Abstract Algebra & Groups

Groups · Cyclic · Rings & Fields

1:00

The four group axioms, cyclic groups, Lagrange's theorem, the group→ring→field ladder, and how finite fields power AES, Reed–Solomon codes, and public-key cryptography.

By the end, you can

State the four group axioms and verify whether a given set-and-operation pair satisfies them.
Compute inverses and identity elements in modular arithmetic groups.
Explain what makes a group cyclic and identify a generator.
Apply Lagrange's theorem to rule out subgroup orders.
Read a Cayley table entry and identify whether a group is abelian from the table's symmetry.
Describe the group → ring → field hierarchy and what each additional level gains.
Explain what GF(p) and GF(2⁸) are and why they appear in AES and Reed–Solomon codes.
Explain how the discrete logarithm problem underpins Diffie–Hellman and RSA security.
Define a homomorphism and give a concrete example.

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