Dr David McGrath
Spine Physician
MB BS (Hons) FAFOM, RACP, FAFMMMaster of Pain Medicine
All elements of K4 belong to their own conjugacy class. eg (HRH=R, VRV=R, IRI=R) The four classes are I,H,R,V
Abelian implies aG=Ga all elements (a) of a subgroup. Thus LEFT coset=RIGHT coset. All subgroups are NORMAL.
HH=VV=RR=I. ie permutations are OWN inverse.
HR=V, RV=H, HV=R
G/H=(R,V) G/R=(H,V) G/V=(R,H) Quotient Groups.
Thus two subgroups sufficient to describe the actions of the group. eg R,H
Replace (xz) by generic (&%). Replace 1,2,3,4 by non ordered (a,b,c,d)
Four sets:
(a=1,b=2,c=3,d=4,&=x,%=z)
(a=2,b=1,c=4,d=3,&=x,%=z)
(a=3,b=4,c=1,d=2,&=z,%=x)
(a=4,b=3,c=2,d=1,&=z,%=x)
Creation b&a=a | Creation b%c=a | |||
Participation c&a=c |
Recursion a%c=a | |||
(a)Neighbourhood | ||||
Participation d&a=c | Recursion a&a=a | |||
Destruction a&b=b | Destruction a%d=b |
1.(b) creates and destroys (a) eg for (1) [1@ 2=2, 1@4=2 ] and [2@1=1, 2 @ 3 =1]
2.(a),(c) recursion modification eg for (1) [1@3=1] and [1@1=1]
3. (a)(d) mutual partners in creating (b)(c) [3@1=3] and [4@1=3]
4. Participation/Recursion preserves total (a)
5. Total (a)+(b) constant. Extreme imbalance implies (a)=0 and no participation in (c) recursion/creation.
N1 | N2 | N3 | N4 | |
N1 | 1 | 2 | 3 | 4 |
N2 | 2 | 1 | 4 | 3 |
N3 | 3 | 4 | 1 | 2 |
N4 | 4 | 3 | 2 | 1 |
Z2 (H) LEFT Action on Neighbourhood Set
ie (12)(34) LEFT Cosets under H Reflection
Identity | 2143 | |
N1 | 1234 | 2143 |
N2 | 2143 | 1234 |
N3 | 3412 | 4321 |
N4 | 4312 | 3412 |
ie (12)(34) RIGHT Cosets under H Reflection. RIGHT=LEFT ie H is Normal Subgroup
Z2(V) Subgroup LEFT Action on Neighbourhood Set
ie (13)(24) Orbits under V Reflection
Identity 1234 | 3412 | |
N1 | 1234 | 3412 |
N2 | 2143 | 4321 |
N3 | 3412 | 1234 |
N4 | 4312 | 2143 |
Z2(R) Subgroup LEFT Action on Neighbourhood Set
ie (14)(23) Orbits under R Rotation
K4 (Z2xZ2) Group Transitive Action on Neighbourhood Set
©Copyright 2007 Dr David McGrath. All rights reserved