the commutator $ghg^{-1}h^{-1}$?

	It equals $g$
	It equals $h$
	It is an element of order two
	It is the identity element

5 Suppose $g$ is an element of order two and $h$ is an element of order three in the symmetric group of order three. What are the orders of the elements $ghg^{-1}$ and $hgh^{-1}$ respectively?

	1 and 1
	2 and 3
	3 and 2
	2 and 2
	3 and 3

Subgroups

See subgroup structure of symmetric group:S3 for background information.

Basic stuff

Summary table on the structure of subgroups: [SHOW MORE]

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

For more information on each automorphism type, follow the link.

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	--	--

View the lattice of subgroups as a picture: [SHOW MORE]

1 What are the possible orders of subgroups of the symmetric group of degree three?

	1 and 6 only
	1, 2, and 6 only
	1, 3, and 6 only
	1, 2, 3, and 6 only
	1, 4, and 6 only
	1, 2, 3, 4, 5, and 6 only

2 What are the possible orders of normal subgroups of the symmetric group of degree three?

	1 and 6 only
	1, 2, and 6 only
	1, 3, and 6 only
	1, 2, 3, and 6 only
	1, 4, and 6 only
	1, 2, 3, 4, 5, and 6 only

3 What are the possible orders of quotient groups of the symmetric group of degree three?

	1 and 6 only
	1, 2, and 6 only
	1, 3, and 6 only
	1, 2, 3, and 6 only
	1, 4, and 6 only
	1, 2, 3, 4, 5, and 6 only

Linear representations

For background, see linear representation theory of symmetric group:S3.

View overall summary: [SHOW MORE]

Item	Value
Degrees of irreducible representations over a splitting field (and in particular over $\mathbb{C}$)	1,1,2 maximum: 2, lcm: 2, number: 3 sum of squares: 6, quasirandom degree: 1
Schur index values of irreducible representations	1,1,1
Smallest ring of realization for all irreducible representations (characteristic zero)	$\mathbb{Z}$
Minimal splitting field, i.e., smallest field of realization for all irreducible representations (characteristic zero)	$\mathbb{Q}$ (hence, it is a rational representation group)
Condition for being a splitting field for this group	Any field of characteristic not two or three is a splitting field. In particular, $\mathbb{Q}$ and $\R$ are splitting fields.
Minimal splitting field in characteristic $p \ne 0,2,3$	The prime field $\mathbb{F}_p$
Smallest size splitting field	field:F5, i.e., the field of five elements.

View representations summary: [SHOW MORE]

Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2 or 3, except in the last two columns, where we consider what happens in characteristic 2 and characteristic 3.

Name of representation type	Number of representations of this type	Degree	Schur index	Criterion for field	Kernel	Quotient by kernel (on which it descends to a faithful representation)	Characteristic 2	Characteristic 3
trivial	1	1	1	any	whole group	trivial group	works	works
sign	1	1	1	any	A3 in S3	cyclic group:Z2	works, same as trivial	works
standard (two-dimensional irreducible)	1	2	1	any	trivial subgroup, i.e., it is faithful	symmetric group:S3	works	indecomposable but not irreducible

View character table: [SHOW MORE]

Representation/Conjugacy class representative	$()$ (identity element) -- size 1	$(1,2,3)$ (3-cycle) -- size 2	$(1,2)$ (2-transposition) -- size 3
Trivial representation	1	1	1
Sign representation	1	1	-1
Standard representation	2	-1	0

Basic stuff

Up to equivalence, there are three irreducible representations of symmetric group:S3 in characteristic zero: the one-dimensional trivial representation, the one-dimensional sign representation (that sends every permutation to its sign), and the standard representation of symmetric group:S3, a two-dimensional representation.

1 Which of the irreducible representations is realized over the field of rational numbers?

	trivial representation only
	trivial representation and sign representation only
	all three representations

2 Which of the irreducible representations can be realized using orthogonal matrices (i.e., matrices in the orthogonal group for the standard dot product) over the field of rational numbers?

	trivial representation only
	trivial representation and sign representation only
	all three representations

3 The tensor product of the standard representation and the sign representation is a representation of the symmetric group of degree three. What representation is it?

	the standard representation
	the sum of the trivial and the sign representation
	the sum of two copies of the sign representation
	the sum of two copies of the trivial representation

4 The symmetric group of degree three can also be viewed as a dihedral group of degree three and order six, acting on a set of size three. The 3-cycles become rotations and the transpositions become reflections. This defines a two-dimensional representation over the real numbers. Which of these is the two-dimensional representation?

	the standard representation
	the sum of the trivial and the sign representation
	the sum of two copies of the sign representation
	the sum of two copies of the trivial representation