New Groups from Existing

DirectProduct(G, H): GrpPC, GrpPC -> GrpPC, [Map], [Map]: The direct product \(K\) of the pc-groups \(G\) and \(H\). The second argument returned is a sequence containing the inclusion maps \(I_G\colon G \rightarrow K\) and \(I_H\colon H \rightarrow K\). The third argument returned is a sequence containing the projection maps \(P_G\colon K \rightarrow G\) and \(P_H\colon K \rightarrow H\). Furthermore, the (user-) presentation of \(K\) is arranged so that the first pc-generators correspond to those of \(G\) and the remaining generators correspond to those of \(H\).

DirectProduct(Q): [GrpPC] -> GrpPC, [ Map ], [ Map ]: The direct product of pc-groups in the non-empty sequence \(Q\), and the inclusion and projection maps.

Extension(G, H, f): GrpPC, GrpPC, [Map] -> GrpPC

The split extension \(K\) of the pc-group \(G\) by the pc-group \(H\), where the action of \(H\) on \(G\) is given by the homomorphism \(\phi: H \rightarrow {\operatorname{Aut}}(G)\) specified by \(f\). The extension \(K\) will have a normal subgroup \(G\mathord{\sim}\) isomorphic to \(G\), while the quotient group \(K/G\mathord{\sim}\) is isomorphic to \(H\).

The homomorphism \(\phi\) is given by the sequence of maps \(f\). Suppose that the pc-generators for \(H\) are \(h_1, \ldots, h_s\). The \(i\)-th entry of \(f\) defines the action of \(h_i\) on \(G\). That is, \(f[i](x) = h_i^{-1}\cdot x\cdot h_i\), for \(x \in G\).

Extension(M, H): ModGrp, GrpPC -> GrpPC: The split extension \(K\) of the \(G\)-module \(M\) by the pc-group \(H\). We use the action of \(H\) on \(M\) to define the action of \(H\) on an elementary abelian \(p\)-group of order \(p^d\) where \(M\) is a \(d\)-dimensional module over \(GF(p)\), \(p\) prime.

Extension(G, H, f, t): GrpPC, GrpPC, [Map], [GrpPCElt] -> GrpPC

Extension(G, H, f, t): GrpPC, GrpPC, [Map], {Tup} -> GrpPC

The non-split extension \(K\) of the pc-group \(G\) by the pc-group \(H\), where the action of \(H\) on \(G\) is given by the homomorphism \(\phi: H \rightarrow {\operatorname{Aut}}(G)\) and the tails for \(H\) are given as the set of tuples \(t\). The extension \(K\) will have a normal subgroup \(G\mathord{\sim}\) isomorphic to \(G\), while the quotient group \(K/G\mathord{\sim}\) is isomorphic to \(H\).

The homomorphism \(\phi\) is given by the sequence of maps \(f\). Suppose that the pc-generators for \(H\) are \(h_1, \ldots, h_s\). The \(i\)-th entry of \(f\) defines the action of \(h_i\) on \(G\). That is, \(f[i](x) = h_i^{-1}\cdot x\cdot h_i\), for \(x \in G\).

The specification of \(t\) involves giving the relations \(h_j^{-1}h_ih_j\) \(= w_{ij}\), where \(w_{ij}\) is a word in \(K\) for \(1 \le j < i \le s\). For \(i = j\), we need the relation \({h_i}^{p_i} = w_{ii}\), where \(w_{ii}\) is a word in \(K\) for \(1 \le i \le s\). Each \(w_{ij}\) is the RHS of the relation from \(H\) with the tail \(x_{ij}\). The tails are given by the sequence \(t\) in the order \(t = [x_{11}, x_{21}, x_{22}, x_{31}, \ldots , x_{ss} ]\). Alternatively, \(t\) can be given as a set of tuples \(< i, j, x_{ij} >\) for non-trivial \(x_{ij}\).

Note that if \(x_{ij} =\)Id(G), for \(1\leq i\leq s\) and \(1\leq j\leq i\), then \(K\) will just be the split extension of \(G\) and \(H\).

Extension(M, H, t): ModGrp, GrpPC, [ModGrpElt] -> GrpPC

The non-split extension \(K\) of the \(G\)-module \(M\) by the pc-group \(H\). We use the action of \(H\) on \(M\) to define the action of \(H\) on an elementary abelian \(p\)-group of order \(p^d\) where \(M\) is a \(d\)-dimensional module over \(GF(p)\), \(p\) prime.

The specification of \(t\) is similar to that for \(t\) in the preceding description.

IsExtension(G, H, f): GrpPC, GrpPC, [Map] -> BoolElt, GrpPC

IsExtension(M, H): ModGrp, GrpPC -> BoolElt, GrpPC

IsExtension(G, H, f, t): GrpPC, GrpPC, [Map], [GrpPCElt] -> BoolElt, GrpPC

IsExtension(G, H, f, t): GrpPC, GrpPC, [Map], {Tup} -> BoolElt, GrpPC

IsExtension(M, H, t): ModGrp, GrpPC, [ModGrpElt] -> BoolElt, GrpPC

For each Extension variation, there is a corresponding function IsExtension which attempts to construct the specified group and returns a boolean value indicating whether or not the construction succeeded. If the construction succeeds, the extension group is also returned.

The Extension functions will generate a runtime error if the specified construction is not legal. The IsExtension function allows the user to detect this error condition and continue.

WreathProduct(G, H): GrpPC, GrpPC -> GrpPC: The wreath product of the pc-groups \(G\) and \(H\), where the regular permutation representation of \(H\) is used to define the action.

