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Subgroup structure of dihedral group:D8

4,851 bytes removed, 19:49, 7 June 2012
Subgroup-defining functions and associated quotient-defining functions
group = dihedral group:D8|
connective = of}}
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The '''dihedral group''' <math>D_8</math>, sometimes called <math>D_4</math>, also called the {{dihedral group}} of order eight or the dihedral group acting on four elements, is defined by the following presentation:
<math>a = (1,2,3,4), \qquad x = (1,3)</math>.
The dihedral group has ten subgroups:==Tables for quick information==
# The trivial subgroup (1)# The [[center]], which is the unique minimal normal subgroup, and is a two-element subgroup generated by <math>a^2</math>. Isomorphic to [[cyclic group:Z2]]. {{further|[[center of dihedral group:D8]]}}(1)# The two-element subgroups generated by <math>x</math>, <math>ax</math>, <math>a^2x</math> and <math>a^3x</math>. Isomorphic to [[cyclic group:Z2]]. These come in two conjugacy classes: the subgroups generated by <math>x</math> and by <math>a^2x</math> are conjugate, and the subgroups generated by <math>ax</math> and by <math>a^3x</math> are conjugate. (4) {{further|[[non-normal subgroups of dihedral groupFile:D8]]}}# The four-element subgroup generated by <math>a^2</math> and <math>x</math>. This comprises elements <math>e,a^2,x,a^2x</math>. It is isomorphic to the [[Klein four-group]]. A similar four-element subgroup is obtained as that generated by <math>a^2</math> and <math>ax</math>. These are both normalD8latticeofsubgroups. (2) {{furtherpng|[[Klein four-subgroups of dihedral group:D8400px]]}}# The four-element subgroup generated by <math>a</math>. Isomorphic to [[cyclic group:Z4]]. (1) {{further|[[Cyclic maximal subgroup of dihedral group:D8]]}}# The whole group. (1)
We study here the properties of each of these ===Table classifying subgroups (except the trivial subgroup and the whole group). We denote the whole group by <math>G</math>.up to automorphisms===
First, a quick summary:{{finite p-group subgroup structure facts to check against}}
* Except the subgroups in (3), all subgroups are normal. Of the subgroups listed in (3), there are two conjugacy classes of subgroups, each comprising two subgroups. Both conjugacy classes are related by an outer automorphism.* The subgroups listed in (1), (2), (5) and (6) are characteristic. The two subgroups listed in (4) are normal, but are automorphs of each other.<section begin="summary"/>
[[Image:D8latticeofsubgroupsIn the "List of subgroups" columns below, a row break within the cell indicates that ''each row represents one conjugacy class of subgroups''.png]]==Tables for quick information== ===Table classifying subgroups up to automorphisms=== {| class="wikitablesortable" border="1"! 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 (=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 group|quotient]] (if existssubgroup is normal) !! [[Subnormal depth ]] (if proper and normal, this equals 1) !! [[Nilpotency class]]|-| trivial subgroup || <math>\{ e \}</math> || [[trivial group]] || 1 || 8 || 1 || 1 || 1 || [[dihedral group:D8]] || 1 || 0
|-
| trivial subgroup [[center of dihedral group:D8|center]] || <math>\{ e,a^2 \}</math> || [[trivial cyclic group:Z2]] || 2 || 4 || 1 || 1 || 1 || [[dihedral Klein four-group:D8]] || 1 || 0 1
