Meta-abelianizations of SL\((2,\mathbb{Z}[{1\over p}])\) and Dennis-Stein symbols (Q1375723)

For a given group \(G\), let \(G^{mab}\) denote the meta-abelianization of \(G\); i.e., \(G^{mab} =G/G''\) where \(G''\) is the second commutator subgroup of \(G\). For a commutative ring \(A\), let \(\text{St} (2,A)\) be the Steinberg group of rank one of \(A\) and let \(K_2(2,A)\) be the kernel of the natural homomorphism \(\pi:\text{St} (2,A) \to\text{SL} (2,A)\). The following results are proved: Theorem 1. Let \(p\) be a prime number. Then \[ \text{SL}\bigl(2, \mathbb{Z}[1/p] \bigr)^{mab} \simeq \begin{cases} \mathbb{Z}_3 \ltimes (\mathbb{Z}_2 \times \mathbb{Z}_2) \quad & p=2 \\ \mathbb{Z}_4 \ltimes\mathbb{Z}_3 \quad & p=3 \\ \mathbb{Z}_{12} \ltimes (\mathbb{Z}_2 \times \mathbb{Z}_6) \quad & p\geq 5 \end{cases}. \] Theorem 2. 1. Suppose \(p=2,3\). Then \(K_2(2, \mathbb{Z}[1/p]) \simeq \mathbb{Z} \times \mathbb{Z}_{p-1}\), and \(K_2(2,\mathbb{Z} [1/p])\) is central in \(\text{St} (2,\mathbb{Z} [1/p] )\). 2. Suppose \(p\geq 5\). Then \(K_2(2, \mathbb{Z}[1/p]) \supset \mathbb{Z}\times \mathbb{Z}\), and \(K_2(2, \mathbb{Z} [1/p])\) is not central. The meta-abelianization of \(\text{SL} (2,\mathbb{Z} [1/p])\) is computed from that of \(\text{St} (2,\mathbb{Z} [1/p])\) together with finding appropriate elements of \(K_2(2, \mathbb{Z} [1/p])\) described by Dennis-Stein symbols.