Our unified group $E_8\otimes E_8$ contains every fundamental structure including spacetime, all chiral fermions, and their three generations, and all bosons. However this comes with some additional degrees of freedom whose states cannot be identified with our usual standard model particles. In this section, we attempt to give certain explanations for these unaccounted degrees of freedom. Each E₈ will give us ten unaccounted states, with five being conjugate (anti-particle) to the other five. We list the five independent unaccounted states from Eqn. (13):

Post symmetry breaking the electric charges corresponding to the above states in that order are 2/3, 0, 2, (2/3 and 1/3 doublet), and 2/3 respectively.

There are a total of 144 unaccounted fermions from each $E_8$, the total from both sides is 288. The total number of physical fermions from our theory are also 144, so there is a 1:2 ratio of accounted and unaccounted fermions. In the subsection below, we argue that the 288 d.o.f. go into the making of two composite Higgs, 144 each. Such a composite Higgs should be looked for at the LHC.

Our theory is a matrix-valued Lagrangian dynamics on an octonionic space, motivated as a pre-quantum, pre-spacetime theory, i.e. a reformulation of quantum field theory which does not depend on an external classical time [36,37]. It is a generalisation of Adler’s theory of trace dynamics, so as to include pre-gravitation. Quantum field theory and classical space-time and gravitation are emergent from this dynamics under suitable approximations. The fundamental universe is a collection of ‘atoms of space-time-matter’, an atom being a 2-brane described by matrix-valued dynamical variables on octonionic space and having the action principle (of the unified interactions) This action defines a 2-brane residing on a split-bioctonionic space. This means that each of the two matrices ${\tilde{Q}}_1$ and ${\tilde{Q}}_2$ have sixteen components, one component per coordinate direction of the split bioctonion. This 16-dim space has $E_8\otimes E_8$ symmetry. We separate the two dynamical variables into bosonic and fermionic parts, each of which again has sixteen coordinate components: Left-Right symmetry breaking separates the bosonic dynamical variable into its left-chiral part $q_B$ defined over 8D octonionic space and the right chiral part ${\dot{q}}_B$ defined over the 8D splt octonionic space. Similarly the fermionic dynamical variable separates into its left chiral part $q_F$ defined over 8D octonionic space and its right chiral part ${\dot{q}}_F$ defined over the split octonionic space. The Yang-Mills coupling constant $\alpha$ arises as a consequence of the symmetry breaking, prior to which the theory is scale invariant, having only an associated length scale L but no other free parameter. The symmetries of the bioctonionic space are spacetime symmetries with the full symmetry being $E_8\otimes E_8$ and with symmetry breaking introducing branching as discussed earlier in the paper, and also introducing interactions and coupling constants.

By defining new variables we can also express the Lagrangian as We can next expand each of these four terms inside of the trace Lagrangian as explored thoroughly in the paper [21], using the definitions of $q_1$ and $q_2$ given above: This form of the Lagrangian displays terms for the bosonic variables (gauge fields), the fermionic part of the action (leading to the Dirac equations upon variation), and terms bilinear in fermionic varaibles (related to the Higgs). A preliminary analysis of this Lagrangian, relating it to three generations, has been carried out in [11]. The bosonic part of the Lagrangian has been discussed in detail in [21], where the weak mixing angle is also derived from first principles. In future work we hope to report as to how the Lagrangian relates to the fermions and bosons reported in the present paper, including the unaccounted for terms discussed in the previous subsection. We note that the (8x3=24) fermions of the three generations contribute 24 x 24 = 576 = 288 x 2 terms to the Lagrangian, which are bilinear in $q_F$ and ${\dot{q}}_F$. It is striking that this is twice the number 288 of unaccounted degrees of freedom. Recall also that there are 144 physical fermions as demonstrated earlier in the paper.The terms bilinear in the $q_F$ and ${\dot{q}}_F$ possibly describe the two Higgs doublets of the theory, and hence it could be that the 288 unaccounted d.o.f. go into constituting two composite Higgs (144 for each Higgs) and hence that there are no new predicted fermions. Such a possibility is currently under investigation. We also note that there are (208-144=64=32x2) non-fermionic d.o.f. and these account for the 32 bosons corresponding to the (2x16=32) space-time d.o.f. (8+8 = 16 dimensional split bioctonioc space and the factor 2 because this space is complexified).

