Finite groups with the set of the number of subgroups of possible order containing exactly two elements |
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Authors: | YANHENG CHEN GUIYUN CHEN |
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Institution: | 1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China 2. School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404100, People’s Republic of China
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Abstract: | Let G be a finite group, and n(G) be the set of the number of subgroups of possible order of G. We investigate the structure of G satisfying that n(G)?=?{1, m} for any positive integer m?>?1. At first, we prove that the nilpotent length of G is less than 2. Secondly, we investigate nilpotent groups with m?=?p?+?1 or p 2?+?p?+?1 (p is a prime), and we get the classification of such kinds of groups. At last, we investigate non-nilpotent groups with m?=?p?+?1 and get the classification of the groups under consideration. |
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