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Finite DC-groups

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arxiv 2001.06832 v1 pith:5QMU2ZJB submitted 2020-01-19 math.GR

classification math.GR
keywords dc-groupfinitedc-groupsgroupmetabelianprovesubgroupabelian
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Let G be a group and DS(G) = { H'| H is any subgroup of G}. G is said to be a DC-group if DS(G) is a chain. In this paper, we prove that a finite DC-group is a semidirect product of a Sylow p-subgroup and an abelian p'-subgroup. For the case of G being a finite p-group, we obtain some properties of a DC-group. In particular, a DC 2-group is characterized. Moreover, we prove that DC-groups are metabelian for p<5 and give an example that a non-abelian DC-group is not be necessarily metabelian for p>3.

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