On the explicit structure of \(K_2({\mathbb F}_pG)\) for \(G\) a finite abelian \(p\)-group
Let \(p\) be a prime number and \(G\) a finite abelian \(p\)-group. Given the structure of \(G\), the authors calculate the structure of the finite abelian \(p\)-group \(K_2(\mathbb{F}_pG)\). Their main result (Theorem 1.2) says the following: Let \(p^e\) be the exponent of \(G\) and let \(r_i\) be the \(p^i\)-rank of \(G\) for \(1\leq i\leq e\). Then \[ K_2(\mathbb{F}_pG)\cong \bigoplus^e_{i=1} C^{s_i- s_{i+1}}_{p^i}, \] where \(C_n\) denotes a cyclic group of order \(n\) and \[ s_i:= \begin{cases} (r_i- 1)(|G^{p^{i-1}}|- |G^{p^i}|),\quad & 1\leq i\leq e,\\ 0\quad & i= e+1.\end{cases} \] This result generalizes previous partial results of \textit{R. K. Dennis}, \textit{M. E. Keating} and \textit{M. R. Stein} [Math. Ann. 223, 97--103 (1976; Zbl 0342.18006)], \textit{B. A. Magurn} [J. Pure Appl. Algebra 209, No. 3, 801--811 (2007; Zbl 1120.19002)] and the second and third authors [J. Pure Appl. Algebra 213, No. 7, 1201--1207 (2009; Zbl 1165.19001)]. As a corollary it follows that \(K_2(\mathbb{F}_pG)\) is an elementary abelian \(p\)-group if and only if the \(p^2\)-rank of \(G\) is at most 1. They prove their result by induction on the \(p\)-rank of \(G\). The starting point is the split short exact sequence \[ 0\to K_2(\mathbb{F}_p G[t]/\langle t^{p^r}\rangle, \langle t\rangle)\to K_2(\mathbb{F}_p[G\times C_{p^r}])\to K_2(\mathbb{F}_p G)\to 0 \] coming from the long exact sequence in \(K\)-theory of the ideal \(\langle t\rangle\). Thanks to the work of \textit{F. Keune} [J. Algebra 54, 159--177 (1978; Zbl 0403.18009)], the relative \(K_2\)-term has a presentation whose generators are Dennis-Stein symbols. The authors use this presentation to calculate a relatively small number of Dennis-Stein symbols which generate this group, and whose order can be bounded. By comparing with the order of the relative \(K_2\)-term, which is known by the results of \textit{R. Oliver} [J. Algebra 94, 425--487 (1985; Zbl 0595.18004)], they deduce the exact structure.
scientific article; zbMATH DE number 6132781