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A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8] and Nicholson [11] from projective semiperfect modules to arbitrary semiperfect modules. [ABSTRACT FROM AUTHOR]

We show that if |Aut(G)| is odd, then without loss of generality we may assume the only primes dividing |G| are those dividing |Aut(G)|. We also classify finite abelian groups A such that |Aut(A)| = |A| and finite abelian groups A such that |Aut(A)| p = |A| p , where p is a prime dividing |A|. [ABSTRACT FROM PUBLISHER]