The Cyclic Group of Order Two

## 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
${\mu }_{2}=\left\{±1\right\}=\left\{+1,-1\right\}$ ordinary multiplication of integers
 $×$ $1$ $-1$ $1$ $1$ $-1$ $1$ $-1$ $1$
$ϕ: C2 → μ2 0 ↦ 1 1 ↦ -1$
${C}_{2}=\left\{0,1\right\}$ 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 $ο\left(g\right)$ Centralizer ${Z}_{g}$
$1$ $1$ ${C}_{2}$
$-1$ $1$ ${C}_{2}$

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

Homomorphism Kernel Image
$\begin{array}{rrcc}{\phi }_{0}:& {C}_{2}& \to & ⟨1⟩\\ & 1& ↦& 1\\ & -1& ↦& 1\end{array}$ $ker {\phi }_{0}={C}_{2}$ $im {\phi }_{0}=⟨1⟩$
$\begin{array}{rrcc}{\phi }_{1}& {C}_{2}& \to & {C}_{2}\\ & 1& ↦& 1\\ & -1& ↦& -1\end{array}$ $ker {\phi }_{1}$=1 $im {\phi }_{1}={C}_{2}$

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)