Post on 02-Jun-2018
8/10/2019 Dihedral Group Segi3
1/25
DIHEDR LGROUP
Dina Fitriya Alwi 13610012 Risna Zulfa Musriroh 13610013
Nur Hidayati 13610087
8/10/2019 Dihedral Group Segi3
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.
8/10/2019 Dihedral Group Segi3
3/25
R1
R2
R3
F1
F2F
3IHEDR LOFTRI NGLE
8/10/2019 Dihedral Group Segi3
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
8/10/2019 Dihedral Group Segi3
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
Then will be written in cycle form as follow :
= (1 3 2)
8/10/2019 Dihedral Group Segi3
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
Then will be written in cycle form as follow :
= 1 2 (3)
8/10/2019 Dihedral Group Segi3
7/25
The following reflection at two dimensional shape(triagle)
And its reflection will be written as follows :
1
2 3
8/10/2019 Dihedral Group Segi3
8/25
Reflection about 1taxis
= (2 3)
Reflection about 2ndaxis
= (1 3)
Reflection about 3daxis
= (1 2)
8/10/2019 Dihedral Group Segi3
9/25
8/10/2019 Dihedral Group Segi3
10/25
ROTATIONOPPOSITEWITHCLOCKWISEDIRECTION
R1
R2
R3
F1
F2F3
1
2 3
8/10/2019 Dihedral Group Segi3
11/25
ROTATIONEQUALCLOCKWISEDIRECTION
R3
R2
R1
F1
F3F2
1
3 2
8/10/2019 Dihedral Group Segi3
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
8/10/2019 Dihedral Group Segi3
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)
8/10/2019 Dihedral Group Segi3
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)
8/10/2019 Dihedral Group Segi3
15/25
BIJECTIONFUNCTION, , *
>
>
>
>
>
>
r
1
s
sr
8/10/2019 Dihedral Group Segi3
16/25
CHARACTERISTIC OF DIHEDRAL GROUP
(Dummit and Foote, 2004)
8/10/2019 Dihedral Group Segi3
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
8/10/2019 Dihedral Group Segi3
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
8/10/2019 Dihedral Group Segi3
19/25
LATTICEDIAGRAMFORSUBGRUPSOFD6
D6= ,
r s sr sr2
1
=
,,2 =
, =
1, sr=sr
1, sr2 =sr2
8/10/2019 Dihedral Group Segi3
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 )
8/10/2019 Dihedral Group Segi3
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
8/10/2019 Dihedral Group Segi3
22/25
HOMORFISME& ISOMORFISME
() = () *()
() = *
() =
=
For all element to preserving propery
8/10/2019 Dihedral Group Segi3
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
8/10/2019 Dihedral Group Segi3
24/25
ORDER
(R)= = | |= 3
(R)= = | |= 3
(R)= | |= 1
(F)= = | |= 2
(F)= = | |= 2
(F)== | |= 2
8/10/2019 Dihedral Group Segi3
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.