Invariant Mappings and Metahomomorphisms Under Group Actions

In the realm of finite group theory, the identification and exploration of nonzero antihomomorphisms are pivotal for comprehending group structures and proving essential theorems. The study introduces the concept of metahomomorphisms and their invariant mappings, offering a novel methodology that utilizes the congruence equation and number theory to uncover invariant mappings. By harnessing these mappings, we aim to provide a streamlined proof of the Schur–Zassenhaus theorem. Our approach diverges from traditional methods by focusing on the invariant mapping sets within the group action on metahomomorphisms, establishing a novel correspondence between antihomomorphisms and invariant mappings. This provides a direct and clear proof technique, enhancing the utility of metahomomorphisms in finite group analysis and contributing to the field’s theoretical development.