Such a pre-spacetime, pre-quantum dynamics is in principle essential at all energy scales, not just at the Planck energy scale [11]. Whenever a physical system has only quantum sources, the coordinate geometry of spacetime is non-commutative; and when that is an octonionic geometry, the spacetime symmetries result in the standard model and its unification with pre-gravitation, as we have seen in the present paper. Fermionic states are spinors (left minimal ideals) of the Clifford algebra $Cl\left(6\right)$ made from octonionic chains. Following the earlier work of other researchers, we constructed such $Cl\left(6\right)$ states for three generations of left chiral and right chiral quarks and leptons [18,19], and their associated bosons. We argued in these papers that pre-gravitation is the right-handed counterpart of the standard model with the associated symmetry group being $E_6\otimes E_6$. This is confirmed by the reps found in the present paper; furthermore the $E_8\otimes E_8$ symmetry unifies the spacetime with the matter and gauge degrees of freedom living on this spacetime, into a common unified description. Doing so has also introduced new so-called unaccounted fermions which are truly beyond the standard model, and whose physical significance and experimental consequences remain to be understood.

The Dirac equation resulting from this Lagrangian dynamics has an $E_6$ symmetry, and its real eigenvalues are determined by the characteristic equation of the exceptional Jordan algebra $J_3\left(8\right)$, whose automorphism group is $F_4$. These eigenvalues can be found for three generations of Dirac quarks and Dirac leptons, and also for three generations of left chiral fermions or right chiral fermions. Since the left chiral fermions are charge eigenstates, and right chiral fermions are square-root mass eigenstates, these eigenvalues help explain the strange mass ratios of quarks and leptons [18,38]. This is because experimental measurements invariably employ charge eigenstates, which are distinct from mass eigenstates.

The octonionic theory has similarities with string theory, but also differs from it in crucial ways. Our theory is also for extended objects, but in 6D = 3 + 3 spacetime and has an $E_8\otimes E_8$ symmetry. However, we build the Fock space of states for particles, not on a 10D Minkowski spacetime vacuum, but on the octonion-valued space, which is a spinor spacetime, as if a ‘square-root’ of the 10D Minkowski spacetime. The non-commutative octonion geometry has to be an essential feature of quantum theory even at low energy scales, not just at the Planck scale. Only then can the standard model be understood. The extra dimensions are complex and not to be compactified - these are the dimensions along which the internal symmetries lie. Finally, the Hamiltonian of the theory is in general not self-adjoint; thus enabling the quantum-to-classical transition, and otherwise becoming self-adjoint in the approximation in which quantum field theory is recovered from trace dynamics. Further evidence for trace dynamics comes from the result that the theory admits supra-quantum nonlocal correlations predicted by the Popescu-Rohrlich bound in the CHSH inequality, but disallowed by quantum mechanics. This constitutes strong evidence that quantum theory is approximate, and emergent from trace dynamics in a coarse-grained approximation [39,40]

The non-associativity of the octonions does bring its own challenges when they are employed in field theory. There is evidence that the challenges are eased when one uses split octonions. See e.g. the work of Gogberashvili [41] where rotations in the space of split octonions have been considered. It was shown therein that Lorentz transformations satisfy associativity when split octonions are employed. In the Freudenthal-Tits magic square as well, the space-time Lorentz algebras arise when one of the two composite algebras under consideration is split. In our own research programme, we employ split bioctonions to construct emergent spacetime. Furthermore, the work of Boyle and Farnsworth [42] extends the Connes noncommutative geometry programme to the non-associative case, rendering it plausible that Connes time is admissible in our octonionic theory.

We also note the Coleman-Mandula theorem: a no-go theorem which states that the space-time symmetry (Lorentz invariance) and internal symmetry of the S-matrix can only be combined in a trivial way, i.e. as a direct product. However, this does not prevent the $E_8\otimes E_8$ unification of gravitation and the standard model, on which the present paper is based. This is because the theorem applies only to the spontaneously broken phase, in which the Minkowski metric is present. The unified phase does not have a metric, and hence not the Minkowski metric either, and hence the Coleman-Mandula theorem does not apply to the unified symmetry.

