GroupTheory/QuasicyclicGroup/CanonicalForm - Maple Help

GroupTheory[QuasicyclicGroup]

 CanonicalForm

 Calling Sequence CanonicalForm( g, G, checkopt )

Parameters

 g - : rational : an element of G G - : QuasicyclicSubgroup : a quasicyclic subgroup checkopt - : identical(check) = truefalse : (optional) option of the form check = t where t is either true (default) or false

Description

 • Since members of an additive quasicyclic group are represented by ordinary Maple rationals (where a rational q represents its coset$q+ℤ$ of the quotient group$ℚ/ℤ$ , it follows that rationals differing by an integer represent the same element of a quasicyclic group. Each coset$q+ℤ$ contains an unique non-negative rational number $r$ for which $r<1$. This rational $r$ is the canonical form of each member of its coset.
 • Two rationals that belong to an additive quasicyclic group represent the same group element if they have the same canonical form. This is equivalent to their difference being an integer.
 • For example, in the quasicyclic group${Z}_{{3}^{\mathrm{\infty }}}$ the elements $\frac{4}{9}$ and $\frac{13}{9}$ represent the same element, as both have
 • Elements of a multiplicative quasicyclic $p$-group are the complex $p$-power roots of unity. These elements have the form ${ⅇ}^{\frac{2I\mathrm{\pi }m}{{p}^{n}}}$, where $m$ and $n$ are non-negative integers, or complex $p$-power roots of unity in rectangular form, such as $-I$ or $\frac{\sqrt{2}}{2}+\frac{I\sqrt{2}}{2}=\left(\frac{1}{2}+\frac{I}{2}\right)\sqrt{2}$. In addition, products and powers of $p$-power roots of unity are considered to belong to the multiplicative quasicyclic $p$-group.
 • Subject to simplifications performed automatically by the exp function, the canonical form of an element of a multiplicative quasicyclic $p$-group is an expression of the form ${ⅇ}^{\frac{2I\mathrm{\pi }m}{{p}^{n}}}$, where $m$ is a non-negative integer such that $m<{p}^{n}$.
 • The CanonicalForm( g, G ) method returns the canonical form of the element g of the quasicyclic group G.
 • If the check = false option is passed, then it is not checked that g is actually an element of G and, if it is not, an incorrect result may be returned. By default, the check option has the value true.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{QuasicyclicGroup}\left(5\right)$
 ${G}{≔}{{ℤ}}_{{{5}}^{{\mathrm{\infty }}}}$ (1)
 > $\mathrm{CanonicalForm}\left(\frac{2}{5},G\right)$
 $\frac{{2}}{{5}}$ (2)
 > $\mathrm{CanonicalForm}\left(\frac{12}{5},G\right)$
 $\frac{{2}}{{5}}$ (3)
 > $\mathrm{Operations}\left(G\right):-\mathrm{=}\left(\frac{2}{5},\frac{12}{5}\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{CanonicalForm}\left(4,G\right)$
 ${0}$ (5)
 > $\mathrm{CanonicalForm}\left(\frac{27}{25},G\right)$
 $\frac{{2}}{{25}}$ (6)
 > $\mathrm{CanonicalForm}\left(\frac{30}{25},G\right)$
 $\frac{{1}}{{5}}$ (7)
 > $\mathrm{evalb}\left(\mathrm{CanonicalForm}\left(\frac{\mathrm{rand}\left(\right)}{25},G\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∈\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\left[\mathrm{seq}\right]\left(\frac{i}{25},i=0..24\right)\right)$
 ${\mathrm{true}}$ (8)
 > $H≔\mathrm{Subgroup}\left(\left[\frac{1}{625}\right],G\right)$
 ${H}{≔}{{ℤ}}_{{{5}}^{{\mathrm{\infty }}}}$ (9)
 > $\mathrm{CanonicalForm}\left(\frac{2}{5},H\right)$
 $\frac{{2}}{{5}}$ (10)
 > $\mathrm{CanonicalForm}\left(\frac{12}{5},H\right)$
 $\frac{{2}}{{5}}$ (11)
 > $\mathrm{CanonicalForm}\left(\frac{30}{25},H\right)$
 $\frac{{1}}{{5}}$ (12)
 > $g≔\mathrm{RandomElement}\left(H\right)$
 ${g}{≔}\frac{{12}}{{25}}$ (13)
 > $\mathrm{CanonicalForm}\left(g,H\right)=\mathrm{CanonicalForm}\left(g,G\right)$
 $\frac{{12}}{{25}}{=}\frac{{12}}{{25}}$ (14)
 > $G≔\mathrm{QuasicyclicGroup}\left(2,'\mathrm{form}'="multiplicative"\right)$
 ${G}{≔}{{C}}_{{{2}}^{{\mathrm{\infty }}}}$ (15)
 > $\mathrm{CanonicalForm}\left(I,G\right)$
 ${I}$ (16)
 > $g≔\frac{1{2}^{\frac{1}{2}}}{2}+\frac{1I{2}^{\frac{1}{2}}}{2}$
 ${g}{≔}\frac{\sqrt{{2}}}{{2}}{+}\frac{{I}{}\sqrt{{2}}}{{2}}$ (17)
 > $g\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}∈\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}G$
 ${\mathrm{true}}$ (18)
 > $\mathrm{CanonicalForm}\left({g}^{6},G\right)$
 ${-I}$ (19)
 > $\mathrm{CanonicalForm}\left({ⅇ}^{\frac{9I\mathrm{Pi}}{16}},G\right)$
 ${{ⅇ}}^{\frac{{9}{}{I}}{{16}}{}{\mathrm{\pi }}}$ (20)
 > $\mathrm{CanonicalForm}\left({ⅇ}^{\frac{27I\mathrm{Pi}}{16}}{ⅇ}^{\frac{3I\mathrm{Pi}}{8}},G\right)$
 ${{ⅇ}}^{\frac{{I}}{{16}}{}{\mathrm{\pi }}}$ (21)