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It is a long standing problem of A.L.S. Corner as to whether there exists a non-transitive fully transitive Abelian p-group with finite Ulm subgroups. The principal result of this paper shows that no such group exists for a special class of p-groups if the Ulm subjgroup is the direct sum of two cyclic groups. Some results on summands of trnsitive and fully transitive groups are obtained.
Carroll, D. and Brendan Goldsmith:On transitive and fully transitive abelian p-groups. Proceedings of the Royal Irish Academy, 96A 1 (1996) pp.33-41.