Let 77 be a finite group and denote by M π the class of all (finitely generated Z-free) 77-modules. In the previous paper [3] we defined an equivalence relation in M π and constructed the abelian semigroup T(77) by giving an addition to the set of all equivalence classes in M π . The investigation of the semigroup T(Π) seems interesting and important, because this gives a classification of the function fields of algebraic tori defined over a field k which split over a Galois extension of k with group 77.The purpose of this paper is to obtain information on the structure of the semigroup Γ(77).We will recall the definitions given in Let k be a field. Let K be a Galois extension of k with group s 77 and let M be a 77-module with a Z-free basis {u u u 2 , ,w n }. Define the action on the rational function field K(X 19 X 2 , , X n ) with n variables X lf X 2 > --,X n over K by putting, for each σe77 and 1 <^ i<>n, σ{Xd = \\ n j=1 X^ when σ Ut = Σ?=i m *Λ > mtjeZ, and denote by K(M) K(X l9 X 2 , • , X Λ ) with this action of 77. It is well known ([7]) that there is a duality between the category of all algebraic tori defined over k which split over K and the category of all 77-modules. In fact, if T is an algebraic torus defined over k which splits over K, then the character

Summary Mycobacterium tuberculosis genome contains four related sets of an operon called mce (mce1-4). The disruption of one of these operons, mce1, causes M. tuberculosis to become hypervirulent, whereas the mce3 and mce4 operon mutants are attenuated in mice. This study examined the phenotype of the mce2 operon mutant. The deletion of mce2 operon in M. tuberculosis H37Rv had no effect on bacterial growth in 7H9 liquid broth or survival within macrophages. However, RAW macrophage-like cells infected with the mutant strain were reduced in their ability to produce TNF-α, IL-6 and MCP-1. In C57BL/6 mouse lungs, the mce2 operon mutant and wild type H37Rv replicated similarly up to 20 weeks of infection. However, by 56 weeks of infection, all mice infected with the wild type H37Rv had died, while 80% of those infected with the mutant remained alive (P< 0.0001). The proportion of affected lung parenchyma in mice infected with the mutant was substantially less than that of mice infected with the wild type for the same time periods of infection. These observations suggest that the mce2 operon mutant is attenuated, and that this attenuation is related not to the bacterial burden but to the mutant’s decreased ability to elicit a type of immune response and lung pathology detrimental to the survival of the animal.

Let G be a group and let k be a field. A K-representation ρ of G is a homomorphism of G into the group of non-singular linear transformations of some finite-dimensional vector space V over k. Let K be the field of fractions of the symmetric algebra S(V) of V, then G acts naturally on K as k-automorphisms. There is a natural inclusion map V→K, so we view V as a k-subvector space of K. Let v1, v2, · · ·, vn be a basis for V, then K is generated by v1, v2, · · ·, vn over k as a field and these are algebraically independent over k, that is, K is a rational field over k with the transcendence degree n. All elements of K fixed by G form a subfield of K. We denote this subfield by KG.