# 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: extension (ex-73458c)` 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. ```magma > C4 := CyclicGroup(GrpPC,4); > E9 := AbelianGroup(GrpPC,[3,3]); > f1 := homE9|[E9.1*E9.2^2, E9.1^2*E9.2^2]>; > f2 := homE9|[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. ```magma > 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 $H$. ```magma > [N`order:N in NormalSubgroups(H)]; [ 1, 9, 81, 162, 324 ] ``` ## `Example: Cossey Hawkes (ex-9a45eb)` In this example we verify an example of Cossey and Hawkes in [[Cossey and Hawkes, 2000](../../references.md#cite-ch-classes)]. 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. ```magma > E := DihedralGroup(GrpPC,5); > A := CyclicGroup(GrpPC,8); ``` Define an action of E on A and create the split extension. ```magma > f1 := homA|A.1->(A.1)^-1>; > f2 := homA|A.1->A.1>; > H := Extension(A, E, [f1, f2]); ``` Then construct a certain H-module… ```magma > QH := SylowSubgroup(H,2); > t := TrivialModule(QH, FiniteField(5)); > B := Induction(t, H); ``` …and form the split extension of H acting on that module. ```magma > 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. ```magma > 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.