Extensions of the Representation Modules of a Prime Order Group
نویسندگان
چکیده
For the ring R of integers of a ramified extension of the field of p-adic numbers and a cyclic group G of prime order p we study the extensions of the additive groups of R-representations modules of G by the group G. Let T be the field of fractions of a principal ideal domain R, F a field which contains R, let G be a finite group and Γ a matrix R-representation of G. Let M be an RG-module, which affords the R-representation Γ of G, and FM = F ⊗R M the smallest linear space over F which contains M and M̂ = FM/M , the factor group of the additive group of the space FM by the additive group of M . Clearly, the group M̂ and the space FM are RG-modules. Put F̂ = F/R. Let f : G → M̂ be a 1-cocycle of G with value in M̂ , i.e. f(xy) = xf(y) + f(x), (x, y ∈ G). Define [g, x] by ( g x 0 1 ) and set Crys(G,M, f) = { [g, x] | g ∈ G, x ∈ f(g) }, where x runs over the cosets f(g) ∈ M̂ for any g ∈ G. Clearly, Crys(G,M, f) is a group, where the multiplication is the usual matrix multiplication. Of course K1 = {[e, x] | e is the unit element of G, x ∈ f(e)} is a normal subgroup of Crys(G,M, f) such that K1 ∼= M + and Crys(G,M, f)/K1 ∼= G. The group Crys(G,M, f) is an extension of the additive group of the RG-module M by G. We are using the terminology of the theory of group representations [1]. A 1-cocycle f : G → M̂ is called coboundary, if there exists an x ∈ FM such that f(g) = (g − 1)x + M for every g ∈ G. The 1-cocycles f1 : G → M̂ and 1991 Mathematics Subject Classification. Primary 16U60, 16W10. The research was supported by OTKA No.T 037202 and No.T 038059
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