Where can all this be used?
Abstracting Common Properties: They provide a powerful way to study systems that share fundamental structural properties, regardless of what the elements actually "are."
Cryptography: Many modern cryptographic systems (like RSA, Diffie-Hellman, Elliptic Curve Cryptography) are built upon the properties of specific groups. The difficulty of certain problems within these groups (e.g., the discrete logarithm problem) forms the basis of their security.
Coding Theory: Error-correcting codes often utilize group structures to detect and correct errors in transmitted data.
Graph Theory: Automorphism groups of graphs describe their symmetries.