The octonionic theory employs the octonions to define a non-commutative coordinate geometry. The symmetries of the geometry dictate the existence of standard model chiral fermions, gauge bosons, and pre-gravitation. Clifford algebras made from octonionic chains define spinorial states for the fermions, and the exceptional Jordan algebra determines values of the standard model parameters. The matrix-valued Lagrangian dynamics determines evolution of these particles. Classical general relativity emerges from pre-gravitation as a result of a quantum-to-classical spontaneous localisation which confines classical systems to our familiar 4D spacetime. However, quantum systems always live in the octonionic space, which has complex extra dimensions which are never compactified. Further work is in progress so as to attempt taking this unification program to completion.

The fundamental action (25) is as it is written in the generalised theory of trace dynamics - a pre-spacetime pre-quantum theory. The dynamical variables are matrix-valued, with Grassmann numbers as matrix entries. Matrices which are even-grade Grassmann are called bosonic, and describe bosons. Matrices which are odd-grade Grassmann are called fermionic, and describe fermions. The variables $q_1$ and $q_2$ in the previous section are a sum of a bosonic part and a fermionic part, as shown in Eqn. (28). Thus the fundamental action describes what we call an ‘atom’ of space-time-matter (STM atom) which by definition is an elementary particle such as an electron (a fermion) along with all the (bosonic) fields it produces. Every particle gives rise to both a gravitational field and the weak force field, whereas only electrically charged particles give rise to electromagnetism, and only electrically charged particles with color give rise to the strong force field. The weak force is the spacetime symmetry of the flipped 4D spacetime $SO(1,3)$, having arisen from the breaking of the electroweak symmetry. General relativity has as its precursor the $SU{\left(2\right)}_R$ gauge symmetry, which is the spacetime symmetry of our 4D spacetime, having arisen from the breaking $SU{\left(2\right)}_R\times U{\left(1\right)}_{Yg}\to U{\left(1\right)}_{grav}$. We say more about this in Section 7.1. On the flipped 4D spacetime, a vector bundle represents the unbroken $SU{\left(3\right)}_c\times U{\left(1\right)}_{em}$ as an internal symmetry, and on our 4D spacetime another vector bundle represents the unbroken $SU{\left(3\right)}_{grav}\times U{\left(1\right)}_{grav}$ as an internal symmetry. The two 4D spacetimes have one common timelike and one common spacelike dimension. Thus overall, our universe has two extra (timelike) dimensions. These appear as internal symmetry directions from the vantage point of our 4D spacetime, whereas in reality they are part of the flipped 4D spacetime.

For every STM atom, the accompanying spacetime is labeled by non-commuting numbers - quaternions and/or octonions. Quaternions are adequate for describing the two 4D spacetimes, whereas octonions are essential for describing the internal symmetries $SU{\left(3\right)}_c$ and $SU{\left(3\right)}_{grav}$. Each STM atom, described by the action (25), is a universe unto itself - a mini-universe, say. The collection of enormously many STM atoms, $N\sim {10}^{80}$ constitutes the observed universe. Entanglement amongst subsets of the enormous collection leads to a quantum-to-classical transition resulting in the formation of classical objects such as stars and galaxies, and this transition is accompanied by the emergence of classical spacetime. Only when such a spacetime is available, the conventional formulation of quantum field theory (QFT) is possible. In principle, even at currently accessible low energies, such critical entanglement need not have taken place, and classical spacetime could be absent even at low energies. Conventional QFT would then not exist; but the trace dynamics (TD) formulation based on Eqn. (25) and on non-commuting coordinates will still exist. That is why the TD formulation is more fundamental, and it is the precise way to describe the standard model, which contains more information than the QFT based description of the standard model, even at energies accessed by the LHC. Thus the TD based description, being more precise, carries an explanation for those properties of the standard model (e.g. origin of the observed gauge symmetries and of values of dimensionless fundamental constants) which cannot be explained by the QFT based description. We now justify how the QFT based standard model can be recovered from the fundamental action, in the limit that dominantly many degrees of freedom in the universe become classical.

