# 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

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

## Field

Definition (field) A field is a set $$F$$ together with two binary operations called addition and multiplication, it has to satisfy the following axioms

• associative: $$a+(b+c) = (a+b)+c, a*(b*c) = (a*b)*c$$
• commutative: $$a+b=b+a, a*b=b*a$$
• identity: $$a+0=a, a*1=a$$
• inverse: $$a+(-a)=0, a*a^{-1}=1$$
• distributive: $$a*(b+c)=(a*b)+(a*c)$$

## Reference

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