0x010 Abstract Algebra

Group

Definition (group) A group is an ordered pair \((G, *)\) where \(G\) is a set and \(*\) is a binary operation on \(G\) satisfying following axioms

• associative: \((a*b)*c = a*(b*c)\)
• identity: \((\exists e \in G) (\forall a \in G) a*e = e*a = a\)
• inverse: \((\forall a \in G) (\exists a^{-1} \in G) a*a^{-1} = a^{-1}*a = e\)

Definition (Abelian) The group is called abelian if \(a*b = b*a\)

Subgroup

Theorem (Lagrange) If \(H\) is a subgroup of a group \(G\), then

\[\left|G\right|=\left[G:H\right]\cdot \left|H\right|\]

Ring

Modules

Field

Galois

Reference

[1] Dummit, David Steven, and Richard M. Foote. Abstract algebra. Vol. 3. Hoboken: Wiley, 2004.