The structure of large subgroups of primary abelian groups

Abstract

Section I IntroductionThe purpose of this paper is to investigate the relation between the structures of primary groups and their large subgroups. All groups are abelian. This research was motivated by a study ot the monumental paper "Homomorphisms of Primary Abelian Groups" [8] by R. S. PIERCE in which the theory of large subgroups was introduced. We shall state some of the fundamental results of this theory in section I.In section II we collect an assortment of facts, some of which appear ill the literature, some of which must be classified as folklore and some of which are new. The principal result of this section is a new representation theorem for large subgroups of p-groups.Section III is devoted to the problem: Give necessary and sufficient conditions for a p-group G to be quasi-isomorphic to every large subgroup of G. The results of this section show that unless the group G satisfies certaiD restrictive conditions on its Ulm invariants it contains a large subgroup L which is not quasi-isomorphic to G.Although a large subgroup ot a group G is not m general quasi-isomorphic to G the theorems of section IV show that the structure ot a p-group G is in many cases preserved by its large subgroups. Some of our results can be stated as follows. If L is a large subgroup of a group G then: 1. L is a direct sum of cyclic groups if and only if G is a direct sum ot cyclic groups; 2. L is totally projective if and only if G is totally projective; 3. L is torsion-complete Jf and only if G is torsion-complete; 4. L is quasi-closed if and only if G is quasi-closed.Our notation is that of FUCHS [1] with the following exceptions. We use A + B and Y, zAz to denote group unions. Direct sums we denote by AOB and | Direct products we denote by H~Aa. We use the symbol Zx to denote the cyclic group generated by the element x.The following definitions and theorems were given by PIERCE in [8]. All groups are p-groups. A large subgroup L of a group G is a fully invariant subgroup of G such that B+L = G for every basic subgroup B of G. It is easily shown that if L is a large subgroup of a group G then pnL is a large subgroup of G for every nonnegative integer n.If x is an element of a group G define the U'sequence of x to be U(x) = = (ha(x), hG(px ), hG(p2x), ...). Our heights are either non-negative integers or the symbol ~. Define a U-sequence for the group G to be any strictly increasing sequence of integers (no, hi, rt2, .-.) such that ni + 1 <hi+ 1 implies fJni) #0.Following Kal'LANSKY [5] we define the m-th Ulm invariant fG(m) by fG(m)= = dim (pmG[p]/pm + 1G [p]). The notion of U-sequences was introduced by KAI~LANSKY Actd 2~4athematlca Academlae Sclentlarl~m Hmzga~icae ~I, z97o

λ-large subgroups ofCλ-groups

“…For each positive integer n, there is an ordinal /x (n) less than /x, the length of p^G, such that <r(n) = a(0) + fi(n)< cr(0) + /x = A. From familiar properties of ordinals, it follows that /x = sup{ju,(n): n < a)} and hence JJL is cofinal with co. for some z G L. Then pz G p" (1) G n J3 C p* mi B, and…”

“…It is shown in [1] that some of the solutions to the open statement "A large subgroup L of G has property P if and only if G has property F" are these properties: direct sum of cyclic groups, direct sum of countable groups, and totally projective group. In this section we study the relation between the structure of A-large subgroups and the structure of the containing groups.…”

“…We first consider the case where G has length A. Our proof is inductive on A; if A = a), then the result follows from Theorem 4.3 in [1]. Thus we assume the conclusion for all limit ordinals j8 less than A where ]8 is a limit ordinal cofinal with <o.…”

A homological characterization of large subgroups

“…In order to do this we shall employ the following characterization of large subgroups, given in [1]. S. WINTHROP (l-n1, 2-n2, 3-n~, ...).…”

“…We mention here only the remarkable theorem of CUTLER and STRINGALL [3] that the completion with respect to the topology generated by the large subgroups is precisely the torsion completion with respect to the p-adic topology. For other interesting work on large subgroups see MEGIBBEN [6], and especially the recent paper of BENABDALLAH, EISENSTADT, IRWIN, and POLUIANOV [1].…”