Variation of this trace action with respect to the matrix-valued degrees of freedom $q_1$ and $q_2$ gives rise to the Lagrange equations of motion: For reference, we mention two other useful and equivalent ways of writing the fundamental action: and This matrix-valued Lagrangian dynamics, or its equivalent Hamiltonian dynamics, is not to be quantized. It is already pre-quantum. This is because, as a result of a global unitary invariance of the trace Lagrangian, the theory possesses a novel conserved charge, denoted $\tilde{C}$ and known as the Adler-Millard charge, and not present in classical nor quantum dynamics: The commutator is over the bosonic matrices, and anti-commutator is over the fermionic matrices. We can also motivate trace dynamics with the following simple example of a collection of non-relativistic particles in classical mechanics: in which we propose to raise the real-number valued $q_i$ to matrices ${\mathbf{q}}_i$ and arrive at the trace Lagrangian: The variation of the trace Lagrangian gives the matrix-valued equations of motion ${\ddot{\mathbf{q}}}_i=0$ and the theory possesses the novel conserved charge mentioned above. The matrix ${\mathbf{q}}_i$ is that matrix of which the corresponding classical variable $q_i$ is an eigenvalue, to be realised in the classical limit of TD. In this sense classical dynamics is recovered from trace dynamics, and in this same sense the Lagrangian for the standard model is recovered from the fundamental trace Lagrangian in (25).

It is assumed that trace dynamics holds at some time scale resolution not accessed by current laboratory experiments, say Planck time ${\tau }_p$. We then ask what is the emergent coarse-grained dynamics, if the system is observed not at Planck time resolution, but at some lower resolution $\tau \gg {\tau }_p$? The standard techniques of statistical thermodynamics are employed to construct a phase space density distribution of the trace dynamical system, whose emergent coarse-grained dynamics is determined by maximising the Boltzmann entropy subject to constraints representing conserved quantities. It is shown that at thermodynamic equilibrium the Adler-Millard charge is equipartitioned over all degrees of freedom so that the canonical average of each commutator $[q_B,p_B]$ and each anti-commutator $\{q_F,p_F\}$ is assumed to be equal to $i\hslash$. This is how the quantum commutation relations emerge from the underlying trace dynamics. Also, in this emergent thermodynamic equilibrium, the canonically averaged Hamilton’s equations of motion become Heisenberg’s equations of motion of quantum theory. Identification of canonical averages of functions of dynamical variables (in their ground state) with Wightman functions in relativistic quantum mechanics enables the transition from trace dynamics to quantum field theory. Quantum field theory is thus shown to be an emergent (equilibrium) thermodynamic phenomenon.

At equilibrium, the Adler-Millard charge is anti-self-adjoint, and the Hamiltonian of the theory is self-adjoint. Statistical fluctuations in this charge, when significant, can drive the quantum system away from equilibrium (the charge is no longer equipartitioned). If these fluctuations are themselves dominantly self-adjoint, the Hamiltonian of the theory picks up an anti-self-adjoint component, which gets amplified if a large number of degrees of freedom are entangled with each other. This drives the system to classicality, via a Ghirardi-Rimini-Weber type of spontaneous collapse process. Thus, macroscopic classical systems, including classical fields, are far from equilibrium emergent states in trace dynamics.

The basis for the fundamental action (25) is the Chamseddine-Connes spectral action principle. Let us first recall the principle and its present application for the case of Einstein-Hilbert action of general relativity. After that we will include Yang-Mills fields. According to this principle, the Einstein-Hilbert action can be expressed in terms of the eigenvalues of the Dirac operator D on a Riemannian geometry, via a truncated heat kernel expansion in powers of $L_p^{-2}$: It has also been shown that these Dirac eigenvalues can be the dynamical observables of general relativity, in place of the metric. In the spirit of trace dynamics, every eigenvalue ${\lambda }_i$ is raised to the status of a bosonic matrix/operator ${\widehat{\lambda }}_i\equiv {\dot{q}}_{Bi}$, this being the very Dirac operator D of which it is an eigenvalue. Thus in generalised trace dynamics (a pre-quantum, pre-spacetime theory) we have a collection of ‘atoms of space-time’, as many atoms as there were Dirac eigenvalues, each atom being associated with a copy of the Dirac operator: ${\dot{q}}_{Bi}\equiv LD$, where L is a newly introduced length parameter which characterises a space-time atom. The Einstein-Hilbert action ${\sum }_i{L}_p^2{\lambda }_i^2$ transits to ${\sum }_{i}Tr(L_p^2/L^2)\left[{\dot{q}}_{Bi}^2\right]$. Since the Dirac eigenvalues have been made operators, space-time is lost, and this is simultaneously a transition to trace dynamics and to non-commutative geometry, hence showing the deep connection between the new geometry and the new dynamics. However, note that what was earlier the (dimensionless) action is now the dimensionless (trace) Lagrangian; the integral over time, which will make it into a trace dynamics action, is missing. Here, Connes time parameter $\tau$, a unique feature of non-commutative geometry, comes into play. The trace dynamics action for atoms of space-time matter, scaled with respect to Planck’s constant ℏ, is given by The recovery of classical general relativity is the reverse of the above : quantum-to-classical transition sends each Dirac operator back to one its eigenvalues; a different eigenvalue for every STM atom, and from the sum total of the squares of eigenvalues, the Einstein-Hilbert action is recovered. The Chamseddine-Connes spectral action principle continues to hold when Yang-Mills gauge fields are included. To get there, we first recall how gauge fields are treated as inner automorphisms of a non-commutative algebra. The seventh axiom of commutative geometry in Connes’ programme states that there exists an antilinear isometry $J:\mathcal{H}\to \mathcal{H}$ such that $Ja{J}^{-1}=a^{*}\forall a\in \mathcal{A}$ and $J^2=ϵ,JD={ϵ}^{\prime }DJ$ and $J\gamma ={ϵ}^{″}\gamma J$ where $ϵ,{ϵ}^{\prime },{ϵ}^{″}\in \{-1,+1\}$.

