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Let $p<q$ be distinct prime numbers. Every group of group order $p^2{q}^2$ is Abelian if and only if $p$ does not divide $q^2-1$.

For sufficiency, the Sylow theorems make the number of Sylow p-subgroups one of $1,q,q^2$, congruent to one modulo $p$. The divisibility hypothesis excludes both nontrivial choices, so that subgroup is normal. The number of Sylow subgroups for $q$ is one of $1,p,p^2$, congruent to one modulo $q$. The choice $p$ is impossible because $p<q$. If $q$ divided $p^2-1=(p-1)(p+1)$, primality and $q>p$ would force $q=p+1$, hence $(p,q)=(2,3)$; this pair violates the hypothesis. Thus the other Sylow subgroup is normal too. Their intersection is trivial, and their group commutators lie in that intersection, so the group is their direct product of groups. A group of prime-square order is abelian: its nontrivial center of a group is either the whole group or has cyclic quotient, and a cyclic central quotient also forces commutativity. Both factors are therefore abelian.

For necessity, suppose $p\mid q^2-1$. The multiplicative group of a finite field is cyclic, so ${\mathbb{F}}_{q^2}^{\times }$ contains $\alpha$ of order $p$. Let the cyclic group $C_{p^2}$ act on the additive group of ${\mathbb{F}}_{q^2}$ by mapping its generator to multiplication by $\alpha$. This is a well-defined nontrivial group action, since ${\alpha }^{p^2}=1$. The resulting semidirect product has order $p^2{q}^2$ and is nonabelian, because the generator does not commute with a nonzero field element. Thus a nonabelian group of this order exists exactly when $p\mid q^2-1$.

For a finite-dimensional vector space $V$, the affine group consists of invertible affine maps $x\mapsto Ax+b$, where $A$ belongs to the general linear group and $b\in V$. Its product is $(A,b)(A^{\prime },b^{\prime })=(A{A}^{\prime },b+A{b}^{\prime })$, making it the semidirect product of the translation vector space and its general linear group. On the real line the positive-dilation subgroup is the real affine group used in the adjacent article.

The affine group of the complex line consists of the mapsComposition identifies it with the semidirect product $(\mathbb{C},+)⋊{\mathbb{C}}^{\times }$, where $a$ acts on a translation coordinate by multiplication.

In the finite group presentation $⟨a,t\mid ta{t}^{-1}=a^2⟩$, let $a$ act by $x\mapsto x+1$ and $t$ by $x\mapsto 2x$. The conjugates $a_i=t^{-i}a{t}^i$ satisfy $a_{i+1}^2=a_i$, so their increasing cyclic subgroups give the additive group $\mathbb{Z}[1/2]$. Every element has the form $a_i^m{t}^k$, and the affine action proves that this form can be the identity only if $k=m=0$. Thus the displayed semidirect product identification holds. This is a finitely presented group with a series having torsion-free abelian factors, but it is not a poly-infinite cyclic group, since $\mathbb{Z}[1/2]$ is not finitely generated.

The Euclidean group $E(n)=O(n)⋉{\mathbb{R}}^n$ is the group of all isometries $x\mapsto Ax+b$ of Euclidean space, with $A$ an orthogonal matrix. Its orientation-preserving subgroup is $SE(n)=SO(n)⋉{\mathbb{R}}^n$, the semidirect product of the special orthogonal group with translations. The Euclidean motion of Euclidean three-space and the proper Euclidean motion of Euclidean three-space are the respective three-dimensional instances.

Two subgroups give an exact factorization of a group when every element has a unique expression $h_1{h}_2$ with $h_i\in H_i$. Existence and trivial intersection are equivalent to this uniqueness: if $h_1{h}_2=k_1{k}_2$, then $k_1^{-1}{h}_1=k_2{h}_2^{-1}\in H_1\cap H_2$, so both sides are the identity. Conversely, an element $a$ in the intersection gives the two factorizations $a\cdot e=e\cdot a$, forcing $a=e$. For finite groups the multiplication map $H_1\times H_2\to G$ is therefore a bijection and $|G|=|H_1||H_2|$. It need not be a group homomorphism: an exact factorization is not automatically a direct product of groups. If $H_1$ is normal, it gives an internal semidirect product.

