dihedral group segi3

8/10/2019 Dihedral Group Segi3

1/25

DIHEDR LGROUP

Dina Fitriya Alwi 13610012 Risna Zulfa Musriroh 13610013

Nur Hidayati 13610087


2/25

DIHEDRAL GROUP

Dehidral group is a group of symmetris

compilation from regular side-n, notated byDn, for each nis the positive integer, n 3.

Regular Poligon with n side has 2n different

Symmetry, it's n ratation symmetry and n

refection symmetry. if n is odd, each axis of symmetry connect

center line to accros line of it.

if n is even there are

symmetry axis

connecting between side and side other and

symmetry axis connecting between angle and

angle other.


3/25

R1

R2

R3

F1

F2F

3IHEDR LOFTRI NGLE


4/25

ROTATION AND REFLECTION ON DIHEDRAL GROUP

For the rotation on , will be shown on triangelflat

And the rotation will be written as follow :

1

2 3

= 120

= 240

= 360


5/25

In will be rotated as far as 120at a center point 0 and opposite with clockwise

So,

1 =2

2 =3 =1 2 3

2 3 1

3 =1

Then will be written in cycle form as follow :

=(1 2 3)

In will be rotated as far as 240at a center point 0 at a center point and opposite

with clockwise

So, (1) = 3

3 =2 =1 3 2

3 2 1

(2) = 1


= (1 3 2)


6/25

In will be rotated as far as 360 at a center point 0 at a center

point an opposite with clockwise

So, (1) = 1

(2) = 2 =1 2 3

1 2 3

(3) = 3


= 1 2 (3)


7/25

The following reflection at two dimensional shape(triagle)

And its reflection will be written as follows :

1

2 3


8/25

Reflection about 1taxis

= (2 3)

Reflection about 2ndaxis

= (1 3)

Reflection about 3daxis

= (1 2)


9/25


10/25

ROTATIONOPPOSITEWITHCLOCKWISEDIRECTION

R1

R2

R3

F1

F2F3

1

2 3


11/25

ROTATIONEQUALCLOCKWISEDIRECTION

R3

R2

R1

F1

F3F2

1

3 2


12/25

Symmetry on Triangle

Example:

- Rotation notated with Rn

- Reflection notated with Fn

P3

= R1,

R2,R3,F1,F2,F3

with operation of compostion form the group.

R3 R1 R2 F1 F2 F3

R3 R3 R1 R2 F1 F2 F3

R1 R1 R2 R3 F3 F1 F2

R2 R2 R3 R1 F2 F3 F1

F1 F1 F2 F3 R3 R1 R2F2 F2 F3 F1 R2 R3 R1

F3 F3 F1 F2 R1 R2 R3


13/25

R1R1= R2

R1= (1 2 3); R2= (1 3 2)

CompostionR1R1(1)=R1 (R1(1))=R1(2)=3

R1R1(2)=R1 (R1(2))=R1(3)=1

R1R1(3)=R1 (R1(3))=R1(1)=2

Permutation

R1R1=

=

Cycle

R1R1= (1 2 3)(1 2 3)

= (1 3 2)

R3F2= F2

R3 = (1) (2) (3) ; F2 = (1 3)

CompostionR3F2(1)=R3 (F2(1))=R3(3)=3

R3F2(2)=R3 (F2(2))=R3(2)=2

R3F2(3)=R3 (F2(3))=R3(1)=1

Permutation

R3F2=

=

Cycle

R3F2= I (1 3)

= (1 3)


14/25

F1R1= F2

R1= (1 2 3); F1= (2 3); F2= (1 3)

CompostionF1R1(1)=F1 (R1(1))=F1(2)=3

F1R1(2)=F1 (R1(2))=F1(3)=2

F1R1(3)=F1 (R1(3))=R1(1)=1

Permutation

R1F1=

=

Cycle

F1R1= (1 2 3)(2 3)

= (1 3)

F3F3= R3

F3 = (1 2); R3 = (1) (2) (3)

CompostionF3F3(1)=F3 (F3(1))=F3(2)=1

F3F3(2)=F3 (F3(2))=F3(1)=2

F3F3(3)=F3 (F3(3))=F3(3)=3

Permutation

F3F3=

=

Cycle

F3F3= (1 2) (1 2)

= (1) (2) (3)


15/25

BIJECTIONFUNCTION, , *

>

>

>

>

>

>

r

1

s

sr


16/25

CHARACTERISTIC OF DIHEDRAL GROUP

(Dummit and Foote, 2004)


17/25

The Relation from P3to D6

1 r r2 s sr sr2

1 1 r r2 s sr sr2

r r r2 1 sr2 s sr

r2 r2 1 r sr sr2 s

s s sr sr2 1 r r2

sr sr sr2 s r2 1 r

sr2 sr2 s sr r r2 1

Implies: - The set of rotation = R1,R2,R3

- The set of reflection =F1,F2,F3

then related R1~ and F1~

D6 =D2.3= 1,r,r2, s, sr, sr2


18/25

rr =r2

rr2=r3=1

r2 r=r3=1

r2 r2=r4=r

rsr=srr1

=sr(r2)

=s

rsr2=sr2r1

=sr2(r2)

=sr

sr=r-1 s

=s (r1)1

=sr

srr2

=(r2

)-1

s=sr (r2)1)1

=sr(r2)

=sr3

=s

ssr=s r

=s r

=1r

=r srsr2=srs r2

=ss r1 r2

=ss r2 r2

=r4=r


19/25

LATTICEDIAGRAMFORSUBGRUPSOFD6

D6= ,

r s sr sr2

1

=

,,2 =

, =

1, sr=sr

1, sr2 =sr2


20/25

= r s

= .= (1, r, , s, sr, )

Subgroup of dihedral groups-6 are:

{1}=1

{1 r +=

*1, s+=s

{1 sr+=sr

{1 +=

r,s

r

s

sr

s

1

LATTICEDIAGRAMFORSUBGRUPSOFD6

{1 )


21/25

8= .= {1, r, , , s, sr, , )

Subgroup of dihedral groups-8 are:

1. D8= r, s

2. {1}=r3. {1, r, , +=r

4. {1, }=

5. {1, s}=s

6. {1, sr+=sr

7. {1, +=

8. {1, }=

9. {1, , s, }=, s

10. {1, , sr, +=, sr

r, s

r2, s

r

, sr

s

sr


22/25

HOMORFISME& ISOMORFISME

() = () *()

() = *

() =

=

For all element to preserving propery


23/25

INVERS

I = r

1= 1 s= s

r= r (sr)= sr

(r)=r (sr)= sr

IDENTITY

rr= r

rr= r

rr= r= 1

So, can be known that identity = r


24/25

ORDER

(R)= = | |= 3

(R)= = | |= 3

(R)= | |= 1

(F)= = | |= 2

(F)= = | |= 2

(F)== | |= 2


25/25

ORDER

= 3 ||=3

2 2 2= 6 3 |2|=3

3= 3

|3|=1

= 3 ||=3 s = 3

|s|=3

2 2 = 3 |s2|=3

|s|=|s2

|=|s3

|=|sn1

|=2

Because s is notation of reflection, sufficiently 2 times reflection to

return original position then order from reflection is 2.

dihedral group segi3

Documents