|-
| [[center non-normal subgroups of dihedral group:D8|centerother subgroups of order two]] || <math>\{e,x \}, \{ e,a^2x \}</math> <br> <math>\{ e,ax \}, \{ e,a^3x \}</math> || [[cyclic group:Z2]] || 1 2 || 4 || 2 || 2 || 1 4 || [[Klein four-group]] - || 1 2 || 1
|-
| [[2Klein four-subnormal subgroups of dihedral group:D8|other Klein four-subgroups of order two]] || <math>\{ e,x,a^2,a^2x \}</math>, <math>\{ e,ax,a^2,a^3x \}</math> || [[cyclic Klein four-group:Z2]] || 4 || 2 || 2 || -- 1 || 2 || [[cyclic group:Z2]] || 1 || 1
|-
| [[Klein four-subgroups cyclic maximal subgroup of dihedral group:D8|Klein four-subgroupscyclic maximal subgroup]] || <math>\{ e,a,a^2,a^3 \}</math> || [[Klein four-cyclic group:Z4]] || 4 || 2 || 1 || 1 || 1 || [[cyclic group:Z2]] || 1 || 1
|-
| whole group || <math>\{ e,a,a^2,a^3,x,ax,a^2x,a^3x \}</math> || [[cyclic maximal subgroup of dihedral group:D8|cyclic maximal subgroup]] || [[cyclic group:Z4]] 8 || 1 || 1 || 1 || 1 || [[cyclic trivial group:Z2]] || 1 0 || 12
|-
| whole group || [[dihedral group:D8]] || 1 || 1 || [[trivial group]] || 0 || 2! Total (6 rows) !! -- !! -- !! -- !! -- !! 8 !! -- !! 10 !! -- !! -- !! --
|}
 
<section end="summary"/>
===Table classifying isomorphism types of subgroups===
The first part of the GAP ID is the order of the group. {| class="wikitablesortable" border="1"! Group name !! GAP ID !! Index !! Occurrences as subgroup (=1 iff [[isomorph-free subgroup]]) !! Conjugacy classes of occurrence as subgroup (=1 iff [[isomorph-conjugate subgroup]]) !! Automorphism classes of occurrence as subgroup (=1 iff [[isomorph-automorphic subgroup]]) !! Occurrences as normal subgroup (=1 iff [[normal-isomorph-free subgroup]]; =Occurrences as subgroup iff [[isomorph-normal subgroup]]) !! Occurrences as characteristic subgroup(=1 iff [[characteristic-isomorph-free subgroup]]; =Occurences as subgroup iff [[isomorph-characteristic subgroup]])
|-
| [[Trivial group]] || <math>(1,1)</math> || 8 || 1 || 1 || 1 || 1 || 1
|-
| [[Cyclic group:Z2]] || <math>(2,1)</math> || 4 || 5 || 3 || 2 || 1 || 1
|-
| [[Cyclic group:Z4]] || <math>(4,1)</math> || 2 || 1 || 1 || 1 || 1 || 1
|-
| [[Klein four-group]] || <math>(4,2)</math> || 2 || 2 || 2 || 1 || 2 || 0
|-
| [[Dihedral group:D8]] || <math>(8,3)</math> || 1 || 1 || 1 || 1 ||1 || 1
|-
| Total || -- || -- || 10 || 8 || 6 || 6 || 4
|}
===Table listing number of subgroups by order===
Note that these orders satisfy the [[congruence condition on number of subgroups of given prime power order]]: the number of subgroups of order <math>p^r</math> is congruent to <math>1</math> modulo <math>p</math>. Here, <math>p = 2</math>, so this means that the number of subgroups of any given order is odd. {| class="wikitablesortable" border="1"! Group order !! Index !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Automorphism classes of occurrences as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
|-
| <math>1</math>|| 8 || 1 || 1 || 1 || 1 || 1
|-
| <math>2</math> || 4 || 5 || 3 || 2 || 1 || 1
|-
| <math>4</math> || 2 || 3 || 3 || 2 || 3 || 1
|-
| <math>8</math> || 1 || 1 || 1 || 1 || 1 || 1
|-
| Total || -- || 10 || 8 || 6 || 6 || 4
|}
==The center (type (2))Defining functions==
