The Klein 4-Group

The Klein 4-Group

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

Last updates: 12 December 2010

The Klein 4-Group

Let us make some shorter notations for the following matrices. 1,1=1001,-1,1=-1001,1,-1=100-1,-1,-1=-100-1.

The Klein 4-group is the group of order 4 defined as in the following table.

Set Operation
C2 × C2=±100±1 Ordinary matrix multiplication

The compete multiplication table for this group is as follows.

Multiplication table
× 1,1 1,-1 -1,1 -1,-1
1,1 1,1 1,-1 -1,1 -1,-1
1,-1 1,-1 1,1 -1,-1 -1,1
-1,-1 -1,-1 -1,1 1,-1 1,1

HW: Show that this group, as definded above, is isomorphic to the direct product of a cyclic group of order two, C2, with another cyclic group of order two, C2.

Center Abelian Conjugacy classes Subgroups
ZG=C2 × C2 Yes 𝒞1,1=1,1 H0=C2 × C2
𝒞1,-1=1,-1 H1=1,1,1,-1
𝒞-1,1=-1,1 H2=1,1,-1,1
𝒞-1,-1=-1,-1 H3=1,1,-1,-1
H4=1,1

Element g Order οg Centralizer Zg Conjugacy Class 𝒞g
1,1 1 C2 × C2 𝒞1,1
1,-1 2 C2 × C2 𝒞1,-1
-1,1 2 C2 × C2 𝒞-1,1
-1,-1 2 C2 × C2 𝒞-1,-1

Generators Relations
x,y x2=1
y2=1
xy=yx

Subgroups Hi Structure Order Hi Index Normal Quotient group
H0=C2 × C2 C2 × C2 4 1 Yes (C2 × C2)/H0≅⟨1⟩
H1=1,1,1,-1 C2 2 2 Yes (C2 × C2)/H1≅C2
H2=1,1,-1,1 C2 2 2 Yes (C2 × C2)/H2≅C2
H3=1,1,-1,-1 C2 2 2 Yes (C2 × C2)/H3≅C2
H4=1,1 ⟨1⟩ 1 4 Yes (C2 × C2)/H1≅C2 × C2

Orders Inclusions 4 2 1 μ2 × μ2 {(1,1),(1,-1)} {(1,1),(-1,-1)} {(1,1),(-1,1)} {(1,1)}

Subgroups Hi Left Cosets Right Cosets
H0 H0=±1,±1 H0=±1,±1
H1 H1=1,1,1,-1 H1=1,1,1,-1
-1,1H1=-1,1,-1,-1 H1-1,1=-1,1,-1,-1
H2 H2=1,1,-1,1 H2=1,1,-1,1
1,-1H2=1,-1,-1,-1 H21,-1=1,-1,-1,-1
H3 H3=1,1,-1,-1 H3=1,1,-1,-1
1,-1H3=1,-1,-1,1 H31,-1=1,-1,-1,1
H4 H4=1,1 H4=1,1
-1,1H4=-1,1 H4-1,1=-1,1
1,-1H4=1,-1 H41,-1=1,-1
-1,-1H4=-1,-1 H4-1,-1=-1,-1

Subgroups Hi Normalizer NHi Centralizer ZHi
H0 H0 H0
H1 H0 H0
H2 H0 H0
H3 H0 H0
H4 H0 H0

Homomorphism Kernel Image
φ0:C2 × C2→⟨1⟩-1,1↦11,-1↦1 ker φ0=C2 × C2 im φ0=⟨1⟩
φ1C2 × C2→C2-1,1↦-11,-1↦1 ker φ1=H1 im φ1=C2
φ2C2 × C2→C2-1,1↦11,-1↦-1 ker φ2=H2 im φ2=C2
φ3C2 × C2→C2-1,1↦-11,-1↦-1 ker φ3=H3 im φ3=C2

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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