JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The College Mathematics Journal. ) earned his Ph.D.from Binghamton University in 2003. He is currently an associate professor of mathematics and holds the endowed Cole Chair in Mathematics at Alfred University in beautiful western New York. His main research focuses on subgroup properties in finite groups. He enjoys teaching, problem solving, playing guitar, and enjoying life with his wife, Lynn, and his son, Julian.

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If a subgroup U of a finite group G has the property that either UH = UK or U ∩ H = U ∩ K for every chief factor H/K of G, then U is said to have the cover-avoidance property in G and is called a CAPsubgroup of G. It is well known that a subgroup U of a direct product G 1 × G 2 is determined by isomorphic sections S 1 of G 1 and S 2 of G 2 and by an isomorphism φ between those sections. We prove that whether U is a CAP-subgroup of G 1 × G 2 depends on the isomorphism φ, but not necessarily on the sections S 1 and S 2 . Equivalently, U is a CAP-subgroup of G 1 × G 2 if and only if UM ∩ G 1 is a CAP-subgroup of G 1 and UN ∩ G 2 is a CAP-subgroup of G 2 for all MPG 2 and N PG 1 . Consequently, subdirect subgroups and CAP-subgroups of direct factors have the cover-avoidance property.

A corollary to the known classification of finite simple groups having chain difference one (Brewster et al., J. Algebra 160, 179-191 (1993)) is that the length of such groups is either four or five. We prove a partial converse of the corollary to obtain a more precise description of these groups, the number of which depends on the unsolved prime k-tuples conjecture. (2000). 20E15. Mathematics Subject Classification

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