S3≅D3: 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=100010001,12=010100001,23=100001010,13=001010100,132=010001100,123=001100010.

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	xiyjxkyl=xi-kmod 3yj+lmod 2

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→S3x↦123y↦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	H2≅C2	2	S3:H2=3	No
H3=1,13	H3≅C2	2	S3:H3=3	No
H4=1,23	H4≅C2	2	S3:H4=3	No
H5=1	H5=⟨1⟩	1	S3:H5=6	Yes	S3/⟨1⟩≅S3

Subgroups Hi	Left Cosets	Right Cosets
H0=S3	S3	S3
H1=1,132,123	H1=1,132,123	H1=1,132,123
	12H1=12,13,23	H112=12,13,23
H2=1,12	H2=1,12	H2=1,12
	23H2=23,132	H223=23,123
	13H2=13,123	H213=13,132
H3=1,13	H3=1,13	H3=1,13
	23H3=23,123	H323=23,132
	12H3=12,132	H312=12,123
H4=1,23	H4=1,23	H4=1,23
	12H4=12,123	H412=12,132
	13H4=13,132	H413=13,123
H5=1	H5=1	H5=1
	12H5=12	H512=12
	23H5=23	H523=23
	13H5=13	H513=13
	132H5=132	H5132=132
	123H5=123	H5123=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↦1s2↦1	ker φ0=S3	im φ0=⟨1⟩
ε:S3→μ2s1↦-1s2↦-1	ker ε=A3	im φ0=μ2
φ2:S3→O312↦01010000123↦100001010	ker φ2=⟨1⟩	im φ2=100010001,010100001,100001010,010001100,001010100,001100010
φ3:S3→O212↦-100123↦121232-12	ker φ3=⟨1⟩	im φ3=1001,-1001,121232-12,-12-1232-12,-1212-32-12,-12-12-32-12
φ4:S3→D312↦y132↦x	ker φ4=⟨1⟩	im φ4=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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