A strict algebraic structure of group actions models the symmetry, permutation and transformation properties in symmetric cryptosystems. In this paper, the reasons why permutation groups are critical and the way they are used to attack the design of block and stream ciphers towards diffusion and confusion are discussed. Group-theoretic properties are pertinent to enhance the strength of cryptography by studying structures like Feistel networks and substitution-permutation networks (SPNs), a fact that the work highlights. Algebraic properties, such as orbit structures and stabilizers are dealt with in respect to their effects on resistance to linear and differential cryptanalysis. Comparison analysis has been determined to introduce well-defined group action that has enhanced resistance to 18-25% against algebraic attacks without a loss in the computational efficiency. These results highlight the importance of the symmetry-breaking towards the creation of secure ciphers. Among the contributions of the study is presenting a single algebraic design of strong symmetric encryption schemes with a better security guarantee.