Abstract algebra
The topics I'll cover in this post are a mere introduction to the topic of abstract algebra, this subject is a really deep branch of mathematics and its goal is to generalise stuff as much as possible. Basically if you prove a nice result in a general context, you won't need to prove the same result again in a more specific case.
Note that I will update the page whenever some aspects are mentioned further in the course.
Algebraic structures
This is a really simple definition, an algebraic structure is a tuple that is formed by a set and a binary operation in this set.
If you wonder what is a binary operation don't worry, this is also really simple: Let \(S\) be a set, let \(*\) a function, \(*\) is a binary operation on \(S\) if and only if it's defined this way: \(*: S \times S \to S\)
By this definition we have that \((S, *)\) is an algebraic structure.
Notation: given an algebraic structure \((S,*)\) and \(s_1, s_2 \in S\), we will say that \(*(s_1,s_2) = s_1*s_2\) to make things look nicer.
Monoid
Let \((S, *)\) be an algebraic structure, it's a monoid if and only if:
- \(\exists e \in S\) \(\forall s \in S\) \(s*e=e*s=s\)
- \(\forall a,b,c \in S\) \(a*(b*c) = (a*b)*c\) Proposition: the element \(e \in S\) such that property 1 holds is unique. Proof: Let \(e_1, e_2 \in S\) be elements such that property 1 holds, by substitution we have:
- \(e_1*e_2 = e_2*e_1 =e_1\) by assumption \(e_2\) follows condition 1, we can obtain this by sobstituting \(s\) with \(e_1\)
- \(e_2*e_1 =e_2\) by assumption \(e_1\) follows condition 1, we can obtain this by sobstituting \(s\) with \(e_2\)
- From the first two passages it follows that \(e_1 = e_2\) by logic (If it's not clear why, see this) Since we proved the identity is unique, we can name it. Given a monoid \((S, *)\) its identity will be represented as \(1_S\)
Group
Let \((G, *)\) be a monoid, this is a group if and only if:
- \(\forall g \in G\) \(\exists g^{-1} \in G\) \(g*g^{-1} = g^{-1}*g = 1_G\)