Homogeneous monomial groups and centralizers

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 Σ_λ(H) and Σ_μ(G) are isomorphic if and only if λ = μ and H?G provided that the splittings of Σ_λ(H) and Σ_μ(G) are regular.