Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2021 / Paper 2 / 8

Write $B(x)$ for the displayed matrix. The hyperbolic addition identities giveThus 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.

For a subgroup $N\le H$ and $h\in H$, a left coset is $hN=\{hn:n\in N\}$ and a right coset is $Nh=\{nh:n\in N\}$. A normal subgroup satisfies $hN{h}^{-1}=N$ for every $h\in H$, equivalently $hN=Nh$.

If $N$ is normal, the quotient group is the set $H/N$ of cosets, with multiplicationTo check independence of representatives, replace $h,k$ by $h{n}_1,k{n}_2$. Thenbecause normality puts $k^{-1}{n}_{1}k$ in $N$. Thus the operation is well defined. Associativity follows from that in $H$, the identity is $N$, and the inverse of $hN$ is $h^{-1}N$. This proves the group axioms for $H/N$ and explains the role of coset multiplication is well-defined exactly for normal subgroups.

Direct multiplication gives $T^2=P^2=I$, $TP=PT=-I$, andThus conjugation by each generator of $G^{\prime }$ preserves $G$; so does conjugation by every product of the generators. Therefore $\boxed{G⊴G^{\prime }}$.

Using $TB(x)=B(-x)T$ and its analogue for $P$, every group word can be reduced to $B(x)S$ with $S\in \{I,T,P,TP\}$. Consequently the quotient has at most four cosets. They are distinct: a nonidentity diagonal matrix among $T,P,-I$ cannot equal $B(x)$, since a diagonal $B(x)$ requires $\sinh x=0$ and hence $B(x)=I$. ThereforeThe three nonidentity cosets all square to $G$, and $TG$ and $PG$ commute because $T,P$ commute. HenceEquivalently, $G^{\prime }$ is the full real $O(1,1)$ group and $G$ its identity component. This is the component quotient of the real Lorentz group in two dimensions; the quotient records the four components, not the continuous boost parameter.

The subgroup generated by $T,P$ is $H=\{I,T,P,-I\}$, a Klein four-group. It is not normal in $G^{\prime }$. Indeed,For $x\ne 0$ the off-diagonal entries are nonzero, whereas every member of $H$ is diagonal. Thus this conjugate of $T$ is outside $H$, givingThe normal subgroup in part (c) is the continuous boost subgroup; choosing one representative from each of its cosets does not make those representatives a normal subgroup.