In this paper, some new notions are defined about the unit group U1(ZG) of a finite group G. Especially, notion of simple unit is defined by using the number of elements in its support and absolutely small coefficients of the unit. Units are classified as monomial, binomial, trinomial and k-nomial, level, unit with level l and simple unit. We have shown triviality of monomial units and nonexistence of binomial units in the unit group U1(ZG) of an arbitrary finite group G. Some basic results and examples are posed about simple units and simple trinomial units in U1(ZCp) of a cyclic group Cp , where p ⩾ 5 .

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Abstract. The structure of V (Z(ZAn)) is known when n ≤ 6. If n = 5 or 6, then a complete set of generators of V (Z(ZAn)) has been determined. In this study, it was shown that V (Z(ZAn)) is trivial when n = 7, 8 or 9 and it is generated by a single unit u when n = 10 or 11. This unit u is characterized explicitly for n = 10 or 11 by using irreducible characters of An.