Vol. 41, No. 1, 1972

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On the endomorphism ring of an abelian p-group, and of a large subgroup

G. S. Monk

Vol. 41 (1972), No. 1, 183–193

For an abelian p-group G, denote the endomorphism ring of G by E(G), the ideal of small endomorphisms by Es(G) and the quotient ring E(G)∕Es(G) by S(G). It is not difficult to show that for a large subgroup L of G, the map that sends an endomorphism of G to its restriction on L induces a monomorphism S(G) S(L). We show that if B1 is a large subgroup of a group B2 which is a direct sum of cyclic p-groups and is of cardinality not more than 20 and R1 and R2 are suitable subgroups of E(B1) and E(B2), then there are groups G1 and G2 having B1 and B2 as basic subgroups such that G1 is large in G2 and S(Gi)Rx(Es(Bi) Rt),(i = 1,2).

Mathematical Subject Classification 2000
Primary: 20K30
Received: 9 June 1970
Published: 1 April 1972
G. S. Monk