Quiz:Alternating group:A4

This quiz is about alternating group:A4, which we take as the alternating group on the set $\{1,2,3,4\}$ of size four.

Elements

See element structure of alternating group:A4 for background information.

Element order and conjugacy class structure

Review the conjugacy class structure: [SHOW MORE]

For a symmetric group, cycle type determines conjugacy class. The statement is almost true for the alternating group, except for the fact that some conjugacy classes of even permutations in the symmetric group split into two in the alternating group, as per the splitting criterion for conjugacy classes in the alternating group, which says that a conjugacy class of even permutations splits in the alternating group if and only if its cycle decomposition comprises odd cycles of distinct length.

Here are the unsplit conjugacy classes:

Partition	Verbal description of cycle type	Elements with the cycle type	Size of conjugacy class	Formula for size	Element order
1 + 1 + 1 + 1	four cycles of size one each, i.e., four fixed points	$()$ -- the identity element	1	$\frac{4!}{(1{)}^4(4!)}$	1
2 + 2	double transposition: two cycles of size two	$(1,2)(3,4)$, $(1,3)(2,4)$, $(1,4)(2,3)$	3	$\frac{4!}{(2{)}^2(2!)}$	2
Total	--	$()$, $(1,2)(3,4)$, $(1,3)(2,4)$ and $(1,4)(2,3)$	4	NA	NA

In this case, the union of the unsplit conjugacy classes is a proper normal subgroup isomorphic to the Klein four-group. Note that this phenomenon is unique to the case $n=4$.

Here is the split conjugacy class:

Partition	Verbal description of cycle type	Elements with the cycle type	Combined size of conjugacy classes	Formula for combined size	Size of each half	First split half	Second split half	Real?	Rational?	Element order
3 + 1	one 3-cycle, one fixed point	$(1,2,3)$, $(1,3,2)$, $(2,3,4)$, $(2,4,3)$, $(3,4,1)$, $(3,1,4)$, $(4,1,2)$, $(4,2,1)$	8	$\frac{4!}{(3)(1)}$	4	$(1,2,3)$, $(4,2,1)$, $(2,4,3)$, $(3,4,1)$	$(1,3,2)$, $(4,1,2)$, $(2,3,4)$, $(3,1,4)$	No	No	3

1 What are the possible orders of elements in the alternating group of degree four?

	1,2,3 only
	1,3,4 only
	1,2,3,4 only
	1,2,3,6 only
	1,2,3,4,6 only
	1,2,3,4,6,12 only

2 Which of the following elements is in the alternating group on the set $\{1,2,3,4\}$ and is not conjugate to its inverse in the alternating group?

	$()$ -- the identity element
	$(1,2)$ -- a 2-transposition
	$(1,2)(3,4)$ -- a double transposition
	$(1,2,3)$ -- a 3-cycle
	$(1,2,3,4)$ -- a 4-cycle

3 How many conjugacy classes of elements are there in the alternating group of degree four?

	1
	2
	3
	4
	5
	6
	7

4 How many orbits are there in the alternating group of degree four under the action of its automorphism group?

	1
	2
	3
	4
	5
	6
	7

Subgroups

See subgroup structure of alternating group:A4 for background information.

Basic stuff

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

Quick summary

Item	Value
Number of subgroups	10
Number of conjugacy classes of subgroups	5
Number of automorphism classes of subgroups	5

Table classifying subgroups up to automorphism

Automorphism class of subgroups	List of subgroups	Isomorphism class	Order of subgroups	Index of subgroups	Number of conjugacy classes	Size of each conjugacy class	Total number of subgroups	Isomorphism class of quotient (if exists)	Subnormal depth (if subnormal)	Note
trivial subgroup	$\{()\}$	trivial group	1	12	1	1	1	alternating group:A4	1	trivial
subgroup generated by double transposition in A4	$\{(),(1,2)(3,4)\}$, $\{(),(1,3)(2,4)\}$, $\{(),(1,4)(2,3)\}$	cyclic group:Z2	2	6	1	3	3	--	2
V4 in A4	$\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$	Klein four-group	4	3	1	1	1	cyclic group:Z3	1	2-Sylow, minimal normal, maximal
A3 in A4	$\{(),(2,3,4),(2,4,3)\}$, $\{(),(1,3,4),(1,4,3)\}$, $\{(),(1,2,4),(1,4,2)\}$, $\{(),(1,2,3),(1,3,2)\}$	cyclic group:Z3	3	4	1	4	4	--	--	3-Sylow, maximal
whole group	all elements	alternating group:A4	12	1	1	1	1	trivial group	1	whole
Total (5 rows)	--	--	--	--	5	--	10	--	--	--

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

1 Which of the following factors of 12 does not occur as the order of a subgroup of the alternating group of degree four?

	1
	2
	3
	4
	6

2 Which of the following groups is not isomorphic to any subgroup of the alternating group of degree four?

	trivial group
	cyclic group:Z2, the cyclic group of order two
	cyclic group:Z3, the cyclic group of order three
	cyclic group:Z4, the cyclic group of order four
	Klein four-group, the non-cyclic group of order four