construct a finite metacyclic group
MetacyclicGroup(m, n, k)
MetacyclicGroup(m, n, k, s)
a positive integer
(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 a normal in G. The group G is then determined by the action of b on a. Since a is normal in G, it follows that the conjugate ab belongs to a so there is a positive integer k for which ab=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 ab=a−k, and where an=1 and bm=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.
In the following example, the first parameter 6 is a proper multiple of the order of the corresponding generator.
a,b ≔ op⁡Generators⁡MetacyclicGroup⁡6,8,4
The GroupTheory[MetacyclicGroup] command was introduced in Maple 17.
For more information on Maple 17 changes, see Updates in Maple 17.
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