Metacyclic \(p\)-groups

Magma contains functions for constructing all metacyclic groups of order \(p^n\). 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.

MetacyclicPGroups(p, n: parameters): RngIntElt, RngIntElt -> SeqEnum

PCGroups: BoolElt                    Default: true

Return a list of the metacyclic groups of order \(p^m\), where \(p\) is a prime and \(n\) is a positive integer.

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.

IsMetacyclicPGroup(P): Grp -> BoolElt

The group \(P\) is a \(p\)-group, either pc- or matrix or permutation group; if \(P\) is metacyclic, then return true, else false.

InvariantsMetacyclicPGroup(P): Grp -> Tup

The group \(P\) 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, \(\langle r, s, t, n \rangle\) has the following meaning: \(P\) has order \(p^{n + s}\); its derived quotient is \(C_{p^r} \times C_{p^s}\); its derived group is cyclic of order \(p^{n - r}\); it has exponent \(p^{n +s - t}\).

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 \(2^{s - 1}\) whose central quotients are both semidihedral, or dihedral and quaternion. The invariant tuple has length at most 6.

StandardMetacyclicPGroup(P): Grp -> GrpPC

The group \(P\) 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.

NumberOfMetacyclicPGroups(p, n): RngIntElt, RngIntElt -> SeqEnum

Return number of metacyclic groups of order \(p^n\).

HasAllPQuotientsMetacyclic(G): GrpFP -> BoolElt, SeqEnum

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.

Example: meta (ex-0140ea)

> 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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