If $N=⟨A\mid R⟩$ and $H=⟨B\mid S⟩$ have finite group presentations and $\varphi :H\to \mathrm{Aut}(N)$ is an action, choose a word $w_{b,a}(A)$ representing $\varphi (b)(a)$. Then $N{⋊}_{\varphi }H$ has the finite presentationThe action automorphisms also let these relations rewrite conjugation by inverse generators. Every word becomes an $A$-word followed by a $B$-word. The natural map to the semidirect product is injective because its triviality forces the $B$-word to be trivial using $S$ and then the $A$-word to be trivial using $R$.

A Gaussian period is a sum of roots of unity indexed by a multiplicative coset in the group of nonzero residues modulo a prime number. Such sums describe subfields of a cyclotomic field and occur as values of induced characters of certain semidirect products.

The group of maps $x\mapsto Ax+b$ of a finite-dimensional vector space, with $A$ invertible. Identifying the underlying points with a regular elementary abelian group identifies its normalizer in the symmetric group with this semidirect product.

These are all the groups of group order $70$ up to group isomorphism, with $D_n$ denoting a dihedral group of order $n$. The Sylow theorems make the Sylow subgroups of orders five and seven unique: the possible counts divide fourteen and ten, respectively, and only one has the required residue. These normal subgroups of coprime order commute, giving a normal subgroup $N\cong C_{35}$. Cauchy's theorem for finite groups provides an element of order two outside $N$, so $G=N⋊C_2$.

The automorphism of a finite cyclic group defining this semidirect product is a unit $a$ modulo $35$ with $a^2\equiv 1$. The Chinese remainder theorem makes its restrictions modulo five and seven independently $+1$ or $-1$. The resulting exponents are $1,34,6,29$, giving the four displayed groups in order. Their center of a group orders are $70,1,5,7$, so no two are isomorphic. Every case exists as the specified semidirect product.

An extension of $A$ by $B$ is a short exact sequence $1\to A\to G\to B\to 1$. It splits if the quotient map has a homomorphic section. Equivalently the embedded $A$ has a subgroup complement isomorphic to $B$, and $G$ is the corresponding semidirect product.

For $t\notin K$ and $a\in K$, both $t$ and $ta$ lie outside $K$, so $t^2=(ta{)}^2=1$. It follows that $tat=a^{-1}$, and $t=t^{-1}$ gives the claimed conjugation identity. Conjugation is an automorphism of $K$, so inversion is an automorphism there. Comparing the image of $ab$ with the product of the images gives $b^{-1}{a}^{-1}=a^{-1}{b}^{-1}$. Therefore $K$ is abelian.

Conversely inversion is an automorphism of any abelian group $A$, of order dividing two. Form the semidirect product $G=A⋊C_2$ with multiplicationFor completeness associativity follows because $b\mapsto b^{(-1{)}^{\epsilon }}$ is an automorphism and these automorphisms compose according to addition modulo two; $(1,0)$ is the identity and inverses follow from this formula. The subgroup $H=A\times \{0\}$ is normal with quotient $C_2$, and every outside element satisfies $(a,1{)}^2=(a{a}^{-1},0)=1$. Thus the generalized dihedral group supplies the required extension, including when $A$ is infinite or has elements of order two.

The class equation implies that a nontrivial finite $p$-group has nontrivial center of a group. Here a center of order $9$ would make $G/Z(G)$ cyclic; a group with cyclic quotient by its center of a group is abelian, contradicting the hypothesis. ThusThe quotient of order nine must be noncyclic for the same reason. For $x\notin Z(G)$, $⟨x,Z(G)⟩$ has order at least nine and lies in the proper centralizer $C_G(x)$. Lagrange's theorem therefore forces $\boxed{|C_G(x)|=9}$. Also $1\ne G^{\prime }\le Z(G)$, so $G^{\prime }=Z(G)$ and all commutators are central of order dividing three.

Classification and splitting. First suppose $a$ has order nine. Then $N=⟨a⟩$ is normal of index three and $Z(G)=⟨a^3⟩$. For $b\notin N$, conjugation on $N$ is nontrivial, since otherwise $a,b$ would generate an abelian group. Its order divides three, so, after replacing $b$ by its inverse if needed, $ba{b}^{-1}=a^4$. Write $b^3=a^{3k}$. Central commutators and $3\mid \left(\genfrac{}{}{0ex}{}{3}{2}\right)$ give $(a^{i}b{)}^3=a^{3i}{b}^3$; choose $i=-k$ modulo three to obtain an element $c=a^{i}b$ of order three outside $N$. This provesThere is a unique resulting isomorphism type, since the two nontrivial actions of order three are interchanged by reversing the complement's generator.

