Abstract
ABSTRACTLet A be a finite group of odd order and let A act on a finite p-group P with |P|>pe, where e is an integer e≥4(e≥5 if p = 2). In this paper we show that P is centralized by if every non-meta-cyclic subgroup of order pe in P is stabilized by Op(A). As applications, some conditions are given for a finite group G with the p-length ≤1 and the p-rank ≤2. We also find a class of finite p-groups, which is not only very useful for the paper but also has its independent meaning.
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