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Homogeneous monomial groups and centralizers

ABSTRACT The construction of homogeneous monomial groups are given and their basic properties are studied. The structure of a centralizer of an element is completely described and the problem of conjugacy of two elements is resolved. Moreover, the classification of homogeneous monomial groups are determined by using the lattice of Steinitz numbers, namely, we prove the following: Let λ and μ be two Steinitz numbers. The homogeneous monomial groups Σλ(H) and Σμ(G) are isomorphic if and only if λ = μ and H≅G provided that the splittings of Σλ(H) and Σμ(G) are regular.