If there is no element of order nine, every nonidentity element has order three. Choose $a,b$ whose images form a basis of $G/Z(G)$ and put $c=[a,b]$. These elements generate $G$, and $c$ generates $Z(G)$. The subgroup $N=⟨a,c⟩\cong C_3^2$ has index three and is normal. The line $⟨b⟩$ meets $N$ trivially and complements it. Conjugation by $b$ fixes $c$ and sends $a$ to $ac$ or $a{c}^{-1}$; choosing the central generator removes this sign. Thus there is one isomorphism type,with the nontrivial unipotent action on ${\mathbb{F}}_3^2$. These presentations are realized by the indicated semidirect products and have order 27. Their exponents are nine and three respectively, so they are not isomorphic. This proves both existence and completeness of the two cases.

Their ordinary character tables are the same, up to relabelling. In either group each of the 24 noncentral elements has conjugacy class of size three, giving eight noncentral classes and three central singleton classes. Each noncentral class is $xZ(G)$, since conjugates of $x$ differ from $x$ by central commutators and its class has size three. There are nine linear characters, arising from $G/G^{\prime }\cong C_3^2$, and eleven irreducible characters altogether. The two remaining degrees satisfy $d_1^2+d_2^2=27-9=18$, so both are three.

Put $Z(G)=⟨z⟩$ and $\omega =e^{2\pi i/3}$. Label the noncentral classes by nonzero $(i,j)\in {\mathbb{F}}_3^2$. For $(r,s)\in {\mathbb{F}}_3^2$ the nine linear characters have valuesAt a noncentral element, column orthogonality gives a squared column norm of nine. The nine linear characters already contribute nine, so both degree-three characters vanish there. By the Schur lemma, a central element has value $3{\omega }^k$ in an irreducible representation of degree three. The central character cannot be trivial, since that would factor the representation through the abelian quotient. Orthogonality of the identity and $z$ columns then forces the two nontrivial central characters to occur once each. The final two rows, on $1,z,z^2$ and the eight other columns, areThis description gives the same entire character table in both cases. The differing element orders are additional power-map data, which an ordinary character table need not determine.

Let $B=H_1\times \cdots \times H_m$, with each $H_i$ a copy of $H$. The permutation group $F\le S_m$ acts on $B$ byDefine the permutation wreath product to be the semidirect productExplicitly,Writing $h_i$ for the element with entry $h$ in coordinate $i$ and identity elsewhere, and writing $\sigma$ for $(1,\sigma )$, conjugation satisfiesThus the top permutation group permutes the coordinate subgroups in exactly its given action. Distinct coordinate subgroups commute, including when $H$ itself is nonabelian.

Write $N=⟨a⟩$. The relation $ba{b}^{-1}=a^2$ lets every element be expressed as $a^i{b}^j$ with $0\le i<7$ and $0\le j<3$. There are 21 such expressions, so the assumption $|G|=21$ makes them distinct. In particular $a$ has order seven, $b$ has order three, and $N$ is a normal subgroup. This is the nonabelian group of order 21, the semidirect product $C_7⋊C_3$ with the action given by multiplication by two modulo seven.

Conjugation by $a$ fixes $N$, while conjugation by $b$ sends $a^i$ to $a^{2i}$. The nonidentity conjugacy classes inside $N$ are thereforeFor the other elements, direct calculation givesWhen $j=1$ or $2$, the coefficient $1-2^j$ is nonzero modulo seven, so letting $t$ vary gives every member of the coset $N{b}^j$. These cosets cannot merge under conjugation, since $G/N$ is an abelian group and conjugation acts trivially on that quotient group. Thus the five conjugacy classes have representatives and sizes

To find the linear characters, note that the commutator subgroup $G^{\prime }$ contains $ba{b}^{-1}{a}^{-1}=a$. The quotient group $G/N\cong C_3$ is abelian, so $G^{\prime }\subseteq N$; hence $G^{\prime }=N$. The abelianization is $C_3$. With $\omega =e^{2\pi i/3}$, its three linear characters areThis exhausts the linear characters because one-dimensional characters factor through the abelianization.

