Difference between revisions of "D8 in S4"

From Groupprops
Jump to: navigation, search
Line 14: Line 14:
  
 
<math>\! H_2 = \{ (), (1,2,4,3), (1,4)(2,3), (1,3,4,2), (1,2)(3,4), (1,3)(2,4), (1,4), (2,3) \}</math>
 
<math>\! H_2 = \{ (), (1,2,4,3), (1,4)(2,3), (1,3,4,2), (1,2)(3,4), (1,3)(2,4), (1,4), (2,3) \}</math>
 +
 +
==Arithmetic functions==
 +
 +
{| class="sortable" border="1"
 +
! Function !! Value !! Explanation !! Comment
 +
|-
 +
| order of the group || 24 || ||
 +
|-
 +
| order of the subgroup || 8 || ||
 +
|-
 +
| [[index of a subgroup|index of the subgroup]] || [[arithmetic function value::index of a subgroup;3|3]] || ||
 +
|-
 +
| size of conjugacy class || 3 || ||
 +
|-
 +
| number of conjugacy classes in automorphism class || 1 || ||
 +
|}
 +
 +
==Effect of subgroup operators==
 +
 +
In the table below, we provide values specific to <math>H</math>.
 +
 +
{| class="sortable" border="1"
 +
! Function !! Value as subgroup (descriptive) !! Value as subgroup (link) !! Value as group
 +
|-
 +
| [[normalizer]] || the subgroup itself || (current page) || [[dihedral group:D8]]
 +
|-
 +
| [[centralizer]] || <math>\{ (), (1,3)(2,4) \}</math> || [[subgroup generated by double transposition in S4]] || [[cyclic group:Z2]]
 +
|-
 +
| [[normal core]] || <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}</math> || [[normal Klein four-subgroup of symmetric group:S4]] || [[Klein four-group]]
 +
|-
 +
| [[normal closure]] || the whole group || -- || [[symmetric group:S4]]
 +
|-
 +
| [[characteristic core]] || <math>\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}</math> || [[normal Klein four-subgroup of symmetric group:S4]] || [[Klein four-group]]
 +
|-
 +
| [[characteristic closure]] || the whole group || -- || [[symmetric group:S4]]
 +
|}
 +
 +
==Subgroup properties==
 +
 +
===Resemblance-based properties===
 +
 +
{| class="sortable" border="1"
 +
! Property !! Meaning !! Satisfied? !! Explanation !! Comment
 +
|-
 +
| [[satisfies property::order-conjugate subgroup]] || [[conjugate subgroups|conjugate]] to any subgroup of the same order || Yes || [[Sylow implies order-conjugate]] ||
 +
|-
 +
| [[satisfies property::order-dominating subgroup]] || any subgroup of the whole group whose order divides the order of <math>H</math> is contained in a conjugate of <math>H</math> || Yes || [[Sylow implies order-dominating]] ||
 +
|-
 +
| [[satisfies property::order-dominated subgroup]] || any subgroup of the whole group whose order is a multiple of the order of <math>H</math> contains a conjugate of <math>H</math> || Yes || [[Sylow implies order-dominated]] ||
 +
|-
 +
| [[satisfies property::order-isomorphic subgroup]] || isomorphic to any subgroup of the group of the same order || Yes || (via order-conjugate) ||
 +
|-
 +
| [[satisfies property::isomorph-automorphic subgroup]] || || Yes || (via order-conjugate) ||
 +
|-
 +
| [[satisfies property::automorph-conjugate subgroup]] || || Yes || (via order-conjugate) ||
 +
|-
 +
| [[satisfies property::order-automorphic subgroup]] || || Yes || (via order-conjugate) ||
 +
|-
 +
| [[satisfies property::isomorph-conjugate subgroup]] || || Yes || (via order-conjugate) ||
 +
|}
 +
 +
===Normality-related properties===
 +
 +
{| class="sortable" border="1"
 +
! Property !! Meaning !! Satisfied? !! Explanation !! Comment
 +
|-
 +
| [[dissatisfies property::normal subgroup]] || equals all its [[conjugate subgroups]] || No || (see other conjugate subgroups) ||
 +
|-
 +
| [[dissatisfies property::subnormal subgroup]] || || No || ||
 +
|-
 +
| [[satisfies property::self-normalizing subgroup]] || equals its [[normalizer]] in the whole group || Yes || ||
 +
|-
 +
| [[satisfies property::abnormal subgroup]] || || Yes || ||
 +
|-
 +
| [[satisfies property::weakly abnormal subgroup]] || || Yes || ||
 +
|-
 +
| [[satisfies property::contranormal subgroup]] || || Yes || ||
 +
|-
 +
| [[satisfies property::maximal subgroup]] || || Yes || ||
 +
|-
 +
| [[satisfies property::pronormal subgroup]] || || Yes || [[Sylow implies pronormal]] ||
 +
|-
 +
| [[satisfies property::weakly pronormal subgroup]] || || Yes || (via pronormal) ||
 +
|}

