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 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)

Generic Neighbourhood Object (a)

	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.

 

Group Matrix on Neighbourhood Set

	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

	N1	N2	N3	N4
Identity (1234)	1234	2143	3412	4321
Horizontal Reflection=H(12)(34)=2143	2143	1234	4321	3412

 ie (12)(34) LEFT Cosets under H Reflection

 

Z2 (H) RIGHT Coset Action

	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

	N1	N2	N3	N4
Identity I=(1234)	1234	2143	3412	4312
Vertical Reflection=V(13)(24)=3412	3412	4321	1234	2143

ie (13)(24) Orbits under V Reflection 

 

Z2(V) RIGHT Coset Action

	Identity 1234	3412
N1	1234	3412
N2	2143	4321
N3	3412	1234
N4	4312	2143

 

Z2(R) Subgroup LEFT Action on Neighbourhood Set

	N1	N2	N3	N4
Identity I=(1234)	1234	2143	3412	4312
Rotation =R(14)(23)=4321	4321	3412	2143	1234

ie (14)(23) Orbits under R Rotation 

 

 

 

 

 

K4 (Z2xZ2) Group Transitive Action on Neighbourhood Set

	N1	N2	N3	N4
Identity=I(1234)	xyx	xyz	zyx	zyz
Horizontal Reflection=H(12)(34)	xy(x)z	xy(x)x	zy(z)z	zy(z)x
Vertical Reflection=V(13)(24)	zy(z)x	zy(z)z	xy(x)x	xy(x)z
Rotation=R(14)(23)	zy(z)z	zy(z)x	xy(x)z	xy(x)x
HH=I	xy(xx)x	xy(zz)z	zy(zz)x	zy(zz)z
VV=I	xy(zz)x	xy(zz)z	zy(xx)x	zy(xx)z
RR=I	xyx	xyz	zyx	zyz
HV=R	zy(xz)z	zy(xz)x	xy(zx)z	xy(zx)x
VH=R	zy(zx)z	zy(zx)x	xy(xz)z	xy(xz)x
HR=V	zy( )x
RH=V

 

 

 

 

©Copyright 2007 Dr David McGrath. All rights reserved