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Subgroup structure of dihedral group:D8

689 bytes added, 19:49, 7 June 2012
Subgroup-defining functions and associated quotient-defining functions
| [[center of dihedral group:D8|center]] || <math>\{ e,a^2 \}</math> || [[cyclic group:Z2]] || 2 || 4 || 1 || 1 || 1 || [[Klein four-group]] || 1 || 1
|-
| [[2non-subnormal normal subgroups of dihedral group:D8|other subgroups of order two]] || <math>\{e,x \}, \{ e,a^2x \}</math> <br> <math>\{ e,ax \}, \{ e,a^3x \}</math> || [[cyclic group:Z2]] || 2 || 4 || 2 || 2 || 4 || -- || 2 || 1
|-
| [[Klein four-subgroups of dihedral group:D8|Klein four-subgroups]] || <math>\{ e,x,a^2,a^2x \}</math>, <math>\{ e,ax,a^2,a^3x \}</math> || [[Klein four-group]] || 4 || 2 || 2 || 1 || 2 || [[cyclic group:Z2]] || 1 || 1
| whole group || <math>\{ e,a,a^2,a^3,x,ax,a^2x,a^3x \}</math> ||[[dihedral group:D8]] || 8 || 1 || 1 || 1 || 1 || [[trivial group]] || 0 || 2
|-
| ! Total (6 rows)|| !! -- || !! -- || !! -- || !! -- || !! 8 || !! -- || !! 10 || !! -- || !! -- || !! --
|}
<section end="summary"/>
 
===Table classifying isomorphism types of subgroups===
| [[Jacobson radical]] || intersection of all [[maximal normal subgroup]]s || [[center of dihedral group:D8]]: <math>\{ e, a^2 \}</math> || [[cyclic group:Z2]] || 2 || ? || [[Klein four-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
|- | [[Fitting subgroupjoin of all abelian normal subgroups]] || join of subgroup generated by all the [[nilpotent abelian normal subgroup]]s || whole group || [[dihedral group:D8]] || 8 || [[Fitting quotient]] ? || [[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
|-
| [[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> || [[trivial group]] || 1 || largest quotient group that is a capable group || [[dihedral group:D8]] || 8
|}
 
Some more notes:
 
* The following subgroup-defining functions are equal to the whole group on account of the group being a [[nilpotent group]]: [[Fitting subgroup]], [[hypercenter]], [[solvable radical]].
* The following subgroup-defining functions are equal to the trivial subgroup on account of the group being a [[solvable group]]: [[hypocenter]], [[nilpotent residual]], [[perfect core]], [[solvable residual]].
 
<section end="sdf summary"/>
| [[Frattini series]] || descending || whole group || [[Frattini subgroup]]: <math>\{ e, a^2 \}</math> -- [[center of dihedral group:D8]] || trivial || trivial || trivial
|-
| [[upper Fitting series]] || ascending || trivial || [[Fitting subgroup]]: whole group || whole group || whole group || whole group
|-
| [[socle series]] || ascending || trivial || [[socle]]: <math>\{ e, a^2 \}</math> -- [[center of dihedral group:D8]] || whole group || whole group || whole group
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