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Magma contains functions for constructing all metacyclic groups of
order pn. It can also decide if a given p-group is metacyclic,
construct invariants which distinguish this metacyclic group from all
others of this order, and construct a standard presentation for the
group.
This section describes the functions for accessing these
algorithms. The functions were developed by
Mike Newman, Eamonn O'Brien, and Michael Vaughan-Lee.
Return a list of the metacyclic groups of order pm,
where p is a prime and n is a positive integer.
PCGroups: BoolElt Default: true
If true, the groups returned are in category GrpPC, otherwise
they are in category GrpFP -- this will be faster if
the groups have large class.
P is a p-group, either pc- or matrix or permutation group; if P
is metacyclic, then return true, else false.
P is a metacyclic p-group, either pc- or matrix or permutation group;
return tuple of invariants which uniquely identify metacyclic p-group P.
This tuple which contains at least four terms, < r, s, t, n >
has the following meaning:
P has order ps; its derived quotient is Cr x Cs;
its derived group is cyclic of order pr;
it has exponent pt.
If p = 2, then additional
invariants are needed to distinguish among the groups.
We record the abelian invariants of the centre of P.
If s = 1 and the centre of P has order 2, then the
2-group is maximal class and we record whether
it is dihedral, quaternion or semidihedral.
If s > 1 then the group has two cyclic
central normal subgroups of order 21 whose
central quotients are both semidihedral, or dihedral
and quaternion. The invariant tuple has length at most 6.
P is a metacyclic p-group, either pc- or matrix or permutation group;
return metacyclic p-group having a canonical pc-presentation which
is isomorphic to P. If two metacyclic p-groups
have the same canonical presentation, then they are isomorphic.
Return number of metacyclic groups of order pn.
HasAllPQuotientsMetacyclic (G, p): GrpFP -> BoolElt
Return true if for all primes p all p-quotients of the
finitely-presented group G are metacyclic; otherwise
return false and a description of the set of primes for which G
has non-metacyclic p-quotient.
If a prime p is supplied as a second argument, then
the function returns true if all p-quotients of
G are metacyclic; otherwise it returns false.
> X := MetacyclicPGroups (3, 6);
> #X;
11
> X[4];
GrpPC of order 729 = 3^6
PC-Relations:
$.1^3 = $.3,
$.2^3 = $.4,
$.3^3 = $.6,
$.4^3 = $.5,
$.5^3 = $.6,
$.2^$.1 = $.2 * $.6^2
> H := SmallGroup (729, 59);
> IsMetacyclicPGroup (H);
true
> I := InvariantsMetacyclicPGroup(H);
> I;
<2, 2, 2, 4, [], , >
> S := StandardMetacyclicPGroup (H);
GrpPC : S of order 729 = 3^6
PC-Relations:
S.1^3 = S.3,
S.2^3 = S.4,
S.3^3 = S.6,
S.4^3 = S.5,
S.5^3 = S.6,
S.2^S.1 = S.2 * S.6^2
> /* find this group in list */
> [IsIdenticalPresentation (S, X[i]): i in [1..#X]];
[ false, false, false, true, false, false, false, false, false, false, false ]
> /* so this group is #4 in list */
> NumberOfMetacyclicPGroups (19, 7);
14
> Q := FreeGroup (4);
> G := quo < Q | Q.2^2, Q.4^3, Q.2 * Q.3 * Q.2 * Q.3^-1, Q.1^9>;
> /* are all p-quotients of G metacyclic? */
> HasAllPQuotientsMetacyclic (G);
false [ 3 ]
> /* the 3-quotient is not metacyclic */
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