For the remaining irreducible characters, put $\zeta =e^{2\pi i/7}$ and take the linear characters ${\theta }_t$ of $N$ defined by ${\theta }_t(a)={\zeta }^t$. The six nontrivial linear characters form two orbits under conjugation, with exponent sets $\{1,2,4\}$ and $\{3,5,6\}$. Define the induced charactersFor $a^i\in N$, the character formula for an induced representation givesBoth vanish outside $N$, since no element outside a normal subgroup can be conjugate into it. This is induction from an abelian normal subgroup with a free character orbit. For completeness, its irreducibility here follows directly from Frobenius reciprocity and character orthogonality on the cyclic group $N$:The same calculation with ${\theta }_3,{\theta }_5,{\theta }_6$ proves that $\overline{\psi }$ is an irreducible character.

LetThe sum of the nontrivial seventh roots of unity gives $\alpha +\beta =-1$. Squaring and reducing exponents modulo seven givesThus $\alpha ,\beta$ are the two polynomial roots of $T^2+T+2$, andTo fix the signs with our specified $\zeta$, observe that $\mathrm{Im}\alpha =\sin (2\pi /7)+\sin (3\pi /7)-\sin (\pi /7)>0$, using the increase of sine on $(0,\pi /2)$. These Gaussian periods for seventh roots of unity make the answer explicit.

The resulting character table isThe two degree-three irreducible characters are distinct because $\alpha \ne \beta$. Together with the three linear characters, they exhaust the irreducible characters: the sum of squares of irreducible degrees is $3\cdot 1^2+2\cdot 3^2=21=|G|$. Equivalently, we have five distinct irreducible characters for the five conjugacy classes.

As a check on part (b), inversion interchanges $K_1$ and $K_2$, and also interchanges the two conjugacy classes outside $N$. Only the identity conjugacy class is real. Correspondingly ${\lambda }_0$ is the only real-valued character among the irreducible characters; ${\lambda }_1,{\lambda }_2$ and $\psi ,\overline{\psi }$ are two conjugate pairs.

Compose permutations from right to left. The indicated cycle lengths immediately give $X^3=P^2=Q^2=e$. Under conjugation, $X^{-1}(ab)X=(X^{-1}a{X}^{-1}b)$. Since $X^{-1}$ sends $1,2,3$ to $3,1,2$ and fixes $4$,Direct composition also gives $PQ=QP=(14)(23)$. Thus $V=\{e,P,Q,PQ\}$ is a Klein four-group. Conjugation by $X$ cycles its three nonidentity elements, so $X$ normalizes $V$.

The cosets $V,VX,V{X}^2$ have four elements each and are distinct. If two coincided, a nontrivial power of $X$ would belong to $V$, impossible because it has order three whereas every nonidentity element of $V$ has order two. Hence the subgroup generated by $X,P,Q$ contains at least twelve elements. These generators are even permutations, so the subgroup lies in the alternating group $A_4$, whose order is $4!/2=12$; multiplying by one fixed transposition pairs even and odd permutations to justify this count. ThereforeThe calculation also describes $A_4$ as the semidirect product $V⋊⟨X⟩$, with the order-three factor permuting the three involutions.

Retain the identification with the left regular action. A group group automorphism $\phi$ of $G$ satisfies $\phi L_g{\phi }^{-1}=L_{\phi (g)}$, so it normalizes $L(G)$ and fixes the identity point.

Conversely let a normalizing permutation $n$ fix the identity. Its conjugation on $L(G)$ induces a group group automorphism $\phi$ characterized by $n{L}_g{n}^{-1}=L_{\phi (g)}$. Evaluating at the identity gives $n(g)=\phi (g)$, so the whole identity stabilizer in the normalizer is exactly $\mathrm{Aut}(G)$. For arbitrary $n$, put $t=n(1)$. Then $L_t^{-1}n$ fixes the identity and is an group automorphism; thus every element has a unique expression $L_t\phi$.

The translation subgroup is normal in its normalizer, its intersection with the group automorphism subgroup is trivial, and the multiplication rule isConsequentlyThe action in this semidirect product is the natural evaluation of group automorphisms on $G$. This normalizer is the holomorph, realized as all permutations $x\mapsto t\phi (x)$. The decomposition is established from the regular group action itself rather than only from an order comparison.

