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 $C_2$ of order two

There are at least two natural ways of defining the group $C_2$. 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
${\mu }_2=\left\{\pm 1\right\}=\left\{+1,-1\right\}$	ordinary multiplication of integers	$\times$ $1$ $-1$ $1$ $1$ $-1$ $1$ $-1$ $1$	$$\begin{array}{clcr}\varphi :& C_2& \to & {\mu }_2\\ & 0& \mapsto & 1\\ & 1& \mapsto & -1\end{array}$$
$C_2=\left\{0,1\right\}$	addition modulo $2$	$+$ $0$ $1$ $0$ $0$ $1$ $1$ $1$ $0$

Center	Abelian	Conjugacy classes	Subgroups
$Z\left(C_2\right)=C_2$	Yes	$\begin{array}{c}{𝒞}_1=\left\{1\right\}\\ {𝒞}_{-1}=\left\{-1\right\}\end{array}$	$\begin{array}{c}{H}_0=C_2\\ H_1=⟨1⟩\end{array}$

Element $g$	Order $o\left(g\right)$	Centralizer $Z_g$
$1$	$1$	$C_2$
$-1$	$1$	$C_2$

Generators	Relations
$g$	$g^2=1$

Homomorphism	Kernel	Image
$\begin{array}{rrcc}{\phi }_0:& C_2& \to & ⟨1⟩\\ & 1& \mapsto & 1\\ & -1& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\phi }_0=C_2$	$im\hspace{0.17em}{\phi }_0=⟨1⟩$
$\begin{array}{rrcc}{\phi }_1& C_2& \to & C_2\\ & 1& \mapsto & 1\\ & -1& \mapsto & -1\end{array}$	$ker\hspace{0.17em}{\phi }_1$=⟨1⟩	$im\hspace{0.17em}{\phi }_1=C_2$

Subgroups $H_i$	Structure	Order $\left|H_i\right|$	Index	Normal	Quotient group
$H_0=C_2$	$C_2$	$2$	$\left[C_2:C_2\right]=1$	Yes	$C_2/H_0\cong ⟨1⟩$
$H_1=⟨1⟩$	$H_1=⟨1⟩$	$1$	$\left[C_2:⟨1⟩\right]=2$	Yes	$C_2/⟨1⟩\cong C_2$

Subgroups $H_i$	Normalizer $N_{H_i}$	Centralizer $Z_{H_i}$
$H_0=C_2$	$C_2$	$C_2$
$H_1=⟨1⟩$	$C_2$	$C_2$

Subgroups $H_i$	Left Cosets	Right Cosets
$H_0=C_2$	$C_2=\left\{1,-1\right\}$	$C_2=\left\{1,-1\right\}$
$H_1=⟨1⟩$	$\begin{array}{c}{H}_1=\left\{1\right\}\\ \left(-1\right)H_1=\left\{-1\right\}\end{array}$	$\begin{array}{c}{H}_1=\left\{1\right\}\\ H_1\left(-1\right)=\left\{-1\right\}\end{array}$

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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