The Cyclic Group of Order Two

The Cyclic Group of Order Two

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

Last updates: 8 December 2010

The cyclic group C2 of order two

There are at least two natural ways of defining the group C2. The isomorphism which shows that these two definitions are the same is given in the righmost column of the following table.

Set Operation Multiplication Table Isomorphism
μ2= ±1 = +1,-1 ordinary multiplication of integers
× 1 -1
1 1 -1
1 -1 1
ϕ: C2 μ2 0 1 1 -1
C2= 0,1 addition modulo 2
+ 0 1
0 0 1
1 1 0

Center Abelian Conjugacy classes Subgroups
ZC2=C2 Yes 𝒞1= 1 𝒞-1= -1 H0=C2 H1=1

Element g Order οg Centralizer Zg
1 1 C2
-1 1 C2

Generators Relations
g g2=1

Homomorphism Kernel Image
φ0: C2 1 1 1 -1 1 kerφ0=C2 imφ0=1
φ1 C2 C2 1 1 -1 -1 kerφ1=1 imφ1=C2

Orders Inclusions 2 1 C2= ±1 1 = 1

Subgroups Hi Structure Order Hi Index Normal Quotient group
H0=C2 C2 2 C2:C2 =1 Yes C2/H0 1
H1= 1 H1= 1 1 C2: 1 =2 Yes C2/ 1 C2

Subgroups Hi Normalizer NHi Centralizer ZHi
H0=C2 C2 C2
H1= 1 C2 C2

Subgroups Hi Left Cosets Right Cosets
H0=C2 C2= 1,-1 C2= 1,-1
H1= 1 H1= 1 -1H1= -1 H1= 1 H1-1= -1

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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