# Finite Group Cohomology This section describes Magma functions for computing the first cohomology group of a finite group with coefficients in a finite (not necessarily abelian) group. These functions are based on [[Haller, 2005](../../references.md#cite-sh)]. Let $\Gamma$ be a group. A group $A$ on which $\Gamma$ acts by group automorphisms from the right, is called a $\Gamma$*-group*. Given a $\Gamma$-group $A$, define $$ H^0(\Gamma,A) := \{ a\in A \mid a^\sigma = a {\hbox{ for all }} \sigma\in\Gamma \}. $$ A $1$*-cocycle of* $\Gamma$ *on* $A$ is a map $$ \alpha: \Gamma \rightarrow A, \quad \sigma \mapsto \alpha_\sigma, $$ such that $$ \alpha_{\sigma\tau} = (\alpha_\sigma)^\tau \alpha_\tau \quad\hbox{ for all } \sigma,\tau\in\Gamma. $$ Two cocycles $\alpha,\beta$ on $A$ are called *cohomologous* (with respect to $a$) if there exists $a\in A$, such that $\beta_\sigma = a^{-\sigma} \cdot \alpha_\sigma \cdot a$ for all $\sigma\in\Gamma$. Note that being cohomologous is an equivalence relation. We denote by $Z^1(\Gamma, A)$ the set of all $1$-cocycles of $\Gamma$ on $A$. We denote by $[\alpha]$ the equivalence class of $\alpha$ and by $H^1(\Gamma, A)$ the set of equivalence classes of $1$-cocycles. $Z^1(\Gamma, A)$ and $H^1(\Gamma, A)$ are *pointed sets*. The constant map $t: \sigma \mapsto 1$ is the distinguished element of $Z^1(\Gamma, A)$, called the *trivial* $1$*-cocycle*. Its cohomology class is the distinguished element of $H^1(\Gamma, A)$. A twisted form $A_\beta$ of $A$ by the cocycle $\beta\in Z^1(\Gamma,A)$ is the same group $A$ but with a different action of $\Gamma$ on it, given by $$ a * \sigma := a^{\sigma \alpha_\sigma} \quad\hbox{ for }\sigma\in\Gamma \hbox{ and } a\in A. $$ ## Creation of Gamma-groups This section describes intrinsics dealing with cocycles and the first cohomology. ### `GammaGroup(Gamma, A, action): Grp, Grp, Map[Grp, GrpAuto] -> GGrp` Given a group $A$ and a group $\Gamma$ acting on it by the map `action`, return the object of type `GGrp`, which is the Group $A$ together with this particular action of $\Gamma$. The map `action` must be a homomorphism from $\Gamma$ to the automorphism group of $A$. If $B$ is a normal subgroup of $A$ and normalised by the action of $\Gamma$ on $A$ (thus a $\Gamma$-group itself), then the action of $\Gamma$ on $A$ induces in the natural way to $A/B$. It is possible to create such a group: ### `InducedGammaGroup(A, B): GGrp, Grp -> GGrp` Given a $\Gamma$-group $A$ and a normal subgroup $B$ normalised by the action of $\Gamma$, return the induced $\Gamma$-group $A/B$. ### `Example: create G Grp (ex-379641)` Let $\Gamma$ act on $A$ by conjugation: ```magma > A := SymmetricGroup(4); > Gamma := sub; > action := hom< Gamma -> Aut(A) | > g :-> iso< A -> A | a :-> a^g, a :-> a^(g^-1) > >; > A := GammaGroup( Gamma, A, action ); > A; Gamma-group: Symmetric group acting on a set of cardinality 4 Order = 24 = 2^3 * 3 (1, 2, 3, 4) (1, 2) Gamma-action: Mapping from: GrpPerm: $, Degree 4 to Set of all automorphisms of GrpPerm: $, Degree 4, Order 2^3 * 3 given by a rule [no inverse] Gamma: Permutation group acting on a set of cardinality 4 (1, 2, 3) (1, 2) > ``` and $B$ be a normal subgroup of $A$: ```magma > B := AlternatingGroup(4); > AmodB := InducedGammaGroup( A, B ); > AmodB; Gamma-group: Symmetric group acting on a set of cardinality 2 Order = 2 (1, 2) (1, 2) Gamma-action: Mapping from: GrpPerm: $, Degree 4, Order 2 * 3 to Set of all automorphisms of GrpPerm: $, Degree 2, Order 2 given by a rule [no