In the theory of operator algebras there is a result of great significance due to Tomita. The theorem states that given a weakly closed * algebra of operators M in Hilbert space $\mathcal{H}$ which admits a cyclic and separating vector, there exists a canonical antilinear isometric involution J from $\mathcal{H}$ to $\mathcal{H}$ such that $JM{J}^{-1}=M^{\prime }$. Here, $M^{\prime }=\{T;Ta=aT\forall a\in M\}$ is the commutant of M. Consequently, M is antiisomorphic to its commutant, where the antiisomorphism is given by the following $\mathbb{C}$-linear map: $a\in M\to J{a}^{*}{J}^{-1}\in M^{\prime }$.

As noted above, axiom 7 of commutative geometry already involves an antilinear isometry J in $\mathcal{H}$, this being complex conjugation. Since $\mathcal{A}$ was assumed commutative, the equality $J{a}^{*}{J}^{-1}=a$ is consistent with the antiisomorphism in Tomita’s theorem. In going over to the non-commutative case the requirement $J{a}^{*}{J}^{-1}=a\forall a\in \mathcal{A}$ is replaced by the condition: $[a,b^0]=0\forall a,b\in \mathcal{A}$ where $b^0=J{b}^{*}{J}^{-1}$. This converts the Hilbert space $\mathcal{H}$ into a module over the algebra $\mathcal{A}\otimes {\mathcal{A}}^0$ this being the tensor product of $\mathcal{A}$ with its opposite algebra ${\mathcal{A}}^0$. And,

We now illustrate the non-commutative case for the algebra $\mathcal{A}=C^{\infty }(M,M_k\left(\mathbb{C}\right))$ of $k\times k$ matrix-valued functions on a smooth compact Riemannian spin manifold M. A metric g is given on M and $\mathcal{H}=L^2(M,S\otimes M_k\left(\mathbb{C}\right))$ is the Hilbert space of $L^2$ sections of the tensor product $S\otimes M_k\left(\mathbb{C}\right)$ of the spinor bundle S with $M_k\left(\mathbb{C}\right)$. The algebra $\mathcal{A}$ acts on $\mathcal{H}$ by left multiplication: $\left(a\xi \right)\left(x\right)=a\left(x\right)\xi \left(x\right),\forall x\in M$. The real structure J is given as $J=C\otimes *$ where C is the charge conjugation operator, and * stands for adjoint of a matrix: $T\to T^{*}$. The adjoint operation converts left multiplication operation $\xi \to a\xi$ to right multiplication $\xi \to \xi a^{*}$ and the axiom $[a,b^0]=0$ is satisfied, where $b^0=J{b}^{*}{J}^{-1}$.