Since $\sigma$ is a group homomorphism, $\sigma (0)=I$, $\sigma (x+y)=\sigma (x)\sigma (y)$, and $\sigma (-x)=\sigma (x{)}^{-1}$. The product is closed on $\mathbb{R}\times {\mathbb{R}}^2$. Its two associative bracketings giveand coincide by the homomorphism property. The identity element is $(0,0)$, and the two-sided inverse isFor example, multiplying on the right gives $v-\sigma (x)\sigma (-x)v=0$, and multiplying on the left gives $-\sigma (-x)v+\sigma (-x)v=0$. Thus this law defines a group, specifically a semidirect product.

If it is an abelian group, comparing $(x,0)\ast (0,w)$ and $(0,w)\ast (x,0)$ gives $\sigma (x)w=w$ for every $w$. Hence $\sigma (x)=I$ for every $x$. Conversely this trivial action makes the product ordinary vector addition. Therefore the unique action giving a commutative group isFor the final condition, compare $(0,a{e}_1)\ast (x,w)$ and $(x,w)\ast (0,a{e}_1)$. They are $(x,a{e}_1+w)$ and $(x,w+\sigma (x)a{e}_1)$ respectively. They coincide for every $a,x,w$ exactly when $\sigma (x)e_1=e_1$ for every $x$. The first column of $\sigma (x)$ is then $(1,0{)}^{\mathsf{T}}$; since it lies in the special linear group, its determinant is one and its other diagonal entry is also one. ThusMatrix multiplication gives $r(x+y)=r(x)+r(y)$. Conversely any such additive function gives a homomorphism into $SL(2,\mathbb{R})$ fixing $e_1$ and therefore makes these translations central. This proves the full characterization of the central translations in a linear semidirect product:No continuity or measurability is assumed, so it would be unjustified to restrict the answer to $r(x)=cx$.

A finitely presented group is one isomorphic to $F(X)/⟨⟨R⟩⟩$, where $X$ and $R$ are finite, $F(X)$ is the free group on $X$, and the double brackets mean normal closure. Equivalently, there is a surjection from a free group of finite rank onto $G$ whose kernel is normally generated by finitely many words. The universal property of the free group extends choices of generator images uniquely, and quotienting imposes the chosen relations.

For the semidirect product, choose finite group presentationsFor each $i,j$, choose an $a$-word $w_{ji}$ representing $\varphi (b_j)(a_i)$. We claim that a finite presentation of a semidirect product isAll relations hold in the desired semidirect product, so the presented group maps onto it. We must also show there are no extra elements in the kernel. For inverse conjugations, choose an $a$-word $v_{ji}$ representing $\varphi (b_j{)}^{-1}(a_i)$. Conjugating this word by $b_j$ using the displayed relations gives an $a$-word representing $\varphi (b_j)\varphi (b_j{)}^{-1}(a_i)=a_i$, so $R$ proves $b_j^{-1}{a}_i{b}_j=v_{ji}$. Thus we can move both positive and negative $b$-letters to the right, rewriting every word as $u(a)v(b)$. If its image in $N⋊H$ is trivial, projection to $H$ makes $v$ trivial using $S$; its remaining image in $N$ then makes $u$ trivial using $R$. The kernel is trivial, proving the claim and finite presentability.

For a poly-infinite cyclic group, write its subnormal series as $G=G_0▹G_1▹\cdots ▹G_n=1$ with infinite cyclic factors. Given a subgroup $K\le G$, intersect the series with $K$. The homomorphism $K\cap G_i\to G_i/G_{i+1}$ has kernel $K\cap G_{i+1}$, soEach factor is trivial or infinite cyclic; deleting repetitions gives a poly-infinite cyclic series for $K$. For an extension $1\to N\to E\to Q\to 1$ with $N,Q$ poly-infinite cyclic groups, lift the series of $Q$ to its full preimages in $E$ and append the series of $N$. The lifted quotient factors are those of $Q$, and the lower factors are those of $N$. Normality at every consecutive step is preserved. This proves closure under both subgroups and extensions.

Finite presentability follows by induction on the length of a poly-infinite cyclic series. The trivial group is finitely presented. For a nontrivial group, its first proper term $G_1$ has a shorter series and $G/G_1\cong \mathbb{Z}$. Lift $1\in \mathbb{Z}$ to an element $t\in G$. Its powers give an injective section of the quotient map, since $t^k$ maps to $k$. Hence $G\cong G_1⋊⟨t⟩$. By induction $G_1$ is finitely presented, and $⟨t⟩\cong \mathbb{Z}$ has a finite presentation. The finite presentation of a semidirect product proved above completes the induction. Thus every poly-infinite cyclic group is finitely presented.