inverse] Gamma: Permutation group acting on a set of cardinality 4 Order = 6 = 2 * 3 (1, 2, 3) (1, 2) Induced from another Gamma-group ``` ### `IsNormalised(B, action): Grp, Map -> BoolElt` Returns `true` if the group $B$ is normalised by the action `action`, where `action` is as above. ### `IsInduced(AmodB): GGrp -> BoolElt, GGrp, GGrp, Map, Map` Returns `true` iff the $\Gamma$-group *AmodB* was created as an induced $\Gamma$-group. If it is, then the $\Gamma$-groups $A$, $B$, the projection and representative maps are returned as well. ## Accessing Information ### `Group(A): GGrp -> Grp` Returns the group $A$ as a `Grp` object to be used in Magma. ### `GammaAction(A): GGrp -> Map[Grp, GrpAuto]` Returns the action of $\Gamma$ on $A$ as a map. ### `ActingGroup(A): GGrp -> Grp` Returns the group $\Gamma$ acting on $A$. ## One Cocycles ### `OneCocycle(A, imgs): GGrp, SeqEnum[GrpElt] -> OneCoC` ### `OneCocycle(A, alpha): GGrp, Map[Grp,Grp] -> OneCoC` ```magma Check: BoolElt Default: true ``` If the map $\alpha:\Gamma \rightarrow A$ or the sequence `imgs` of images of the generators $\Gamma.1,...,\Gamma.n$ defines a $1$-cocycle, return the $1$-cocycle. By default, the map is checked to define a $1$-cocycle. If it doesn’t, `OneCocycle` will abort with an error. This check can be disabled by setting the optional argument `Check` to `false`. ### `TrivialOneCocycle(A): GGrp -> OneCoC` Return the trivial $1$-cocycle. ### `IsOneCocycle(A, imgs): GGrp, SeqEnum[GrpElt] -> BoolElt, OneCoC` ### `IsOneCocycle(A, alpha): GGrp, Map[Grp,Grp] -> BoolElt, OneCoC` Return true if the map $\alpha:\Gamma \rightarrow A$ or the sequence `imgs` of images of the generators $\Gamma.1,...,\Gamma.n$ defines a $1$-cocycle and `false` otherwise. If `true`, return the cocycle as the second argument. Note that `IsOneCocycle` does not abort with an error in contrast to `OneCocycle` if the map does not define a cocycle. ### `AreCohomologous(alpha, beta): OneCoC, OneCoC -> BoolElt, GrpElt` Return `true` if and only if the $1$-cocycles $\alpha$ and $\beta$ are cohomologous. If they are, return the intertwining element as the second return value. ### `CohomologyClass(alpha): OneCoC -> SetIndx[OneCoC]` Return the cohomology class of the $1$-cocycle $\alpha$. ### `InducedOneCocycle(AmodB, alpha): GGrp, OneCoC -> OneCoC` ### `InducedOneCocycle(A, B, alpha): GGrp, Grp, OneCoC -> OneCoC` Given a $1$-cocycle on $A$, return the induced $1$-cocycle on *AmodB*. The second version will generate the induced $\Gamma$-group $A/B$ first. ### `ExtendedOneCocycle(alpha): OneCoC -> SetEnum[OneCoC]` ```magma OnlyOne: BoolElt Default: false ``` Given a $1$-cocycle on an induced $\Gamma$-group $A/B$, return the set of all non-cohomologous $1$-cocycles on $A$, which induce to $\alpha$. If the optional argument `OnlyOne` is `true`, the set will contain at most one $1$-cocycle. If $\alpha$ is not extendible, the returned set is empty. ### `ExtendedCohomologyClass(alpha): OneCoC -> SetEnum[OneCoC]` Given a $1$-cocycle on an induced $\Gamma$-group $A/B$, return the the set of all non-cohomologous $1$-cocycles on $A$, which induce to a cocycle in the cohomology class of $\alpha$. If no such cocycles on $A$ exist, the returned set is empty. ### `GammaGroup(alpha): OneCoC -> GGrp` Return the $\Gamma$-group on which $\alpha$ is defined. ### `CocycleMap(alpha): OneCoC -> Map` Return the `Map` object corresponding to $\alpha$. ## Group Cohomology ### `Cohomology(A, n): GGrp, RngIntElt -> SetEnum[OneCoC]` Given a finite $\Gamma$-group $A$ and an integer $n$ (currently restricted to being $1$) return the $n$-th cohomology group $H^n(\Gamma, A)$. Since the group $A$ is not assumed to be abelian, only $n=0,1$ can be used. Currently, only $n=1$ implemented. (The zero cohomology of $A$ is the subgroup of $A$ centralised by $\Gamma$ and can be constructed using group theoretical methods available in Magma.) ### `OneCohomology(A): GGrp -> SetEnum[OneCoC]` Return the first cohomology $H^1(\Gamma, A)$. as a set of representatives of all cohomology classes. If the group $A$ is abelian, existing code by Derek Holt is used (see Chapter [Cohomology and Extensions](index-cohomology-and-extensions.md#grpcohom-main)). Otherwise use [[Haller, 2005](../../references.md#cite-sh)]. ### `TwistedGroup(A, alpha): GGrp, OneCoC -> GGrp` Given the $\Gamma$-group $A$ and a $1$-cocycle $\alpha$ on it, return the twisted $\Gamma$-group $A_\alpha$. ### `Example: large example (ex-82c924)` First, we create the group $A=D_8$. The returned group is the usual permutation group on the octagon. $\Gamma$ is the Normaliser of $A$ in $S_8$ and is acting by conjugation. ```magma > A := DihedralGroup(8); > Gamma := sub< Sym(8) | (1, 2, 3, 4, 5, 6, 7, 8), > (1, 8)(2, 7)(3, 6)(4, 5), (2, 4)(3, 7)(6, 8) >; > A^Gamma eq A; true > Gamma; Permutation group Gamma acting on a set of cardinality 8 Order = 32 = 2^5 (1, 2, 3, 4, 5, 6, 7, 8) (1, 8)(2, 7)(3, 6)(4, 5) (2, 4)(3, 7)(6, 8) > action := hom< Gamma -> Aut(A) | > g :-> iso< A -> A | a :-> a^g, a :-> a^(g^-1) > >; > A := GammaGroup( Gamma, A, action ); ``` Now let $B$ be the center of $A$ and create the induced $\Gamma$-group $A/B$: ```magma > B := Center(Group(A)); > AmodB := InducedGammaGroup(A, B); ``` Create the trivial $1$-cocycle on $A/B$ and compute its cohomology class: ```magma > triv := TrivialOneCocycle(AmodB); > CohomologyClass( triv ); {@ One-Cocycle defined by [ Id($), Id($), Id($) ], One-Cocycle defined by [ Id($), (1, 4)(2, 7)(3, 8)(5, 6), (1, 4)(2, 7)(3, 8)(5, 6) ], One-Cocycle defined by [ (1, 4)(2, 7)(3, 8)(5, 6), Id($), (1, 4)(2, 7)(3, 8)(5, 6) ], One-Cocycle defined by [ (1, 4)(2, 7)(3, 8)(5, 6), (1, 4)(2, 7)(3, 8)(5, 6), Id($) ] @} ``` Pick one of the cocycles in this class and compute the intertwining element: ```magma > alpha := Random($1);alpha; One-Cocycle defined by [ (1, 4)(2, 7)(3, 8)(5, 6), (1, 4)(2, 7)(3, 8)(5, 6), Id($) ] > bo, a := AreCohomologous(alpha,triv); > bo; a; true (1, 5)(2, 8)(3, 7)(4, 6) ``` Now create another cocycle on $A/B$ and extend it to $A$: ```magma > alpha := OneCocycle( AmodB, > [Group(AmodB)| (1, 7, 4, 2)(3, 5, 8, 6), > (1, 2, 4, 7)(3, 6, 8, 5), > 1 ] ); > ExtendedOneCocycle(alpha); { One-Cocycle defined by [ (1, 4, 7, 2, 5, 8, 3, 6), (1, 2, 3, 4, 5, 6, 7, 8), Id($) ], One-Cocycle defined by [ (1, 8, 7, 6, 5, 4, 3, 2), (1, 6, 3, 8, 5, 2, 7, 4), Id($) ] } ``` Pick a cocycle $\beta$ in this set and check if it really induces to $\alpha$: ```magma > beta := Rep($1); > InducedOneCocycle(AmodB, beta) eq alpha; true ``` Finally, create the twisted group $A_\beta$: ```magma > A_beta := TwistedGroup(A, beta); > A_beta; Gamma-group: Permutation group acting on a set of cardinality 8 Order = 16 = 2^4 (1, 2, 3, 4, 5, 6, 7, 8) (1, 8)(2, 7)(3, 6)(4, 5) Gamma-action: Mapping from: GrpPerm: $, Degree 8, Order 2^5 to Set of all automorphisms of GrpPerm: $, Degree 8, Order 2^4 given by a rule [no inverse] Gamma: Permutation group acting on a set of cardinality 8 Order = 32 = 2^5 (1, 2, 3, 4, 5, 6, 7, 8) (1, 8)(2, 7)(3, 6)(4, 5) (2, 4)(3, 7)(6, 8) > ```