{{further|[[center of dihedral group:D8]]}}===Subgroup-defining functions and associated quotient-defining functions===
This is <section begin="sdf summary"/>{| class="sortable" border="1"! [[Subgroup-defining function]] !! What it means !! Value as subgroup !! Value as group !! Order !! Associated quotient-defining function !! Value as group !! Order (= index of subgroup)|-| [[center]] || elements that commute with every group element || [[center of dihedral group:D8]]: <math>\{ e, a^2 \}</math> || [[cyclic group:Z2]] || 2 || [[inner automorphism group]] || [[Klein four-group]] || 4|-| [[derived subgroup]] || subgroup generated by all [[commutator]]s || [[center of dihedral group:D8]]: <math>\{ e, a^2 \}</math> || [[cyclic group:Z2]] || 2 || [[abelianization]] || [[Klein four-group]] || 4|-| [[Frattini subgroup]] || intersection of all [[maximal subgroup]]s || [[center of dihedral group:D8]]: <math>\{ e, a^2 \}</math> || [[cyclic group:Z2]] || 2 || [[Frattini quotient]] || [[Klein four-group]] || 4|-| [[Jacobson radical]] || intersection of all [[maximal normal subgroup]]s || [[center of dihedral group:D8]]: <math>\{ e, a two^2 \}</math> || [[cyclic group:Z2]] || 2 || ? || [[Klein four-element group]] || 4|-| [[socle]] || join of all [[minimal normal subgroup ]]s || [[center of dihedral group:D8]]: <math>\{ e, a^2\}</math> || [[cyclic group:Z2]] || 2 || [[socle quotient]] || [[Klein four-group]] || 4|-| [[Baer norm]] || intersection of [[normalizer]]s of all subgroups || [[center of dihedral group:D8]]: <math>\{ e, a^2 \}</math> || [[cyclic group:Z2]] || 2 || ? || [[Klein four-group]] || 4|-| [[join of all abelian normal subgroups]] || subgroup generated by all the [[abelian normal subgroup]]s || whole group || [[dihedral group:D8]] || 8 || ? || [[trivial group]] || 1|-| [[join of abelian subgroups of maximum order]] || join of all abelian subgroups of maximum order among abelian subgroups || whole group || [[dihedral group:D8]] || 8 || ? || [[trivial group]] || 1|-| [[join of abelian subgroups of maximum rank]] || join of all abelian subgroups of maximum rank among abelian subgroups || whole group || [[dihedral group:D8]] || 8 || ? || [[trivial group]] || 1|-| [[join of elementary abelian subgroups of maximum order]] || join of all elementary abelian subgroups of maximum order among elementary abelian subgroups || whole group || [[dihedral group:D8]] || 8 || ? || [[trivial group]] || 1|-| [[ZJ-subgroup]] || center of the [[join of abelian subgroups of maximum order]] || [[center of dihedral group:D8]]: <math>\{ e, a^2 \}</math> || [[cyclic group:Z2]] || 2 || ? || [[Klein four-group]] || 4|-| [[epicenter]] || intersection of images of centers for all central extensions || trivial subgroup: <math>\{ e\}</math>. It || [[trivial group]] || 1 || largest quotient group that is a capable group || [[characteristic subgroupdihedral group:D8]].|| 8|}
In the permutation representation, it is given by the setSome more notes:
<math>\{ (1* The following subgroup-defining functions are equal to the whole group on account of the group being a [[nilpotent group]]: [[Fitting subgroup]],3)(2[[hypercenter]],4)[[solvable radical]].* The following subgroup-defining functions are equal to the trivial subgroup on account of the group being a [[solvable group]]: [[hypocenter]], () \}</math>[[nilpotent residual]], [[perfect core]], [[solvable residual]].
