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Subgroups of p 5 -Bounded Groups

By Fred Richman and Elbert A. Walker


Each v-module B with B(5) = 0 is a direct sum of simply presented v-modules and copies of two v-modules which come from (…nite) hung trees. There are in…nite-rank indecomposable v-modules B with B(6) = 0. 1 Valuated modules By a module we will mean a module over a …xed discrete valuation domain with prime p. The reader is assumed to be familiar with the notion of a valuated module, or v-module, which is a module B together with a …ltration B = B(0) B(1) B(2) such that pB(n) B(n+1). We need not consider arbitrary ordinal values of n because we are interested in the case B(5) = 0. If x 2 B(n) and x =2 B(n + 1) we write vx = n and say the value of x is n. If B is a subgroup of a p 5-bounded group G, then B is naturally a module over the ring of integers localized at p, and the module B is …ltered by setting B(n) = B\p n G. Classifying such subgroups, up to isomorphism of G, is equivalent to classifying the associated v-modules, because bounded modules (with the …ltration B(n) = p n B) are injective in the category of v-modules [5, Theorem 9]. We classify v-modules by writing them as direct sums. These direct sums must respect values, that is, they must respect the …ltration: (A B)(n) = A(n) B(n). The following lemma aids in verifying that a sum respects values. Lemma 1 (respect value) If A and H are submodules of a torsion v-module, and A \ H = 0, then A H respects values provided whenever n is an Ulm invariant of A

Year: 1999
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