1. Group theory plays a central role in error-correcting codes by offering the algebraic structure needed to design codes, the coset partitioning used in decoding, the symmetry principles that guide optimization, and representation theory as a tool for deeper analysis.

2. The importance of group theory in coding theory lies in its ability to provide the algebraic basis for building codes, to organize decoding through cosets, to exploit symmetries for optimization, and to use representation theory for studying code properties.

3. Group theory supports error-correcting codes in multiple ways: it gives the algebraic foundation for code construction, helps with decoding through coset structures, improves efficiency through symmetry, and enables analysis with representation-theoretic methods.

I know very litle about error-correction except that it is based on finite field theory in high dimensions rather than group theory. Of course, group theory is almost certainly used like any other modern mathemetical theory.