<section end===Subgroup-defining functions yielding this subgroup==="sdf summary"/>
There are many subgroup===Subgroup series-defining functions that yield this subgroup, for instance:===
* The [[center]]: <math>Z(G) {| class="sortable" border= \{ a^2, e \}</math>. These are the only two elements that commute with every element. It is also equal to <math>\Omega^"1(Z(G))</math>, the subgroup generated by elements of order two in the center."! Series-defining function !! Type !! Zeroth member !! First member !! Second member !! Third member !! Stable member* The [[commutator subgroup]]: <math>[G,G] = \{ a^2, e \}</math>. The quotient group is isomorphic to the [[Klein four|-group]].* The | [[Frattini subgroupupper central series]]: It is the intersection of three maximal subgroups, each of order four. (these are covered in points (4) and (5) in the list).* The || ascending || trivial || [[soclecenter]]: In fact, <math>\{ e, a^2, e\}</math> is the ''unique'' -- [[minimal normal subgroup]].* The [[agemo subgroups center of a dihedral group of prime power order|first agemo subgroup]]: <math>\{ a^2, e \}</math> is the subgroup generated by all squares, and is hence <math>\mho^1(G)</math>.* The [[ZJ-subgroupD8]]: It is the || [[second center]] of the [[join of abelian subgroups of maximum order]]. ===Subgroup properties satisfied by this subgroup=== On account of being an agemo subgroup as well as on account of being the commutator subgroup, the center is a [[verbal subgroup]] -- it is a subgroup generated by words of a certain form (in the agemo description, these words are squares; in the commutator subgroup description, these words are commutators). Thus, it satisfies the following properties: * [[Fully invariant subgroup]]: It is invariant under any endomorphism of the whole group. {{further|[[Verbal implies fully invariant]]}}* [[Image-closed fully invariant subgroup]]: Its image under any surjective homomorphism is fully characteristic in the image. {{further|[[Verbal implies image-closed fully invariant]]}}* [[Image-closed characteristic subgroup]]: Its image under any surjective homomorphism is characteristic in the image. {{furtherwhole group ||[[Verbal implies image-closed characteristic]]}}* [[Characteristic subgroup]]: It is characteristic in the whole group.|-For obvious reasons, it satisfies the following properties: * | [[Central subgrouplower central series]], [[Central factor]]: These follow from its being equal to the center.* [[Simple normal subgroup]], [[Transitively normal subgroup]]* [[Base diagonal of a wreath product]] ===Subgroup properties not satisfied by this subgroup=== * [[Homomorph|| descending || -containing subgroup]]: There are homomorphic images of this subgroup that are not contained in it.* [[Isomorph-free subgroup]]: There are other subgroups of the || whole group isomorphic to it, namely, the subgroups of type (3) in the list.* || [[Intermediately characteristic derived subgroup]]: The subgroup <math>\{ a^2, e \}</math> is ''not'' characteristic in every intermediate subgroup. In particular, it is not characteristic in subgroups of the type (4), such as <math>\{ e, a^2, x, a^2x \}</math>. {{further|[[Characteristicity does not satisfy intermediate subgroup condition]], [[center not is intermediately characteristic]], [[commutator subgroup not is intermediately characteristic]], [[Frattini subgroup not is intermediately characteristic]]}}* [[Complemented normal subgroup]], [[Lattice-complemented subgroup]], [[Permutably complemented subgroup]]: There is no complement to the center. This is a phenomenon for all nilpotent groups, and follows from the fact that - [[nilpotent implies center is normality-large]]. ==The four-element characteristic subgroup (type (5))== {{further|[[Cyclic maximal subgroup of dihedral group:D8]]}}|| trivial || trivial  This subgroup is the set <math>\{ a, a^2, a^3, e \}</math>. In the permutation representation, it is given by <math>\{ (1,2,3,4), (1,3)(2,4), (1,4,2,3), ()\}</math>. ===Subgroup|-defining functions yielding this subgroup=== None