By Samiou E.

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**Additional resources for 2-step nilpotent Lie groups of higher rank**

**Example text**

A) If A is the direct sum [product] of the groups Bi (i E I ) , and Ci are subgroups of Bi, and if C is the direct sum [product] of the C i , then C is a subgroup of A such that A/C is the direct sum [product] of the quotient groups Bi/Ci. (b) If 0 -+ Ai* Bi* Ci -+ 0 are exact sequences (i E I ) , then so are O + @Ai 3 @ B i o+n~ and 881 @Ci+O ~B 3, ci-+o. A n~ 3. A is a direct sum of its subgroups A i (i E I ) if and only if CAi= A i and n A: i =O where A:=CAj. j+i (a) The direct sum of p-groups [torsion groups] is again a p-group [torsion group].

Both will be discussed here, along with their basic properties. The external definition leads to unrestricted direct sums, called direct products, which, too, are extremely useful for us. We discuss pullback and pushout diagrams as well. Also, we are going to define direct and inverse limits, which have begun to play an increasing role in the theory of abelian groups. We shall often have occasion to use them in various context. In the final section of this chapter, completions under a linear topology are dealt with.

N The equivalence with the definition given in (b) follows from results in Chapter 111. More precisely, U ( A ) is called t h e j r s t Ulrn subgroup of A , while the oth iterated functor U"(A) = A" yields the 0th Ulrn subgroup of A . The quotient group U"(A)/U"+'(A)= U"(A) = A , (o = 0, 1,2, . ) is said to be the 0th Ulrn factor of A ; in particular, A/A' = A , is the 0th Ulrn factor of A . For a discussion of Ulrn subgroups and factors, we refer to 37. Notice that if F , , F , are subfunctors of the identity, and F , 5 F , , then the inclusion maps 6" : F,(A) F,(A) yield a natural transformation cD : F , -+ F , , and the corresponding maps 42 : A / F , ( A ) --* A / F , ( A ) define a natural transformation cD* : FT -+ FT.