If \(k\) is a field of characteristic \(p>0\), an endotrivial module of a group is a finitely generated module whose \(k\)-endomorphism ring is isomorphic to a trivial module in the stable category. This paper discusses the groups of endotrivial modules of finite groups, in particular, of Lie type and proves that Alperin's theorem holds on the rank of the group of endotrivial modules for all finite groups. The paper proves, for groups which are not of type \(A_1(p)\), \(p>2\), \(^2A_2(p)\), or \(^2B_2(2^{1/2})\), the torsion subgroup \(TT(U)\) of \(T(U)\) is trivial; for groups which are not of type \(A_1(q)\), \(q=p^a\) and \(p>2\), \(^2A_2(q)\), \(q=p^a\), \(^2B_2(2^{a+\tfrac 12})\), or \(^2G_2(3^{a+\tfrac 12})\), the torsion subgroup \(TT(G)\) of \(T(G)\) is trivial. Exceptional cases include, for the case of \(A_1(2)\), \(TT(U)=T(U)=0\); for the case of \(A_1(p)\), \(p>2\), or \(^2B_2(2^{1/2})\), \(TT(U)=T(U)\cong\mathbb{Z}/2\mathbb{Z}\); for the case of \(^2A_2(2)\), \(TT(U)=T(U)\cong\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}\); with \(TT(U)=0\) otherwise. The paper also proves that the torsion-free subgroups \(TF(U)\), \(TF(B)\), \(TF(G)\) are dependent entirely on the number of conjugacy classes of maximal elementary Abelian \(p\)-subgroups of \(p\)-rank 2 in the groups, \(U\), \(B\), and \(G\), respectively, with their ranks equal to one, except in the cases of \(A_1(p)\), \(p>2\), or \(^2B_2(2^{1/2})\), all such torsion subgroups are trivial; and in the cases of \(A_2(p)\), \(B_2(p)\), \(G_2(p)\), \(^2A_2(p)\), \(^2B_2(2^{a+1/2})\), \(a\geq 1\), and \(^2G_2(3^{a+1/2})\), \(a\geq 0\), ranks of such torsion-free subgroups are given in details in the paper.