Subgroup structure of alternating group:A4: Difference between revisions

Latest revision as of 03:37, 15 November 2013

This article gives specific information, namely, subgroup structure, about a particular group, namely: alternating group:A4.
View subgroup structure of particular groups | View other specific information about alternating group:A4

The alternating group on ${1, 2, 3, 4}$ is a group of order 12.

There is no subgroup of order $6$ . The alternating group of degree four is the group of smallest possible order (in this case $12$ ) not having subgroups of all orders dividing the group order.

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	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	--	--	--

Table classifying isomorphism types of subgroups

Group name	Order	Second part of GAP ID (first part is order)	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Automorphism classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
trivial group	1	1	1	1	1	1	1
cyclic group:Z2	2	1	3	1	1	0	0
cyclic group:Z3	3	1	4	1	1	0	0
Klein four-group	4	2	1	1	1	1	1
alternating group:A4	12	3	1	1	1	1	1
Total	--	--	10	5	5	3	3

Table listing numbers of subgroups by group property

Group property	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Automorphism classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
cyclic group	8	3	3	1	1
abelian group	9	4	4	2	2
nilpotent group	9	4	4	2	2
solvable group	10	5	5	3	3

@@ Line 6: / Line 6: @@
 [[File:A4latticeofsubgroups.png|thumb|400px|right]]
-The alternating group on <math>\{ 1,2,3,4 \}</math> has the following subgroups (clubbed together by conjugacy):
+The alternating group on <math>\{ 1,2,3,4 \}</math> is a group of order 12.
-# The trivial subgroup. (1)
-# [[subgroup generated by double transposition in A4]]: Three subgroups of order two, each generated by a double transposition, such as <math>(1,2)(3,4)</math>. These are all isomorphic to the [[cyclic group:Z2|cyclic group of order two]]. (3)
-# [[V4 in A4]]:A subgroup of order four, comprising the identity element and the three double transpositions: <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}</math>. This is isomorphic to the [[Klein four-group]]. It is the unique proper nontrivial normal subgroup. The quotient group is isomorphic to [[cyclic group:Z3]]. (1)
-# [[A3 in A4]]: Four subgroups of order three, each generated by a <math>3</math>-cycle, such as <math>(1,2,3)</math>. These are all isomorphic to the [[cyclic group:Z3|cyclic group of order three]]. These can be thought of as alternating groups on subsets of size three. (4)
-# The whole group. (1)
 There is no subgroup of order <math>6</math>. The alternating group of degree four is the group of smallest possible order (in this case <math>12</math>) ''not'' [[group having subgroups of all orders dividing the group order|having subgroups of all orders dividing the group order]].
 ==Tables for quick information==
+{{finite solvable group subgroup structure facts to check against}}
+<section begin="summary"/>
+===Quick summary===
+{| class="sortable" border="1"
+! 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===
 {| class="sortable" border="1"
-! Automorphism class of subgroups !! Isomorphism class !! Number of conjugacy classes !! Size of each conjugacy class !! Isomorphism class of quotient (if exists) !! [[Subnormal depth]] (if subnormal)
+! Automorphism class of subgroups !! List of subgroups !! Isomorphism class !! [[Order of a group|Order]] of subgroups !! [[Index of a subgroup|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 || 1 || [[alternating group:A4]] || 1
+| trivial subgroup || <math>\{ () \}</math> || [[trivial group]] || 1 || 12 || 1 || 1 || 1 || [[alternating group:A4]] || 1 || trivial
 |-
-| [[subgroup generated by double transposition in A4]] || [[cyclic group:Z2]] || 1 || 3 || -- || 2
+| [[subgroup generated by double transposition in A4]] || <math>\{ (), (1,2)(3,4) \}</math>, <math>\{ (), (1,3)(2,4) \}</math>, <math>\{ (), (1,4)(2,3) \}</math> || [[cyclic group:Z2]] || 2 || 6 || 1 || 3 || 3 ||  -- || 2 ||
 |-
-| [[V4 in A4]] || [[Klein four-group]] || 1 || 1 || [[cyclic group:Z3]] || 1
+| [[V4 in A4]] || <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}</math> || [[Klein four-group]] || 4 || 3 || 1 || 1 || 1 || [[cyclic group:Z3]] || 1 || 2-Sylow, minimal normal, maximal
 |-
-| [[A3 in A4]] || [[cyclic group:Z3]] || 1 || 4 || -- || --
+| [[A3 in A4]] || <math>\{ (), (2,3,4), (2,4,3) \}</math>, <math>\{ (), (1,3,4), (1,4,3)\}</math>, <math>\{ (), (1,2,4), (1,4,2)\}</math>, <math>\{ (), (1,2,3), (1,3,2) \}</math> || [[cyclic group:Z3]] || 3 || 4 || 1 || 4 || 4 ||  -- || -- || 3-Sylow, maximal
 |-
-| whole group || [[alternating group:A4]] || 1 || 1 || [[trivial group]] || 1
+| whole group || all elements || [[alternating group:A4]] || 12 || 1 || 1 || 1 || 1 || [[trivial group]] || 1 || whole
+|-
+! Total (5 rows) || -- || -- || -- || -- || 5 || -- || 10 || -- || -- || --
 |}
+<section end="summary"/>
 ===Table classifying isomorphism types of subgroups===
@@ Line 43: / Line 56: @@
 | [[cyclic group:Z2]] || 2 || 1 || 3 || 1 || 1 || 0 || 0
 |-
-| [[cyclic group:Z3]] || 3 || 1 || 4 || 1 || 1 | 0 || 0
+| [[cyclic group:Z3]] || 3 || 1 || 4 || 1 || 1 || 0 || 0
 |-
 | [[Klein four-group]] || 4 || 2 || 1 || 1 || 1 || 1 || 1