GroupTheory

construct a finite metacyclic group


Calling Sequence


MetacyclicGroup(m, n, k)
MetacyclicGroup(m, n, k, s)


Parameters


m



a positive integer

n



a positive integer

k



a positive integer

s



optional equation: form= "fpgroup" or form = "permgroup" (default)





Description


•

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.



Examples


>

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

>

$\mathrm{MetacyclicGroup}\left(6\,8\,5\right)$

$\u27e8\left({1}{\,}{9}{\,}{10}{\,}{11}{\,}{12}{\,}{13}\right)\left({2}{\,}{15}{\,}{16}{\,}{17}{\,}{18}{\,}{19}\right)\left({3}{\,}{20}{\,}{21}{\,}{22}{\,}{23}{\,}{24}\right)\left({4}{\,}{25}{\,}{26}{\,}{27}{\,}{28}{\,}{29}\right)\left({5}{\,}{30}{\,}{31}{\,}{32}{\,}{33}{\,}{34}\right)\left({6}{\,}{35}{\,}{36}{\,}{37}{\,}{38}{\,}{39}\right)\left({7}{\,}{40}{\,}{41}{\,}{42}{\,}{43}{\,}{44}\right)\left({8}{\,}{14}{\,}{45}{\,}{46}{\,}{47}{\,}{48}\right){\,}\left({1}{\,}{2}{\,}{3}{\,}{4}{\,}{5}{\,}{6}{\,}{7}{\,}{8}\right)\left({9}{\,}{15}{\,}{20}{\,}{25}{\,}{30}{\,}{35}{\,}{40}{\,}{14}\right)\left({10}{\,}{16}{\,}{21}{\,}{26}{\,}{31}{\,}{36}{\,}{41}{\,}{45}\right)\left({11}{\,}{17}{\,}{22}{\,}{27}{\,}{32}{\,}{37}{\,}{42}{\,}{46}\right)\left({12}{\,}{18}{\,}{23}{\,}{28}{\,}{33}{\,}{38}{\,}{43}{\,}{47}\right)\left({13}{\,}{19}{\,}{24}{\,}{29}{\,}{34}{\,}{39}{\,}{44}{\,}{48}\right)\u27e9$
 (1) 
>

$\mathrm{MetacyclicGroup}\left(6\,8\,5\,\'\mathrm{form}\'\=''permgroup''\right)$

$\u27e8\left({1}{\,}{9}{\,}{10}{\,}{11}{\,}{12}{\,}{13}\right)\left({2}{\,}{15}{\,}{16}{\,}{17}{\,}{18}{\,}{19}\right)\left({3}{\,}{20}{\,}{21}{\,}{22}{\,}{23}{\,}{24}\right)\left({4}{\,}{25}{\,}{26}{\,}{27}{\,}{28}{\,}{29}\right)\left({5}{\,}{30}{\,}{31}{\,}{32}{\,}{33}{\,}{34}\right)\left({6}{\,}{35}{\,}{36}{\,}{37}{\,}{38}{\,}{39}\right)\left({7}{\,}{40}{\,}{41}{\,}{42}{\,}{43}{\,}{44}\right)\left({8}{\,}{14}{\,}{45}{\,}{46}{\,}{47}{\,}{48}\right){\,}\left({1}{\,}{2}{\,}{3}{\,}{4}{\,}{5}{\,}{6}{\,}{7}{\,}{8}\right)\left({9}{\,}{15}{\,}{20}{\,}{25}{\,}{30}{\,}{35}{\,}{40}{\,}{14}\right)\left({10}{\,}{16}{\,}{21}{\,}{26}{\,}{31}{\,}{36}{\,}{41}{\,}{45}\right)\left({11}{\,}{17}{\,}{22}{\,}{27}{\,}{32}{\,}{37}{\,}{42}{\,}{46}\right)\left({12}{\,}{18}{\,}{23}{\,}{28}{\,}{33}{\,}{38}{\,}{43}{\,}{47}\right)\left({13}{\,}{19}{\,}{24}{\,}{29}{\,}{34}{\,}{39}{\,}{44}{\,}{48}\right)\u27e9$
 (2) 
>

$\mathrm{MetacyclicGroup}\left(6\,8\,5\,\'\mathrm{form}\'\=''fpgroup''\right)$

$\u27e8{}{a}{\,}{b}{}{\mid}{}{{a}}^{{6}}{\,}{{b}}^{{8}}{\,}{{b}}^{{1}}{}{a}{}{b}{}{{a}}^{{5}}{}\u27e9$
 (3) 
In the following example, the first parameter $6$ is a proper multiple of the order of the corresponding generator.
>

$a\,b\u2254\mathrm{op}\left(\mathrm{Generators}\left(\mathrm{MetacyclicGroup}\left(6\,8\,4\right)\right)\right)$

${a}{\,}{b}{\u2254}\left({1}{\,}{9}{\,}{10}\right)\left({2}{\,}{12}{\,}{13}\right)\left({3}{\,}{14}{\,}{15}\right)\left({4}{\,}{16}{\,}{17}\right)\left({5}{\,}{18}{\,}{19}\right)\left({6}{\,}{20}{\,}{21}\right)\left({7}{\,}{22}{\,}{23}\right)\left({8}{\,}{11}{\,}{24}\right){\,}\left({1}{\,}{2}{\,}{3}{\,}{4}{\,}{5}{\,}{6}{\,}{7}{\,}{8}\right)\left({9}{\,}{13}{\,}{14}{\,}{17}{\,}{18}{\,}{21}{\,}{22}{\,}{24}\right)\left({10}{\,}{12}{\,}{15}{\,}{16}{\,}{19}{\,}{20}{\,}{23}{\,}{11}\right)$
 (4) 
>

$\mathrm{PermOrder}\left(a\right)$

>

$\mathrm{PermOrder}\left(b\right)$



Compatibility


•

The GroupTheory[MetacyclicGroup] command was introduced in Maple 17.







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