= Solution
Write $B(x)$ for the displayed matrix. The hyperbolic addition identities give
$$
B(x)B(y)=\begin{pmatrix}\cosh(x+y)&-\sinh(x+y)\\-\sinh(x+y)&\cosh(x+y)\end{pmatrix}=B(x+y).
$$
Thus <matrix multiplication> is closed in $G$. It is associative because ordinary <matrix multiplication> is associative. The identity is $B(0)=I$, and $B(x)B(-x)=I$ gives the inverse $B(x)^{-1}=B(-x)$. Hence $\boxed{G\text{ is a group}}$.
Moreover, $x\mapsto B(x)$ is an injective homomorphism from the additive real numbers, since $\sinh x$ determines $x$. Therefore $\boxed{G\cong(\mathbb R,+)}$; in particular it is abelian. These matrices are the connected boost <subgroup> relevant to the <component quotient of the real Lorentz group in two dimensions>.
Back to article page
