Since members of an additive quasicyclic group are represented by ordinary Maple rationals (where a rational q represents its coset$q\&plus;\mathbb{Z}$ of the quotient group$\mathbb{Q}\&sol;\mathbb{Z}$ , it follows that rationals differing by an integer represent the same element of a quasicyclic group. Each coset$q\&plus;\mathbb{Z}$ 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 ${\&ExponentialE;}^{\frac{2\mathrm{I}\mathrm{\pi }m}{p^n}}$, where $m$ and $n$ are non-negative integers, or complex $p$-power roots of unity in rectangular form, such as $\mathrm{-I}$ or $\frac{\sqrt{2}}{2}+\frac{\mathrm{I}\sqrt{2}}{2}=\left(\frac{1}{2}+\frac{\mathrm{I}}{2}\right)\sqrt{2}$. 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 ${\&ExponentialE;}^{\frac{2\mathrm{I}\mathrm{\pi }m}{p^n}}$, 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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\&colon;$

$G≔\mathrm{QuasicyclicGroup}\left(5\right)$

$G≔{\mathbb{Z}}_{5^{\mathrm{\infty }}}$

$\mathrm{CanonicalForm}\left(\frac{2}{5}\&comma;G\right)$

$\frac{2}{5}$

$\mathrm{CanonicalForm}\left(\frac{12}{5}\&comma;G\right)$

$\frac{2}{5}$

$\mathrm{Operations}\left(G\right):-\mathrm{`=`}\left(\frac{2}{5}\&comma;\frac{12}{5}\right)$

$\mathrm{true}$

$\mathrm{CanonicalForm}\left(4\&comma;G\right)$

$0$

$\mathrm{CanonicalForm}\left(\frac{27}{25}\&comma;G\right)$

$\frac{2}{25}$

$\mathrm{CanonicalForm}\left(\frac{30}{25}\&comma;G\right)$

$\frac{1}{5}$

$\mathrm{evalb}\left(\mathrm{CanonicalForm}\left(\frac{\mathrm{rand}\left(\right)}{25}\&comma;G\right)\mathbf{in}\left[\mathrm{seq}\right]\left(\frac{i}{25}\&comma;i=0..24\right)\right)$

$\mathrm{true}$

$H≔\mathrm{Subgroup}\left(\left[\frac{1}{625}\right]\&comma;G\right)$

$H≔{\mathbb{Z}}_{5^{\mathrm{\infty }}}$

$\mathrm{CanonicalForm}\left(\frac{2}{5}\&comma;H\right)$

$\frac{2}{5}$

$\mathrm{CanonicalForm}\left(\frac{12}{5}\&comma;H\right)$

$\frac{2}{5}$

$\mathrm{CanonicalForm}\left(\frac{30}{25}\&comma;H\right)$

$\frac{1}{5}$

$g≔\mathrm{RandomElement}\left(H\right)$

$g≔\frac{12}{25}$

$\mathrm{CanonicalForm}\left(g\&comma;H\right)=\mathrm{CanonicalForm}\left(g\&comma;G\right)$

$\frac{12}{25}=\frac{12}{25}$

$G≔\mathrm{QuasicyclicGroup}\left(2\&comma;'\mathrm{form}'=''multiplicative''\right)$

$G≔C_{2^{\mathrm{\infty }}}$

$\mathrm{CanonicalForm}\left(\mathrm{I}\&comma;G\right)$

$\mathrm{I}$

$g≔\frac{1}{2}{2}^{\frac{1}{2}}+\frac{1}{2}\mathrm{I}{2}^{\frac{1}{2}}$

$g≔\frac{\sqrt{2}}{2}+\frac{\mathrm{I}\sqrt{2}}{2}$

$g\mathbf{in}G$

$\mathrm{true}$

$\mathrm{CanonicalForm}\left(g^6\&comma;G\right)$

$\mathrm{-I}$

$\mathrm{CanonicalForm}\left(\exp \left(\frac{9\mathrm{I}\mathrm{\pi }}{16}\right)\&comma;G\right)$

${\&ExponentialE;}^{\frac{9\mathrm{I}}{16}\mathrm{\pi }}$

$\mathrm{CanonicalForm}\left(\exp \left(\frac{27\mathrm{I}\mathrm{\pi }}{16}\right)\exp \left(\frac{3\mathrm{I}\mathrm{\pi }}{8}\right)\&comma;G\right)$

${\&ExponentialE;}^{\frac{\mathrm{I}}{16}\mathrm{\pi }}$