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Hereditary Upper Radical Properties and Dual Supplementing Radicals of Hereditary Radicals of Groups

One aim of this paper is to prove that for any hereditary radical property 𝒫, there exists a radical property 𝒫′ supplementing 𝒫. Certain hereditary upper radical properties of groups are constructed, and some of them are applied to formulate the necessary and sufficient condition for a hereditary radical 𝒫 and its supplementing radical 𝒫′ forming a dual pair.