The operator D is taken as $D={\gamma }^{\mu }{\partial }_{\mu }\otimes I$ which is the tensor product of the Dirac operator on M by the identity on $M_k\left(\mathbb{C}\right)$. This gives a non-commutative geometry and conveys important properties of non-commutative geometries in general. The group $\mathrm{Aut}\left(\mathcal{A}\right)$ of * automorphisms of the algebra $\mathcal{A}$ is also the diffeomorphism group $\mathrm{Diff}\left(M\right)$ of diffeomorphisms of the manifold M. It has a natural normal subgroup $\mathrm{Int}\left(M\right)\subset \mathrm{Aut}\left(M\right)$ of inner automorphisms, these being automorphisms of the form ${\alpha }_u\left(x\right)=ux{u}^{*}\forall x\in \mathcal{A}$. Here, u is an arbitrary element of the unitary group The action of this group of internal diffeomorphisms on the geometry can be expressed by replacing the Dirac operator D with where the A are Yang-Mills gauge potentials. This generalisation of the Dirac operator is a special case of $A=A^{*}$ with $A={\sum }_i{a}_i[D,b_i]$. In the commutative case this generalisation vanishes because $A=A^{*}$ implies that $A+JAJ{A}^{-1}=0$. In the non-commutative case the generalisation is non-trivial. In the present example, $\mathrm{Int}\left(M\right)=C^{\infty }(M,SU\left(k\right))$ is the group of local gauge transformations for an $SU\left(k\right)$ gauge theory on M. The internal perturbations of the metric are parametrised by $SU\left(k\right)$ gauge potentials.

It is striking to observe the parallels between non-commutative geometry and trace dynamics. In both cases, making the algebra non-commutative brings in significant new features (which are not there in the commutative case) which influence applications of the theory. For NCG, it is the inner automorphisms (responsible for standard model gauge fields arising naturally), and also the concept of Tomita-Takesaki-Connes time. For trace dynamics it is the Adler-Millard charge, which is responsible for trace dynamics being a pre-quantum theory. We have therefore suggested a merger of trace dynamics and non-commutative geometry, wherein eigenvalues of the Dirac operator will themselves be raised to the status of operators, thus generalising trace dynamics to a pre-quantum, pre-spacetime theory. This theory will then have a conserved Adler-Millard charge which incorporates pre-gravitational degrees of freedom, and it will have Connes time.

We can now return to the spectral action principle which incorporates Yang-Mills gauge fields using Dirac eigenvalues. We recall that a canonical triple $(\mathcal{A},\mathcal{H},D)$ over a manifold is defined as follows. $\mathcal{A}=C^{\infty }\left(M\right)$ is the algebra of complex-valued smooth functions on M. $\mathcal{H}=L^2(M,S)$ is the Hilbert space of square-integrable sections of the irreducible spinor bundle over M. And D is the Dirac operator of a Clifford connection. Assuming it to be an even spectral triple, a real structure (a generalised charge conjugation) is introduced through the antilinear isometry J on $\mathcal{H}$ and is crucial in deriving the spectral action for Yang-Mills fields. Thus, We also need to define the product of two real spectral triples $({\mathcal{A}}_1,{\mathcal{H}}_1,D_1,J_1)$ and $({\mathcal{A}}_2,{\mathcal{H}}_2,D_2,J_2)$. The product triple $(\mathcal{A},\mathcal{H},D,J)$ is given by In the celebrated Chamseddine-Connes toy model for describing the bosonic sector of the standard model (at tree level) coupled to gravity, they proposed the purely geometric action Here, $D_A=D+A+JA{J}^{*}$ and A is the gauge potential $A={\sum }_j{a}_j[D,b_j]$. ${\mathrm{Tr}}_{\mathcal{H}}$ is the usual trace in Hilbert space (this will be our point of contact with trace dynamics), $\Lambda$ is a cutoff parameter, and the function $\chi$ cuts off all eigenvalues of $D_A^2$ larger than ${\Lambda }^2$. The action is exclusively in terms of the spectrum of the self-adjoint operator $D_A$ and is motivated by the above-discussed interpretation that the inner automorphisms of $\mathcal{A}$ are the gauge degrees of freedom. As noted above, $\mathrm{Diff}\left(M\right)$ is isomorphic to $\mathrm{Aut}\left(C^{\infty }\left(M\right)\right)$ and we define $\mathrm{Out}\left(\mathcal{A}\right)=\mathrm{Aut}\left(\mathcal{A}\right)/\mathrm{Int}\left(\mathcal{A}\right)$. Then we have the following short exact sequence of groups: