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Additional info for 3-Selmer groups for curves y^2 = x^3 + a
But, as a matter of fact, when dealing with contravariant functors a from F to Ab (cf.
4 in [28, Ch. 5]), whereas it need not be the case for the condition deﬁning the F-selfcentralizing subgroups (cf. 8 below). 5 Assume that F is a Frobenius P -category. A subgroup Q of P which contains an F-nilcentralized subgroup R is F-nilcentralized too. If moreover R is fully centralized in F , then Q is fully centralized in F too. 7, there is an F-morphism ξ : Q·CP (Q) → P such that ξ(R) is both fully centralized and fully normalized in F ; hence, up to replacing R and Q by ξ(R) and ξ(Q) , we may assume that R is fully centralized in F .
1 FP (P ) is a Sylow p-subgroup of F(P ) . 2 If Q is an F-intersected subgroup of P , R is a subgroup of NP (Q) containing Q and ϕ : Q → P is an F-morphism fulﬁlling ϕ FR (Q) ⊂ FP ϕ(Q) then there is an F-morphism ψ : R → P extending ϕ . 3 Any divisible P -category F fulﬁlling F (P, Q) ⊃ F(P, Q) for every F-intersected subgroup Q of P contains F . 2 are clearly necessary. 3 is necessary too since any F -morphism to P from a subgroup Q of P fully centralized in F can be extended to IF (Q) (cf. 1) which is an F -intersected subgroup (cf.