Braid relations, meta-abelianizations and the symbols \(\{p,-1\}\) in \(K_ 2(2,\mathbb{Z}[1/p])\) (Q1907750)

We will determine the meta-abelianization of \(St (2, \mathbb{Z} [1/p])\), which is \(\mathbb{Z}_3 \ltimes (\mathbb{Z}_2 \times \mathbb{Z}_2)\) if \(p = 2\); \(\mathbb{Z}_8 \ltimes \mathbb{Z}_3\) if \(p = 3\); \(\mathbb{Z}_{p^2 - 1} \ltimes (\mathbb{Z} \times \mathbb{Z})\) otherwise. Also we will study the Steinberg symbols \(\{p, - 1\}\). That is, \(\{p, - 1\}^2 \neq 1\) if \(p \neq 2,3,5,11\). Furthermore, we will establish \(K_2 (2, \mathbb{Z} [1/p]) \not \simeq \mathbb{Z} \times \mathbb{Z}_{p - 1}\) if \(p \neq 2,3,5,11\).