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By Bergman G.M.

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Extra resources for A companion to S.Lang's Algebra 4ed.

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52, last two paragraphs [<]: Lang doesn’t seem to have made up his mind whether he wants to tell us about the completion of a group G with respect to an arbitrary directed system of normal subgroups, or only about the special case where is the set of all normal subgroups of finite index, so what he says is a bit inconsistent. The latter construction, called the ‘‘profinite completion’’ of G, is important; but since he doesn’t go into the theory of that completion, let us assume is a general directed system of normal subgroups.

One can obtain a countable cofinal chain, H1 ⊇ H1 ∩ H 2 ⊇ H1 ∩ H 2 ∩ H 3 ⊇ ... , and that the inverse limit over that chain is isomorphic to the inverse limit of the original system. This last statement is not obvious, but, as Lang says, it makes a nice exercise. (Actually, one can show from the cofinality assumption that there is a subsequence (Hi ) of (Hi ) m ⊆ Hi for all m, and is still cofinal in which satisfies Hi . This gives the desired chain without m m+1 taking intersections, and the statements Lang makes hold with Hi in place of Hi .

If a a1 ≠ 1. Finally (iv) If a ≠ 1 and belongs to the same group as a1 , and if a a1 = 1, then we let a(a1 , ... , an ) = (a 2 , ... , an ) (understood to mean () if n =1). Restricting attention to the action of elements a ∈A, we find that the map A × X → X given by the above ‘‘multiplication’’ is indeed a group action. ) Similarly, the map B × X → X is an action of B. We define A ° B to be the group of permutations of X generated by the images of these two actions. Now if we write a- for the image of a ∈A ∪ B in Perm(X), then, simply from the fact that the group A ° B is generated by the images of A and B under certain homomorphisms a → a- , it is not hard to verify that every element of A ° B can be reduced to an expression a-1 ...

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