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By P.J. Hilton

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By (a), Hi = ∅. j. Therefore, Since ϕij is an epimorphism, ϕij (Hi ) ⊂ Hj , whenever i {Hi , ϕij , I} is an inverse system of nonempty finite sets. 4, lim H = ∅. ←− i∈I Let (Hi ) ∈ lim Hi . Then Hi is a π-Hall subgroup of Gi for each i ∈ I, and ←− {Hi , ϕij , I} is an inverse system of finite groups. 4, H = lim Hi is a π-Hall subgroup of G, as desired. ←− (b ) Let H be a π-subgroup of G. Then, ϕi (H) is a π-subgroup of Gi (i ∈ I). 3 The Order of a Profinite Group and Sylow Subgroups 37 Si = {S | ϕi (H) ≤ S ≤ Gi , S is a π-Hall subgroup of Gi } is nonempty.

In most cases we leave the proofs as exercises, although we shall remind the reader of the necessary corresponding properties of finite groups. If G is a finite nilpotent group, then it has a unique p-Sylow subgroup for each prime p; moreover, G is the direct product of its p-Sylow subgroups. These properties characterize finite nilpotent groups (cf. 4). 8 A profinite group G is pronilpotent if and only if for each prime number p, G contains a unique p-Sylow subgroup. Denote by Gp the unique p-Sylow subgroup of a pronilpotent group G.

I i0 Since ηi (Gi ×Gi ) is contained in the finite set H ×H and since I0 is a directed poset, it follows that η(G × G) = ηk (Gk × Gk ), for some k ∈ I0 . Next observe that since β is a homomorphism, η(G × G) ⊆ Δ = {(h, h) | h ∈ H}. Therefore ηk (Gk × Gk ) ⊆ Δ; thus ηk is a homomorphism. Put β = ηk . 2 Direct or Inductive Limits In this section we study direct (or inductive) systems and their limits. 1; however there some specific results for direct limits that we want to emphasize. Again, we shall not try to develop the theory under the most general conditions; we are mainly interested in direct limits of abelian groups (or modules).

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