Hangyang Meng, Xiuyun Guo
April 28, 2015
In this paper we first investigate the relationship between the absolute center L ( G ) and the Frattini subgroup Φ( G ) for a finite group G , and then we describe the structure of finite groups G satisfying L ( G ) ≤ Φ( G ). For example, we prove that a finite group G is an ℱ-group if and only if G / L ( G ) is an ℱ-group, where ℱ is a saturated formation containing Z 2 . Next we determine all finite groups X such that X / L ( X ) is isomorphic to Z p ω × Z p , where ω is a positive integer and Z p ω is a cyclic group of order p ω .