Crossover operators are very important tools in Evolutionary Computation. Here we are interested in crossovers for the permutation representation that find applications in combinatorial optimization problems such as the permutation flowshop scheduling and the traveling salesman problem. We introduce three families of permutation crossovers based on algebraic properties of the permutation space. In particular, we exploit the group and lattice structures of the space. A total of 14 new crossovers is provided. Algebraic and semantic properties of the operators are discussed, while their performances are investigated by experimentally comparing them with known permutation crossovers on standard benchmarks from four popular permutation problems. Three different experimental scenarios are considered and the results clearly validate our proposals.