construct a finite metacyclic group - Maple Help

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GroupTheory[MetacyclicGroup] - 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 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=ak. 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=ak, 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.

Examples

withGroupTheory:

MetacyclicGroup6,8,5

1,9,10,11,12,132,15,16,17,18,193,20,21,22,23,244,25,26,27,28,295,30,31,32,33,346,35,36,37,38,397,40,41,42,43,448,14,45,46,47,48,1,2,3,4,5,6,7,89,15,20,25,30,35,40,1410,16,21,26,31,36,41,4511,17,22,27,32,37,42,4612,18,23,28,33,38,43,4713,19,24,29,34,39,44,48

(1)

MetacyclicGroup6,8,5,'form'=permgroup

1,9,10,11,12,132,15,16,17,18,193,20,21,22,23,244,25,26,27,28,295,30,31,32,33,346,35,36,37,38,397,40,41,42,43,448,14,45,46,47,48,1,2,3,4,5,6,7,89,15,20,25,30,35,40,1410,16,21,26,31,36,41,4511,17,22,27,32,37,42,4612,18,23,28,33,38,43,4713,19,24,29,34,39,44,48

(2)

MetacyclicGroup6,8,5,'form'=fpgroup

a,ba6,b8,b-1aba5

(3)

In the following example, the first parameter 6 is a proper multiple of the order of the corresponding generator.

a,b:=opGeneratorsMetacyclicGroup6,8,4

a,b:=1,9,102,12,133,14,154,16,175,18,196,20,217,22,238,11,24,1,2,3,4,5,6,7,89,13,14,17,18,21,22,2410,12,15,16,19,20,23,11

(4)

PermOrdera

3

(5)

PermOrderb

8

(6)

See Also

GroupTheory[CyclicGroup], GroupTheory[DicyclicGroup]


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