Logarithmic descriptions of \(K'_ 1({\hat {\mathbb{Z}}}_ pG)\) and class- groups of symmetric groups
A logarithmic description of \(K'_ 1({\hat {\mathbb{Z}}}_ pG)=K_ 1({\hat {\mathbb{Z}}}_ pG)/SK_ 1({\hat {\mathbb{Z}}}_ pG)\) is given for any finite group G and any prime p. This is based on a homomorphism \[ \Gamma_ G: K'\!_ 1({\hat {\mathbb{Z}}}_ pG)\to H_ 0(G;{\hat {\mathbb{Z}}}_ pG)={\hat {\mathbb{Z}}}_ pG/<x-gxg^{-1};\quad x\in {\hat {\mathbb{Z}}}_ pG,\quad g\in G> \] induced by setting \(\Gamma_ G(u)=\log u-p^{-1}\Phi (\log u)\quad (\Phi (a_ ig_ i)=\sum a_ ig^ p_ i)\) for any \(u\in 1+rad({\hat {\mathbb{Z}}}_ pG).\) In particular, \(Ker(\Gamma_ G)\) and \(Co\ker (\Gamma_ G)\) are seen to be finite groups of the same order. As one application, the following formula for \(| \tilde K_ 0({\mathbb{Z}}S_ n)|\) is derived, where \(S_ n\) denotes the symmetric group: Theorem. For \(r\in {\mathbb{Q}}\), set \(f(r)=(-1)^{k-1}\) if \(r=(3k^ 2\pm k)/2\), \(k\in {\mathbb{Z}}\); \(f(r)=0\) otherwise. For any prime p and any \(k\geq 1\), set \[ a_ p(k)= ord_ p(k)\sigma_ 0(k)-f(k/p)+\sum_{p<q\leq k,\quad q prime}(ord_ p(q-1)\cdot f(k/q)). \] Then for any prime p and any \(n\geq 2:\) \[ ord_ p| \tilde K_ 0({\mathbb{Z}}S_ n)| = \begin{cases} \sum^{n}_{k=1} a_ p(k)p(n-k) &\text{ if p is odd;} \\ 2-p(n)+ \sum^{n}_{k=1}a_ p(k)p(n-k) &\text{ if p=2.}\end{cases} \] Here \(p(.)\) denotes the partition function and \(\sigma_ 0(.)\) the number of divisors. For any integer \(k>0\), and any prime p, \(ord_ p(k)=a\) if \(p^ a| k\), \(p^{a+1}\nmid k\).
scientific article; zbMATH DE number 3887885
scientific article