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Finite groups in which primary subgroups have cyclic cofactors∗1. School of Math. and Informational Science, Shandong Institute of Business and Technology,2. Department of Mathematics, Zhejiang University, Hangzhou 310007, ChinaIn this paper, we prove the following theorem: Let G be a group, q be the largest prime divisorof |G| and π = π(G) \ {q}. Suppose that the factor group X/coreGX is