On the socles of fully invariant subgroups of Abelian p-groups

Abstract

The classification of the fully invariant subgroups of a reduced Abelian p-group is a difficult long-standing problem when one moves outside of the class of fully transitive groups. In this work we restrict attention to the socles of fully invariant subgroups and introduce a new class of groups which we term socle-regular groups; this class is shown to be large and strictly contains the class of fully transitive groups. The basic properties of such groups are investigated but it is shown that the classification of even this simplified class of groups, seems extremely difficult.

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

“…It obviously follows that h(y (11) ij ) ≥ h 11 , e(y (11) ij ) ≤ e 11 , h(y (11) ij ) + e(y (11) ij ) = i, i = 1, 2, . .…”

“…Let us consider f x 11 ; it is obvious that it belongs to T and therefore has the form (11) ij , E(f x 11 ) = h(y (11) ij ), e(y (11) ij ) i=1,2,..., j=1,2,...,r 11 .…”

On projection-invariant subgroups of Abelian 𝑝-groups

“…In fact, consider the example based on an idea of Megibben that has been used in [3] and [4] However, as observed in Proposition 1.1, separable p-groups are always projectively socle-regular and so one would expect that the addition of a separable summand to a projectively socle-regular group would result in a projectively socle-regular group; this is, indeed, the case: if G is projectively socle-regular and H is separable, then A = G ⊕ H is projectively socle-regular. The proof follows exactly as the proof of the corresponding statement for strongly socleregular groups -see [4,Proposition 3.2].…”

“…In fact in some situations the notions coincide: Hausen [9] and Megibben [14] have established that for separable p-groups, and for transitive, fully transitive groups In a different direction, the authors have recently investigated the socles of fully invariant and characteristic subgroups of Abelian p-groups. This led to the notions of socle-regular and strongly socle-regular groups, see [3,4]. Recall the definitions: a group G is said to be socle-regular (strongly socle-regular) if for all fully invariant (characteristic) subgroups F of G, there exists an ordinal α (depending on F ) such that F [p] = (p α G) [p].…”

“…In [3] the following ad hoc terminology was introduced; it has been useful in [3,4,5] and we find it convenient to use it again here:…”