formopt : option of the form form = "multiplicative" or form = "additive" (the default)

A quasicyclic group is an infinite abelian group $G$ in which each element has order a power of a single prime number $p$, and such that every proper subgroup of $G$ is a finite cyclic $p$-group. They are also called Prüfer $p$-groups.

Abstractly, a quasicyclic group can be defined as a direct limit of the system of finite cyclic groups of the form $C_{p^k}$, for positive integers $k$, in which $C_{p^k}$ is embedded naturally in $C_{p^{k\+1}}$.

The QuasicyclicGroup( p ) calling sequence constructs a quasicyclic p-group, where p is a prime number. By default, an additive representation that models the Sylow p-subgroup of the quotient group $\mathbb{Q}\/\mathbb{Z}$ is generated. A multiplicative version of the group can be realized by passing the option form = "multiplicative".

The default form of a quasicyclic $p$-group $G$ is an additive group that represents the Sylow $p$-subgroup of the quotient group $\mathbb{Q}\/\mathbb{Z}$ of the additive group of rationals by the integers. Formally, the elements of $G$ are cosets $q\+\mathbb{Z}$ where $q$ is a rational number whose denominator is a power of the prime $p$. Maple, however, uses rational representatives as group elements, so that the group operation is ordinary addition of rationals modulo $1$. That is, two rationals represent the same group element if their difference is an integer. In particular, any integer is a representative of the identity element of the group.

Each rational of the form $\frac{k}{p^m}$, where $k$ and $m$ are integers, has an unique equivalent representative where $k$ and $m$ satisfy $0\le m$ and $k$ is a non-negative integer with $k<p^m$.

The Operations module for the group implements these conventions. Moreover, the CanonicalForm method of the group returns the unique canonical form of any rational representative of a group element.

An alternative multiplicative form of a quasicyclic $p$-group is the group of complex $p$-power roots of unity. These are the complex numbers of the form ${\&ExponentialE;}^{\frac{2\mathrm{I}k\mathrm{Pi}}{p^n}}$, where $n$ is a non-negative integer, and $k$ is a positive integer such that $k<p^n$. The group operation is then just ordinary multiplication of complex numbers.

Other more complicated expressions can also represent complex roots of unity. You can use the CanonicalForm method of a quasicyclic group to obtain an expression in the form above, subject to simplifications automatically performed by the exp function.

The Operations module for the group implements these conventions.

Since a proper subgroup of a quasicyclic group is not itself quasicyclic (rather, it is a finite cyclic group of prime power order), in order that its elements remain elements of the parent quasicyclic group, it is represented as a QuasicyclicSubgroup object. The group operations are inherited from the parent quasicyclic group. (In fact, full quasicyclic groups are also represented as QuasicyclicSubgroup objects.)

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

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

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

$\mathrm{type}\left(G\&comma;'\mathrm{QuasicyclicGroup}'\right)$

$\mathrm{true}$

$\mathrm{type}\left(G\&comma;'\mathrm{QuasicyclicSubgroup}'\right)$

$\mathrm{true}$

$\mathrm{type}\left(G\&comma;'\mathrm{Group}'\right)$

$\mathrm{true}$

$\mathrm{IsFinite}\left(G\right)$

$\mathrm{false}$

$\mathrm{IsFinitelyGenerated}\left(G\right)$

$\mathrm{false}$

$\mathrm{GroupOrder}\left(G\right)$

$\mathrm{\infty }$

$\mathrm{IsAbelian}\left(G\right)$

$\mathrm{true}$

$\mathrm{IsCyclic}\left(G\right)$

$\mathrm{false}$

$\frac{12}{25}\mathbf{in}G$

$\mathrm{true}$

$\mathrm{ElementOrder}\left(\frac{12}{25}\&comma;G\right)$

$25$

$\frac{3}{5}\mathbf{in}G$

$\mathrm{true}$

$\frac{12}{25}+\frac{3}{5}\mathbf{in}G$

$\mathrm{true}$

$\mathrm{CanonicalForm}\left(\frac{12}{25}+\frac{3}{5}\&comma;G\right)$

$\frac{2}{25}$

$\mathrm{Operations}\left(G\right):-\mathrm{`.`}\left(\frac{12}{25}\&comma;\frac{3}{5}\right)$

$\frac{2}{25}$

$\frac{2}{3}\mathbf{in}G$

$\mathrm{false}$

$7\mathbf{in}G$

$\mathrm{true}$

$\mathrm{CanonicalForm}\left(7\&comma;G\right)$

$0$

$\mathrm{IsPGroup}\left(G\right)$

$\mathrm{true}$

$\mathrm{PGroupPrime}\left(G\right)$

$5$

$H≔\mathrm{Subgroup}\left(\left[\frac{12}{125}\&comma;\frac{3}{5}\right]\&comma;G\right)$

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

$\mathrm{type}\left(H\&comma;'\mathrm{QuasicyclicGroup}'\right)$

$\mathrm{false}$

$\mathrm{type}\left(H\&comma;'\mathrm{QuasicyclicSubgroup}'\right)$

$\mathrm{true}$

$\mathrm{IsFinite}\left(H\right)$

$\mathrm{true}$

$\mathrm{IsCyclic}\left(H\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(H\right)$

$125$

$\mathrm{DrawSubgroupLattice}\left(H\right)$

$M≔\mathrm{QuasicyclicGroup}\left(5\&comma;'\mathrm{form}'=''multiplicative''\right)$

$M≔C_{5^{\mathrm{\infty }}}$

$\mathrm{AreIsomorphic}\left(G\&comma;M\&comma;'\mathrm{assign}'='\mathrm{iso}'\right)$