of the standard choices of subgroup-defining functions yields this subgroup. It can be described using the following: * It is the unique | [[maximal among abelian characteristic subgroupsderived series]]. Note that for a general || descending || whole group, there need not exist a unique maximal among abelian characteristic subgroups. {{proofat||[[Maximal among abelian characteristic subgroups may be multiple and isomorphic]]}}* It is the unique [[constructibly critical subgroup]], i.e., the only subgroup that could arise through an application of the constructive procedure in [[Thompson's critical subgroup theorem]]. However, it is ''not'' the only critical subgroup. ===Subgroup properties satisfied by this subgroup=== The subgroup is a [[cyclic maximal subgroup]]. It satisfies the following properties: * [[Isomorph-free subgroup]], [[Isomorph-containing subgroup]]* [[Prehomomorph-contained subgroup]]* [[Maximal characteristic subgroup]]* [[Intermediately characteristic subgroup]]: This subgroup is characteristic in every intermediate subgroup.* [[Complemented normal subgroup]]* [[Regular kernel]]* [[Self-centralizing subgroup]]: It is not centralized by any outside element.* [[Coprime automorphism-faithful subgroup]]* [[Critical subgroup]]* [[Constructibly critical subgroup]] Some maximality properties of note: * [[Abelian subgroup of maximum order]]* [[Maximal among abelian subgroups]]* [[Maximal among abelian normal subgroups]]* [[Maximal among abelian characteristic subgroups]]* [[Centrally large subgroup]], [[centralizer-large subgroup]], [[minimal CL-subgroup]]* [[Subgroup with abelianization of maximum order]] ===Subgroup properties not satisfied by this subgroup=== * [[Fully characteristic subgroup]], [[retraction-invariant derived subgroup]]: There exists a retraction with kernel <math> \{ e, a^2, x, a^2x \}</math> and image <math>\{e, ax \}</math> that does ''not'' leave this subgroup invariant. Hence, it is not fully characteristic or retraction-invariant.* - [[Verbal subgroup]]: Since the subgroup is not fully characteristic, it is not verbal.* [[Image-closed characteristic subgroup]]: Quotienting out by the center of the whole dihedral group gives a subgroup that is not characteristic in the image. {{further|[[Characteristicity does not satisfy image condition:D8]]}}|| trivial || trivial || trivial  ==The non-characteristic four|-element subgroups (type (4))== {{further|[[Klein four-subgroups of dihedral group:D8Frattini series]]}} These two subgroups are related by an outer automorphism, but are ''not'' conjugate (in fact, both are normal subgroups). Since they're || descending || whole group || [[automorphFrattini subgroup]]s, they in particular satisfy and dissatisfy the same subgroup properties. The two subgroups are: <math>\{ e, a^2, x, a^2x \}</math> and <math>\{ e, a^2, ax, a^3x\}</math>. In terms of permutations, they are given by: <math>\{ (), (1,3)(2,4), (1,3), (2,4) \}</math> and <math>\{ (), (1,3)(2,4), (1,2)(3,4), (1,4)(2,3) \}</math>. Note that these two subgroups, while automorphs inside the dihedral group, are not automorphs inside the whole symmetric group. ===Subgroup properties satisfied by these subgroups=== * [[Maximal normal subgroup]]: They are normal subgroups of index two.* [[Isomorph-automorphic subgroup]]* [[Self-centralizing subgroup]] Properties related to maximality and abelianness: * [[Maximal among abelian subgroups]]* [[Abelian subgroup center of maximum orderdihedral group:D8]]|| trivial || trivial || trivial * [[Abelian subgroup of maximum rank]]* [[Elementary abelian subgroup of maximum order]]* [[Maximal among abelian normal subgroups]]* [[Centrally large subgroup]], [[centralizer|-large subgroup]], [[minimal CL-subgroup]]* | [[Subgroup with abelianization of maximum orderupper Fitting series]] ===Subgroup properties not satisfied by these subgroups=== * || ascending || trivial || [[Automorph-conjugate Fitting subgroup]]: Although the two subgroups are automorphs of each other, they are not conjugate. Hence, neither of them is an automorph-conjugate subgroup.