Symmetries and stabilisers in modular invariant flavour models

The idea of modular invariance provides a novel explanation of flavour mixing. Within the context of finite modular symmetries Gamma (N) and for a given element gamma is an element of Gamma (N), we present an algorithm for finding stabilisers (specific values for moduli fields tau (gamma) which remain unchanged under the action associated to gamma). We then employ this algorithm to find all stabilisers for each element of finite modular groups for N = 2 to 5, namely, Gamma (2) similar or equal to S-3, Gamma (3) similar or equal to A(4), Gamma (4) similar or equal to S-4 and Gamma (5) similar or equal to A(5). These stabilisers then leave preserved a specific cyclic subgroup of Gamma (N). This is of interest to build models of fermionic mixing where each fermionic sector preserves a separate residual symmetry.