WreathProduct(G, H, f): GrpPC, GrpPC, Map -> GrpPC
WreathProduct(G, H, f): GrpPC, GrpPC, [GrpPermElt] -> GrpPC: The wreath product of the pc-groups \(G\) and \(H\) where the action of \(H\) is given by \(f\), which may be either a homomorphism from \(H\) into a permutation group \(P\) or a sequence of permutations defining a homomorphism from \(H\) into \(P\). If \(f\) is a sequence, the homomorphism \(\phi: H \rightarrow P\) is defined by \(H.i \rightarrow f[i]\) for \(i = 1, \ldots, s\).

Example: Extensions of an Elementary Abelian Group (ex-73458c)

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To demonstrate some of the versions of Extension we first build a split extension of a cyclic group of order 4 acting on an elementary abelian group of order 9.

> C4 := CyclicGroup(GrpPC,4);
> E9 := AbelianGroup(GrpPC,[3,3]);
> f1 := hom<E9->E9|[E9.1*E9.2^2, E9.1^2*E9.2^2]>;
> f2 := hom<E9->E9|[E9.1^2,E9.2^2]>;
> G := Extension(E9,C4,[f1,f2]);
> G;
GrpPC : G of order 36 = 2^2 * 3^2
PC-Relations:
    G.1^2 = G.2,
    G.2^2 = Id(G),
    G.3^3 = Id(G),
    G.4^3 = Id(G),
    G.3^G.1 = G.3 * G.4^2,
    G.3^G.2 = G.3^2,
    G.4^G.1 = G.3^2 * G.4^2,
    G.4^G.2 = G.4^2

Then, we define a module for this group and use it to build a nonsplit extension.

> MR := MatrixRing(GF(3),2);
> m1 := MR![1,1,1,2];
> m2 := MR![2,0,0,2];
> V := GModule(G,[m1,m2,Id(MR),Id(MR)]);
> IsIrreducible(V);
true
> v0 := V!0;
> tails := [v0,v0,v0,v0,V![1,0],V![2,0],V![1,2],V![0,2],v0,V![0,1]];
> H := Extension(V,G,tails);
> H;
GrpPC : H of order 324 = 2^2 * 3^4
PC-Relations:
    H.1^2 = H.2,
    H.2^2 = Id(H),
    H.3^3 = H.5^2,
    H.4^3 = H.6,
    H.5^3 = Id(H),
    H.6^3 = Id(H),
    H.3^H.1 = H.3 * H.4^2,
    H.3^H.2 = H.3^2 * H.5,
    H.4^H.1 = H.3^2 * H.4^2 * H.5 * H.6^2,
    H.4^H.2 = H.4^2 * H.6^2,
    H.5^H.1 = H.5 * H.6,
    H.5^H.2 = H.5^2,
    H.6^H.1 = H.5 * H.6^2,
    H.6^H.2 = H.6^2

Notice that the relations of \(H\) involving the first four generators are those of \(G\) with the specified tails appended. We are then ready to compute various properties of \(H\).

> [N`order:N in NormalSubgroups(H)];
[ 1, 9, 81, 162, 324 ]

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Example: Cossey-Hawkes Conjugacy Class Sizes (ex-9a45eb)

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In this example we verify an example of Cossey and Hawkes in [CH00]. The paper shows that the largest size of a conjugacy class in an abelian by nilpotent finite group is at least as large as the product of the largest class sizes for the Sylow subgroups. The example is a group having derived length 3 in which this fails.

We start with a dihedral group of order 10 acting on a cyclic group of order 8.

> E := DihedralGroup(GrpPC,5);
> A := CyclicGroup(GrpPC,8);

Define an action of E on A and create the split extension.

> f1 := hom<A->A|A.1->(A.1)^-1>;
> f2 := hom<A->A|A.1->A.1>;
> H := Extension(A, E, [f1, f2]);

Then construct a certain H-module…

> QH := SylowSubgroup(H,2);
> t := TrivialModule(QH, FiniteField(5));
> B := Induction(t, H);

…and form the split extension of H acting on that module.

> G := Extension(B, H);
> print G;
GrpPC : G of order 250000 = 2^4 * 5^6
PC-Relations:
    G.1^2 = Id(G),
    G.2^5 = Id(G),
    G.3^2 = G.4,
    G.4^2 = G.5,
    G.5^2 = Id(G),
    G.6^5 = Id(G),
    G.7^5 = Id(G),
    G.8^5 = Id(G),
    G.9^5 = Id(G),
    G.10^5 = Id(G),
    G.2^G.1 = G.2^4,
    G.3^G.1 = G.3 * G.4 * G.5,
    G.4^G.1 = G.4 * G.5,
    G.6^G.2 = G.10,
    G.7^G.1 = G.10,
    G.7^G.2 = G.6,
    G.8^G.1 = G.9,
    G.8^G.2 = G.7,
    G.9^G.1 = G.8,
    G.9^G.2 = G.8,
    G.10^G.1 = G.7,
    G.10^G.2 = G.9
> print DerivedLength(G);
3

Now check the relevant class sizes.

> P := SylowSubgroup(G,5);
> Q := SylowSubgroup(G,2);
> print  Maximum({x[2]:x in Classes(G)});
1250
> print  Maximum({x[2]:x in Classes(P)});
625
> print  Maximum({x[2]:x in Classes(Q)});
4

Note that 1250 is less than the product 625*4.

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New Groups from Existing#

New Groups from Existing