Vol. 37, No. 2, 1971

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Generalized final rank for arbitrary limit ordinals

Doyle Otis Cutler and Paul F. Dubois

Vol. 37 (1971), No. 2, 345–351

Let G be a p-primary Abelian group. The final rank of G can be obtained in two equivalent ways: either as inf nω{r(pnG)} where !r(pnG) is the rank of pnG; or as sup{7⋅(G ∕B)|B is a basic subgroup of G }. In fact it is known that there exists a basic subgroup of G such that r(G∕B) is equal to the final rank of G. In this paper are displayed two appropriate generalizations of the above definitions of final rank, rα(G) and sα(G), where α is a limit ordinal. It is shown that the two cardinals rα(G) and sα(G) are indeed the same for any limit ordinal α. In this context one can think of the usual final rank as ω-final rank”.

Mathematical Subject Classification
Primary: 20K05
Received: 20 January 1970
Published: 1 May 1971
Doyle Otis Cutler
Paul F. Dubois