* [[p-normal-extensible automorphism-invariant subgroup]]: These subgroups are ''not'' invariant under all the automorphisms of the whole group that are [[p-normal-extensible automorphism|p-normal-extensible]], i.e., those automorphisms that can be extended to automorphisms for any <math>2</math>-| whole group || whole group containing the dihedral || whole group as a normal subgroup. This follows from the fact that ''every'' automorphism is [[p-normal-extensible automorphism|p-normal-extensible]]. {{further|[[Finite p-group with center of prime order and inner automorphism group maximal in p-Sylow-closure of automorphism group implies every p-automorphism is p-normal-extensiblesocle series]]}} ==The two-element non-normal subgroups (type (3))== {{further|| ascending || trivial ||[[non-normal subgroups of dihedral group:D8socle]]}} There are four of these: <math>\{ e, x \}</math>, <math>\{ e, ax \}</math>, <math>\{ e,a^2x \}</math>, and <math>\{ e,a^3x \}</math>. In terms of permutations, these are the subgroups <math>\{ (), (1,3) \}</math>, <math>\{ (), (1,2)(3,4) \}</math>, <math>\{ (), (2,4) \}</math> and <math>\{ (), (1,4)(2,3)</math>. The subgroups <math>\{ e, x \}</math> and <math>\{ e,a^2x \}</math> are conjugate to each other, and the subgroups <math>\{ e, ax\}</math> and <math>\{ e, a^3x \}</math> are conjugate to each other. These two pairs of subgroups are not conjugate, but they are related by an outer automorphism of the dihedral group -- one that does not extend to the symmetric group on four letters. ===Subgroup properties satisfied by these subgroups=== * [[2-subnormal subgroup]]* [[2-hypernormalized subgroup]]* [[Conjugate-permutable subgroup]]* [[Core-free subgroup]]* [[Base center of a wreath productdihedral group:D8]]|| whole group || whole group || whole group ===Subgroup properties not satisfied by these subgroups=== * [[Permutable subgroup]]* [[Automorph-permutable subgroup]]* [[Automorph-conjugate subgroup]]|}
==Lattice of subgroups==
[[Image:D8latticeofsubgroups.png|1000px]]
===The entire lattice===
===The sublattice of normal subgroups===
 
[[File:D8latticeofnormalsubgroups.png|1000px]]
For the sublattice of normal subgroups, we delete the four 2-subnormal subgroups of order two, leaving only the center. The center is the unique minimal normal subgroup (i.e., a [[monolith]]) and is contained in three maximal normal subgroups of order four. Note that this lattice is isomorphic to the lattice of normal subgroups of the [[quaternion group]], but the quaternion group has no non-normal subgroups.
Some aspects of this generalize to arbitrary <math>p</math>-groups: if the center is of order <math>p</math> it is the unique minimal normal subgroup. In fact, more general results include: [[prime power order implies center is normality-large]], [[minimal normal implies central in nilpotent]], [[omega-1 of center is normality-large in nilpotent p-group]]. The upshot of all these results is that the minimal normal subgroups of a finite <math>p</math>-group are ''precisely'' the subgroups of order <math>p</math> in the center.
The lattice of normal subgroups is isomorphic to that for the [[quaternion group]]. In fact, these are both [[groups having the same Hall-Senior genus]], namely <math>8\Gamma_2a</math>. The picture is given by:
 
[[File:8Gamma2a.png|1000px]]
===The sublattice of characteristic subgroups===
The sublattice of characteristic subgroups is totally ordered, comprising the identity, the center, the four-element cyclic subgroup, and the whole group. Note that there is one characteristic subgroup of every order dividing the group order. This differs from the quaternion group case, which has no characteristic subgroup of order four.
===The sublattice of fully characteristic invariant subgroups===
The sublattice of fully characteristic invariant subgroups comprises the identity element, the center, and the whole group. It is totally ordered.
===Abelian subgroups and elementary abelian subgroups===
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