$\mathrm{true}$

$\mathrm{iso}\left(\frac{3}{25}\right)$

${\&ExponentialE;}^{\frac{6\mathrm{I}}{25}\mathrm{\pi }}$

$H≔\mathrm{Subgroup}\left(\left[\mathrm{iso}\left(\frac{4}{25}\right)\right]\&comma;M\right)$

$H≔C_{25}$

$\mathrm{IsSubgroup}\left(H\&comma;M\right)$

$\mathrm{true}$

$\mathrm{IsSubgroup}\left(H\&comma;G\right)$

$\mathrm{false}$

$\mathrm{GroupOrder}\left(H\right)$

$25$

$\mathrm{IdentifySmallGroup}\left(H\right)$

$25,1$

$\mathrm{AreIsomorphic}\left(H\&comma;\mathrm{CyclicGroup}\left(25\right)\right)$

$\mathrm{true}$

$L≔\mathrm{Subgroup}\left(\left\{\frac{4}{25}\right\}\&comma;G\right)$

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

$\mathrm{AreIsomorphic}\left(H\&comma;L\right)$

$\mathrm{true}$

$\mathrm{Elements}\left(H\right)$

$\left\{1\&comma;-{\&ExponentialE;}^{\frac{\mathrm{I}}{5}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{\mathrm{I}}{25}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{3\mathrm{I}}{5}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{3\mathrm{I}}{25}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{7\mathrm{I}}{25}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{9\mathrm{I}}{25}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{11\mathrm{I}}{25}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{13\mathrm{I}}{25}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{17\mathrm{I}}{25}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{19\mathrm{I}}{25}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{21\mathrm{I}}{25}\mathrm{\pi }}\&comma;-{\&ExponentialE;}^{\frac{23\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{2\mathrm{I}}{5}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{2\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{4\mathrm{I}}{5}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{4\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{6\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{8\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{12\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{14\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{16\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{18\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{22\mathrm{I}}{25}\mathrm{\pi }}\&comma;{\&ExponentialE;}^{\frac{24\mathrm{I}}{25}\mathrm{\pi }}\right\}$

$\mathrm{Elements}\left(L\right)$

$\left\{0\&comma;\frac{1}{5}\&comma;\frac{1}{25}\&comma;\frac{2}{5}\&comma;\frac{2}{25}\&comma;\frac{3}{5}\&comma;\frac{3}{25}\&comma;\frac{4}{5}\&comma;\frac{4}{25}\&comma;\frac{6}{25}\&comma;\frac{7}{25}\&comma;\frac{8}{25}\&comma;\frac{9}{25}\&comma;\frac{11}{25}\&comma;\frac{12}{25}\&comma;\frac{13}{25}\&comma;\frac{14}{25}\&comma;\frac{16}{25}\&comma;\frac{17}{25}\&comma;\frac{18}{25}\&comma;\frac{19}{25}\&comma;\frac{21}{25}\&comma;\frac{22}{25}\&comma;\frac{23}{25}\&comma;\frac{24}{25}\right\}$

$\mathrm{Elements}\left(G\right)$

Error, (in GroupTheory:-Generators) group is not finitely generated

$$\begin{gathered} \mathbf{for}g\mathbf{in}G\mathbf{do} \\ \mathbf{if}1000<\mathrm{denom}\left(g\right)\mathbf{then} \\ \mathbf{break} \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do}\&colon; \end{gathered}$$

$\left\{\mathrm{seq}\right\}\left(\mathrm{ElementOrder}\left(x\&comma;L\right)\&comma;x=L\right)$

$\left\{1\&comma;5\&comma;25\right\}$

$\mathrm{PermutationGroup}\left(L\right)$

$⟨\left(1\&comma;3\&comma;5\&comma;7\&comma;9\&comma;2\&comma;10\&comma;11\&comma;12\&comma;13\&comma;4\&comma;14\&comma;15\&comma;16\&comma;17\&comma;6\&comma;18\&comma;19\&comma;20\&comma;21\&comma;8\&comma;22\&comma;23\&comma;24\&comma;25\right)⟩$

$\mathrm{CayleyTableGroup}\left(L\right)$

$\mathrm{<\; a\; Cayley\; table\; group\; with\; 25\; elements\; >}$

$\mathrm{FPGroup}\left(L\right)$

$⟨\mathrm{\_g}\mid {\mathrm{\_g}}^{25}⟩$

$F≔\mathrm{FrattiniSubgroup}\left(M\right)$

$F≔C_{5^{\mathrm{\infty }}}$

$\mathrm{IsSubgroup}\left(M\&comma;F\right)\mathbf{and}\mathrm{IsSubgroup}\left(F\&comma;M\right)$

$\mathrm{true}$

$T≔\mathrm{QuasicyclicGroup}\left(2\&comma;'\mathrm{form}'=''multiplicative''\right)$

$T≔C_{2^{\mathrm{\infty }}}$

$U≔\mathrm{Subgroup}\left(\left[\mathrm{I}\right]\&comma;T\right)$

$U≔C_4$

$\mathrm{IsSubgroup}\left(U\&comma;T\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(U\right)$

$4$

$\mathrm{Elements}\left(U\right)$

$\left\{\mathrm{-1}\&comma;1\&comma;\mathrm{-I}\&comma;\mathrm{I}\right\}$

$\mathrm{AreIsomorphic}\left(T\&comma;M\right)$

$\mathrm{false}$

$\mathrm{AreIsomorphic}\left(G\&comma;T\right)$

$\mathrm{false}$