Revision as of 23:02, 17 December 2010

This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) dihedral group:D8 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

This article is about the subgroup H in the group G, where G is symmetric group:S4, i.e., the symmetric group on the set \{ 1,2,3,4 \}, and H is the subgroup:

\! H = \{ (), (1,2,3,4), (1,3)(2,4), (1,4,3,2), (1,2)(3,4), (1,4)(2,3), (1,3), (2,4) \}

H is a 2-Sylow subgroup of G, and has two other conjugate subgroups, which are given below:

\! H_1 = \{ (), (1,3,2,4), (1,2)(3,4), (1,4,2,3), (1,3)(2,4), (1,4)(2,3), (1,2), (3,4) \}

and:

\! H_2 = \{ (), (1,2,4,3), (1,4)(2,3), (1,3,4,2), (1,2)(3,4), (1,3)(2,4), (1,4), (2,3) \}

Arithmetic functions

Function Value Explanation Comment
order of the group 24
order of the subgroup 8
index of the subgroup 3
size of conjugacy class 3
number of conjugacy classes in automorphism class 1

Effect of subgroup operators

In the table below, we provide values specific to H.

Function Value as subgroup (descriptive) Value as subgroup (link) Value as group
normalizer the subgroup itself (current page) dihedral group:D8
centralizer \{ (), (1,3)(2,4) \} subgroup generated by double transposition in S4 cyclic group:Z2
normal core \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \} normal Klein four-subgroup of symmetric group:S4 Klein four-group
normal closure the whole group -- symmetric group:S4
characteristic core \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \} normal Klein four-subgroup of symmetric group:S4 Klein four-group
characteristic closure the whole group -- symmetric group:S4

Subgroup properties

Resemblance-based properties

Property Meaning Satisfied? Explanation Comment
order-conjugate subgroup conjugate to any subgroup of the same order Yes Sylow implies order-conjugate
order-dominating subgroup any subgroup of the whole group whose order divides the order of H is contained in a conjugate of H Yes Sylow implies order-dominating
order-dominated subgroup any subgroup of the whole group whose order is a multiple of the order of H contains a conjugate of H Yes Sylow implies order-dominated
order-isomorphic subgroup isomorphic to any subgroup of the group of the same order Yes (via order-conjugate)
isomorph-automorphic subgroup Yes (via order-conjugate)
automorph-conjugate subgroup Yes (via order-conjugate)
order-automorphic subgroup Yes (via order-conjugate)
isomorph-conjugate subgroup Yes (via order-conjugate)

Normality-related properties

Property Meaning Satisfied? Explanation Comment
normal subgroup equals all its conjugate subgroups No (see other conjugate subgroups)
subnormal subgroup No
self-normalizing subgroup equals its normalizer in the whole group Yes
abnormal subgroup Yes
weakly abnormal subgroup Yes
contranormal subgroup Yes
maximal subgroup Yes
pronormal subgroup Yes Sylow implies pronormal
weakly pronormal subgroup Yes (via pronormal)