Group theory analysis for two-dimensional elemental systems related to phosphorene is presented, including (i) graphene, silicene, germanene and stanene, (ii) dependence on the number of layers and (iii) two stacking arrangements. Departing from the
most symmetric $D_{6h}^{1}$ graphene space group, the structures are found to have a group-subgroup relation, and analysis of the irreducible representations of their lattice vibrations makes it possible to distinguish between the different allotropes. The analysis can be used to study the effect of strain, to understand structural phase transitions, to characterize the number of layers, crystallographic orientation and nonlinear phenomena.
Transition metal dichalcogenides (TMDCs) have emerged as a new two dimensional materials field since the monolayer and few-layer limits show different properties when compared to each other and to their respective bulk materials. For example, in some
cases when the bulk material is exfoliated down to a monolayer, an indirect-to-direct band gap in the visible range is observed. The number of layers $N$ ($N$ even or odd) drives changes in space group symmetry that are reflected in the optical properties. The understanding of the space group symmetry as a function of the number of layers is therefore important for the correct interpretation of the experimental data. Here we present a thorough group theory study of the symmetry aspects relevant to optical and spectroscopic analysis, for the most common polytypes of TMDCs, i.e. $2Ha$, $2Hc$ and $1T$, as a function of the number of layers. Real space symmetries, the group of the wave vectors, the relevance of inversion symmetry, irreducible representations of the vibrational modes, optical selection rules and Raman tensors are discussed.
