The Nonabelian Group of Order Six

S3D3: The Nonabelian Group of Order Six

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 14 December 2010

The nonabelian group of order six

Let 1= 1 0 0 0 1 0 0 0 1 , 12= 0 1 0 1 0 0 0 0 1 , 23= 1 0 0 0 0 1 0 1 0 , 13= 0 0 1 0 1 0 1 0 0 , 132= 0 1 0 0 0 1 1 0 0 , 123= 0 0 1 1 0 0 0 1 0 .

The groups S3 and D3 are as in the following table.

Set Operation
S3= 1,12, 23,13, 132, 123 Ordinary matrix multiplication
D3= 1,x,x2, y,xy,x2y xiyj xkyl= xi-kmod3 yj+lmod2

The complete multiplication tables for these groups are as follows.

Multiplication tables
S3 1 12 23 13 132 123
1 1 12 23 13 132 123
12 12 1 123 132 13 12
23 23 132 1 123 12 13
13 13 123 132 1 23 12
132 132 23 13 12 123 1
123 123 13 12 23 1 132
D3 1 y x2y xy x2 x
1 1 y x2y xy x2 x
y y 1 x x2 xy x2y
x2y x2y x2 1 x y xy
xy xy x x2 1 x2y y
x2 x2 x2y xy y x 1
x x xy y x2y 1 x2

HW: Prove that the group homomorphism given as in the following table is an isomorphism.

Isomorphism
Φ: D3 S3 x 123 y 12

Center Abelian Conjugacy classes Subgroups
ZS3= 1 No 𝒞13 = 1 H0= S3
𝒞21 = 12, 23, 13 H1= 1, 132, 123
𝒞3 = 123, 132 H2= 1, 12
H3= 1, 13
H4= 1, 23
H5= 1

Element g Order οg Centralizer Zg Conjugacy Class 𝒞g
1,1 1 S3 𝒞13
12 2 H2 𝒞21
23 2 H4 𝒞21
13 2 H3 𝒞21
132 3 H1 𝒞3
123 3 H1 𝒞3

Generators Relations Realization
D3 x,y x3=y2=1 x=123
xy2=1 y=12
S3 s1,s2 s12=s22=1 s1=y=12
s1s2s1= s2s1s2 s2=x2y= 23

Subgroups Hi Structure Order Hi Index Normal Quotient group
H0= S3 H0=S3 6 S3:S3 = 1 Yes S3 /H0 1
H1= 1, 132, 123 H1 C3 A3 3 S3:H1 = 2 Yes S3 /H1 C2
H2= 1, 12 H2C2 2 S3:H2 =3 No
H3= 1, 13 H3C2 2 S3:H3 =3 No
H4= 1, 23 H4C2 2 S3:H4 =3 No
H5= 1 H5= 1 1 S3:H5 =6 Yes S3/ 1 S3

Orders Inclusions 6 3 2 1 S3 1, 123, 132 1, 13 1, 23 1, 13 1

Subgroups Hi Left Cosets Right Cosets
H0=S3 S3 S3
H1 = 1, 132, 123 H1 = 1, 132, 123 H1 = 1, 132, 123
12 H1 = 12, 13, 23 H1 12 = 12, 13, 23
H2= 1, 12 H2= 1, 12 H2= 1, 12
23 H2= 23, 132 H2 23= 23, 123
13 H2= 13, 123 H2 13= 13, 132
H3= 1, 13 H3= 1, 13 H3= 1, 13
23 H3= 23, 123 H3 23= 23, 132
12 H3= 12, 132 H3 12= 12, 123
H4= 1, 23 H4= 1, 23 H4= 1, 23
12 H4= 12, 123 H4 12= 12, 132
13 H4= 13, 132 H4 13= 13, 123
H5= 1 H5= 1 H5= 1
12 H5= 12 H5 12= 12
23 H5= 23 H5 23= 23
13 H5= 13 H5 13= 13
132 H5= 132 H5 132= 132
123 H5= 123 H5 123= 123

Subgroups Hi Normalizer NHi Centralizer ZHi
H0=S3 H0=S3 H5= 1
H1= 1, 132, 123 H0= S3 H1= 1, 123, 132
H2= 1, 12 H2= 1, 12 H2= 1, 12
H3= 1, 13 H3= 1, 13 H3= 1, 13
H4= 1, 23 H4= 1, 23 H4= 1, 23
H5= 1 H0=S3 H0=S3

Homomorphism Kernel Image
φ0: S3 1 s1 1 s2 1 kerφ0= S3 imφ0=1
ε: S3 μ2 s1 -1 s2 -1 kerε= A3 imφ0 =μ2
φ2: S3 O3 12 0 1 0 1 0 0 0 0 1 23 1 0 0 0 0 1 0 1 0 kerφ2= 1 imφ2= 1 0 0 0 1 0 0 0 1 , 0 1 0 1 0 0 0 0 1 , 1 0 0 0 0 1 0 1 0 , 0 1 0 0 0 1 1 0 0 , 0 0 1 0 1 0 1 0 0 , 0 0 1 1 0 0 0 1 0
φ3: S3 O2 12 -1 0 0 1 23 12 12 32 -12 kerφ3= 1 imφ3= 1 0 0 1 , -1 0 0 1 , 12 12 32 -12 , -12 -12 32 -12 , -12 12 -32 -12 , -12 -12 -32 -12
φ4: S3 D3 12 y 132 x kerφ4= 1 imφ4=D3

References

[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)

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