GroupTheory
MetacyclicGroup
construct a finite metacyclic group
Calling Sequence
Parameters
Description
Examples
Compatibility
MetacyclicGroup(m, n, k)
MetacyclicGroup(m, n, k, s)
m

a positive integer
n
k
s
(optional) equation of the form form= "fpgroup" or form = "permgroup" (default)
A group metacyclic if it has a cyclic normal subgroup the quotient by which is also cyclic. Every such group $G$ can be generated by two elements $a$ and $b$, with the subgroup $\u27e8a\u27e9$ normal in $G$. The group $G$ is then determined by the action of $\u27e8b\u27e9$ on $\u27e8a\u27e9$. Since $\u27e8a\u27e9$ is normal in $G$, it follows that the conjugate ${a}^{b}$ belongs to $\u27e8a\u27e9$ so there is a positive integer $k$ for which ${a}^{b}={a}^{k}$. Thus, a finite metacyclic group $G$ is completely determined by the orders of $a$ and $b$ and the integer $k$.
The MetacyclicGroup( m, n, k ) command constructs a metacyclic group with generators $a$ and $b$ as described above, such that ${a}^{b}={a}^{k}$, and where ${a}^{n}=1$ and ${b}^{m}=1$.
Note that the generators $a$ and $b$ need not have orders $n$ and $m$, respectively, but that their orders are necessarily divisors of $n$ and $m$.
By default, a permutation group is returned, but you can create a finitely presented group by passing the 'form' = "fpgroup" option.
$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$
$\mathrm{MetacyclicGroup}\left(6\,8\,5\right)$
$\u27e8\left({1}{\,}{2}{\,}{6}{\,}{14}{\,}{9}{\,}{3}\right)\left({4}{\,}{7}{\,}{15}{\,}{25}{\,}{19}{\,}{10}\right)\left({5}{\,}{8}{\,}{16}{\,}{26}{\,}{20}{\,}{11}\right)\left({12}{\,}{17}{\,}{27}{\,}{36}{\,}{31}{\,}{21}\right)\left({13}{\,}{18}{\,}{28}{\,}{37}{\,}{32}{\,}{22}\right)\left({23}{\,}{29}{\,}{38}{\,}{44}{\,}{41}{\,}{33}\right)\left({24}{\,}{30}{\,}{39}{\,}{45}{\,}{42}{\,}{34}\right)\left({35}{\,}{40}{\,}{46}{\,}{48}{\,}{47}{\,}{43}\right){\,}\left({1}{\,}{4}{\,}{12}{\,}{23}{\,}{35}{\,}{24}{\,}{13}{\,}{5}\right)\left({2}{\,}{7}{\,}{17}{\,}{29}{\,}{40}{\,}{30}{\,}{18}{\,}{8}\right)\left({3}{\,}{10}{\,}{21}{\,}{33}{\,}{43}{\,}{34}{\,}{22}{\,}{11}\right)\left({6}{\,}{15}{\,}{27}{\,}{38}{\,}{46}{\,}{39}{\,}{28}{\,}{16}\right)\left({9}{\,}{19}{\,}{31}{\,}{41}{\,}{47}{\,}{42}{\,}{32}{\,}{20}\right)\left({14}{\,}{25}{\,}{36}{\,}{44}{\,}{48}{\,}{45}{\,}{37}{\,}{26}\right)\u27e9$
$\mathrm{MetacyclicGroup}\left(6\,8\,5\,'\mathrm{form}'=''permgroup''\right)$
$\mathrm{MetacyclicGroup}\left(6\,8\,5\,'\mathrm{form}'=''fpgroup''\right)$
$\u27e8{}{a}{\,}{b}{}{\mid}{}{{a}}^{{6}}{\,}{{b}}^{{8}}{\,}{{b}}^{{1}}{}{a}{}{b}{}{{a}}^{{5}}{}\u27e9$
In the following example, the first parameter $6$ is a proper multiple of the order of the corresponding generator.
$a,b\u2254\mathrm{op}\left(\mathrm{Generators}\left(\mathrm{MetacyclicGroup}\left(6\,8\,4\right)\right)\right)$
${a}{,}{b}{\u2254}\left({1}{\,}{2}{\,}{3}\right)\left({4}{\,}{8}{\,}{6}\right)\left({5}{\,}{9}{\,}{7}\right)\left({10}{\,}{12}{\,}{14}\right)\left({11}{\,}{13}{\,}{15}\right)\left({16}{\,}{20}{\,}{18}\right)\left({17}{\,}{21}{\,}{19}\right)\left({22}{\,}{23}{\,}{24}\right){,}\left({1}{\,}{4}{\,}{10}{\,}{16}{\,}{22}{\,}{17}{\,}{11}{\,}{5}\right)\left({2}{\,}{6}{\,}{12}{\,}{18}{\,}{23}{\,}{19}{\,}{13}{\,}{7}\right)\left({3}{\,}{8}{\,}{14}{\,}{20}{\,}{24}{\,}{21}{\,}{15}{\,}{9}\right)$
$\mathrm{PermOrder}\left(a\right)$
${3}$
$\mathrm{PermOrder}\left(b\right)$
${8}$
The GroupTheory[MetacyclicGroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
See Also
GroupTheory[CyclicGroup]
GroupTheory[DicyclicGroup]
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