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 = 1 0 0 1 , -1,1 = -1 0 0 1 , 1,-1 = 1 0 0 -1 , -1,-1 = -1 0 0 -1 .

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

Set Operation
C2×C2= ±1 0 0 ±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 C2×C2 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
-1,1 H1= -1,1, -1,-1 H1 -1,1= -1,1, -1,-1
H2 H2= 1,1, -1,1 H2= 1,1, -1,1
1,-1 H2= 1,-1, -1,-1 H2 1,-1= 1,-1, -1,-1
H3 H3= 1,1, -1,-1 H3= 1,1, -1,-1
1,-1 H3= 1,-1, -1,1 H3 1,-1= 1,-1, -1,1
H4 H4= 1,1 H4= 1,1
-1,1 H4= -1,1 H4 -1,1= -1,1
1,-1 H4= 1,-1 H4 1,-1= 1,-1
-1,-1 H4= -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 1 1,-1 1 kerφ0= C2×C2 imφ0=1
φ1 C2×C2 C2 -1,1 -1 1,-1 1 kerφ1= H1 imφ1=C2
φ2 C2×C2 C2 -1,1 1 1,-1 -1 kerφ2= H2 imφ2=C2
φ3 C2×C2 C2 -1,1 -1 1,-1 -1 kerφ3= H3 imφ3=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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