- Let G be a group and a and b be any two elements in G. Then the equations and have unique solutions in G.
- There is a unique
e(identity element) => , its order is 1, the inverses in a group are also unique
- The right and left cancellation laws are true in groups
Group theory - Wikiwand
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse.
abelian or commutative
A group G with the property that a ◦ b = b ◦ a for all a, b ∈ G is called abelian or commutative.
the general linear group
The set of invertible matrices forms a group called the general linear group.
- The order of a finite group is the number of elements it contains.
- The order of a element is the minimum positive number such that , . If there is no such number, then has infinite order.
If the group is or , we write the group operation additively and the exponential operation multiplicatively; that is, we write instead of .
Groups are essentially a Set