Groups

Examples

• \( (\mathbb{Z}, +)\)  is a conmutative or Abelian group.
• \( (\mathbb{Q}, +)\) is a conmutative or Abelian group. Also \( (\mathbb{Q}^{*}, .)\)
• \( (\mathbb{R}, +)\) is a conmutative or Abelian group. Also \( (\mathbb{R}^{*}, .)\)
• \( (\mathbb{C}, +)\) is a conmutative or Abelian group. Also \( (\mathbb{C}^{*}, .)\)
• Given a vector space V, then \( (\mathbb{V}, +)\) is an conmutative or Abelian group.

Subgroups

Lets G a Group, lets \( S \subset \) of G. It is said that S is a subgroup of G if S with the operation . restricted to S is a group.

Proof
\(\Rightarrow \) \(a, b \in S \Rightarrow b^{-1} \in S \Rightarrow a, b^{-1} \in S \Rightarrow a.b^{-1} \in S.\)
\(\Leftarrow \) \(\exists x \in S \Rightarrow 1.x = x \in S \Rightarrow 1 \in S \Rightarrow 1. x^{-1} \in S \Rightarrow 1 \in S \)

When the group G is finite, the number of elements of G is called order of G and denoted |G|. In this case, there is an important theorem called Lagrange’s Theorem wich asserts the following:

In other simple words, the order of a subgroup divides the order of the group.

Corrollary
Given G group, if order of G=p prime, then the only subgroups of G are {1} and G.

These subgroups are called impropial subgroups.

Generated Subgroups

Any given subset S of a group (G,.), can generates a subgroup of G by operate over itself. This subgroup is denoted by <S>, ie
\(<S> = { x_{1}.x_{2}. … .x_{n} : x_{i} \in G } \).
When S is a finite subset, it is said that <S> is finitely generated.
When S is formed by only an unique element of F, ie, S={a}, it is said that <S> is cyclical.