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what is a group? And construct three example of groups.
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Examples of groups:
1. All integer numbers with the operation "adding" on them.
2. All real numbers with the operation "adding" on them.
3. All real numbers except zero with the operation "multiplication" on them.
4. All complex numbers with the operation "adding" on them.
5. All complex numbers except zero with the operation "multiplication" on them.
From Wikipedia ( https://en.wikipedia.org/wiki/Group_(mathematics) )
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation
that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms,
namely closure, associativity, identity and invertibility.
One of the most familiar examples of a group is the set of integers together with the addition operation,
but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group
and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract
algebra and beyond to be handled in a flexible way while retaining their essential structural aspects.
The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.
Axioms of groups:
A group is an algebraic structure (G,∘) which satisfies the following four conditions: ( https://proofwiki.org/wiki/Definition:Group_Axioms )
(G0) : Closure ∀a,b∈G: a∘b∈G
(G1) : Associativity ∀a,b,c∈G: a∘(b∘c)=(a∘b)∘c
(G2) : Identity ∃e∈G:∀a∈G: e∘a=a=a∘e
(G3) : Inverse ∀a∈G:∃b∈G: a∘b=e=b∘a
