Group/New page
From Maths
< Group
Contents
[<hidetoc>]Definition
Let G be a set and a binary operation (a function) ∗:G×G→G, with the following properties[1]:
- ∀g,h,k∈G[(g∗h)∗k=g∗(h∗k)] - is associative
- ∃e∈G ∀g∈G[g∗e=e∗g=g] - there exists an identity element of G[Note 1]
- ∀g∈G∃h∈G[g∗h=h∗g=e] - for each element there exists an inverse element in G[Note 2]
If ∗ satisfies these 3 properties than the tuple, (G,∗:G×G→G) - or just (G,∗) as mathematicians are lazy, is called a group.
Claims: (see below for proof)
- Claim 1: The identity element is unique
- Claim 2: The inverse element of an element is unique
Abelian group
If, additionally, a group (G,∗) satisfies and additional property:
- ∀g,h∈G[g∗h=h∗g] - the operation is commutative
then we call the group an Abelian group or commutative group
Terminology and notations
Proof of claims
Group/Claim 1: The identity element of a group is unique Group/Claim 2: The inverse element for each element of a group is unique
Notes
- <cite_references_link_accessibility_label> ↑ At this point we do not know that the identity element is unique, there could be more than one such e - but one exists. In fact it is unique, as we will see later
- <cite_references_link_accessibility_label> ↑ Again, we do not know there is a unique inverse, or for which of the e elements the equality refers to (if there are even more than one).
- There is actually only one identity element, only one e∈G and also only one inverse for each element, but this requires proof.
References
| ||||||||||||||||||||
