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.

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

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	$\mathrm{l}\mathrm{c}\mathrm{m}\{1,1,1\}$
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	$\mathrm{l}\mathrm{c}\mathrm{m}\{2,1\}$
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	$\mathrm{l}\mathrm{c}\mathrm{m}\{3\}$
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 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

4 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