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Until now, various methods have been demonstrated to solve theFriedmann-Lama\'{\i}tre-Robertson-Walker (FLRW) equations in the spatially flat$(k=0)$ cosmological model. In this study, in order to solve the fieldequations of the spatially flat FLRW cosmological model in the presence of$\Lambda$, a new method based on the invariants of the symmetry groups which wecalled ISG-method, is presented. This method is based on the extendedPrelle-Singer (PS) method and it uses the Lie point symmetry,$\lambda$-symmetry and Darboux polynomials (DPs). We employ this method toextract systematically the two independent first integrals (or invariants) suchas $I_1 (a,\dot{a})=c_1$ and $I_2 (t,a,\dot{a})=c_2$ associated to the group ofthe Lie point transformations keeping the Friedmann-Einstein dynamical equation(DE), $\ddot{a}=\phi(t,a,\dot{a})$, invariant. The obtained solutions fromsolving the DEs of the FLRW cosmological model by the ISG-method are explicitlywritten, so that they are suitable for cosmological applications. Finally, asan application of the solutions we look at the age of universe in the presenceof cosmological constant where the dominant matter of the universe isconsidered to be fluid with the state parameter $w$. In this regard, wecalculate the age of universe when the dominant matter is dust.

Exact FLRW cosmological solutions via invariants of the symmetry groups