Category:Group Generated by Reciprocal of z and 1 minus z

Group Generated by $1 / z$ and $-z$

Let:

$S = \set {f_1, f_2, f_3, f_4, f_5, f_6}$

where $f_1, f_2, \ldots, f_6$ are complex functions defined for all $z \in \C \setminus \set {0, 1}$ as:

$\ds \map {f_1} z$

$=$

$\ds z$

$\ds \map {f_2} z$

$=$

$\ds \dfrac 1 {1 - z}$

$\ds \map {f_3} z$

$=$

$\ds \dfrac {z - 1} z$

$\ds \map {f_4} z$

$=$

$\ds \dfrac 1 z$

$\ds \map {f_5} z$

$=$

$\ds 1 - z$

$\ds \map {f_6} z$

$=$

$\ds \dfrac z {z - 1}$

Then $\struct {S, \circ}$ is the group generated by $\dfrac 1 z$ and $1 - z$.

The following 2 pages are in this category, out of 2 total.