For the final example, take the dyadic affine Baumslag-Solitar groupwhere the generator of $\mathbb{Z}$ acts on the additive group $\mathbb{Z}[1/2]$ by multiplication by two. Here is a verification of this identification, rather than an assumption about the presentation. Put $a_i=t^{-i}a{t}^i$ for $i\ge 0$. The relation gives $a_{i+1}^2=a_i$, soTheir union $A$ is abelian and is normal: conjugation by $t^{-1}$ shifts $a_i$ to $a_{i+1}$, while conjugation by $t$ shifts $a_i$ to $a_{i-1}$ for $i\ge 1$ and sends $a_0$ to $a_0^2$. Every word can consequently be written $a_i^m{t}^k$ for some $i\ge 0$ and $m,k\in \mathbb{Z}$.

Let $a$ act on the rational line by $x\mapsto x+1$ and $t$ by $x\mapsto 2x$. These affine transformations satisfy the relation, and $a_i^m{t}^k$ acts by $x\mapsto 2^{k}x+m/2^i$. It is the identity only when $k=m=0$. Thus the action is faithful and identifies $A$ with all dyadic translations. We have the required seriesBoth factors are torsion-free abelian groups, and $G$ is visibly finitely presented. But $\mathbb{Z}[1/2]$ is not finitely generated, since finite sets have bounded powers of two in their denominators. Every poly-infinite cyclic group has the maximal condition on subgroups: its factors $\mathbb{Z}$ have that condition, and Question 1 proves closure under extensions. Therefore $G$ cannot be poly-infinite cyclic. This gives a finitely presented counterexample to replacing infinite cyclic factors by arbitrary torsion-free abelian factors.

Take a recursively enumerable set $S\subseteq {\mathbb{N}}_{>0}$ that is not recursive, for example the diagonal halting set from Question 2 shifted by one. To obtain a finitely generated group with recursive presentation but undecidable word problem for a group, first define the countable Right-angled Artin groupTranslation of the indices gives an automorphism $a_i\mapsto a_{i+1}$. The semidirect product with its infinite cyclic shift is generated by $a=a_0,t$ and has presentationTo verify this presentation, the semidirect product satisfies its relators, while the reverse map sends $a_i$ to $t^{i}a{t}^{-i}$; all translated commutation relations follow by conjugation. These maps are inverse on their generators. Thus $B_S$ embeds in $G_S$.

For $n\in S$, the word $[a_0,a_n]$ is trivial. For $n\notin S$, map $a_0,a_n$ to the free generators $x,y$ of $F(x,y)$ and every other $a_i$ to one. Every defining commutator of $B_S$ maps to one, since the only possible relation involving both surviving generators would have separation $n$, which is absent. The resulting group homomorphism sends $[a_0,a_n]$ to the nontrivial reduced word $[x,y]$. HenceThis proves the shift-invariant graph groups encode arbitrary sets construction rather than supposing that a missing relator automatically means a nontrivial word. A word algorithm for $G_S$ would decide $S$, so none exists. Since $S$ is enumerable, the displayed two-generator presentation is a recursive presentation of a group. By the Higman embedding theorem, $G_S$ embeds in a finitely presented group $G_0$. A word algorithm for $G_0$, composed with substitution of the two fixed generator-image words, would solve the word problem for $G_S$. Therefore $\boxed{G_0\text{ is finitely presented with insoluble word problem}}$.

Let $v=(x,y{)}^T$ and $B(\rho )=\left(\begin{array}{cc}\cosh \rho & \sinh \rho \\ \sinh \rho & \cosh \rho \end{array}\right)$. Matrix multiplication givesThus $G$ is a semidirect product of the translation plane by the one-parameter Lorentz boost group. Differentiating left multiplication at the identity gives the left-invariant vector fieldsThey obey $[X_0,X_1]=X_2$, $[X_0,X_2]=X_1$ and $[X_1,X_2]=0$.

Their dual coframe is $d\rho$, $\cosh \rho dx-\sinh \rho dy$, and $-\sinh \rho dx+\cosh \rho dy$. The Haar measure from a left-invariant coframe is their absolute wedge product. Since $\det B(\rho )=1$,One can also check this directly: left multiplication has Jacobian determinant one. Right multiplication has a block triangular Jacobian with determinant one too, so the same measure is both left and right Haar measure.