Group (mathematics)

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$\frac{d}{dx} \sin x=?\,$

This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

A group is a mathematical structure consisting of a set of elements combined with a binary operator which satisfies four conditions:

Closure: applying the binary operator to any two elements of the group produces a result which itself belongs to the group
Associativity: $(A B) C = A (B C)$ where $A$ , $B$ and $C$ are any element of the group
Existence of Identity: there must exist an identity element $I$ such that $I A = A I = A$ ; that is, applying the binary operator to some element $A$ and the identity element $I$ leaves $A$ unchanged
Existence of Inverse: for each element $A$ , there must exist an inverse $A - 1$ such that $A A - 1 = A - 1 A = I$

A group with commutative binary operator is known as Abelian.

Examples

the set of integers $\mathbb{Z}$ under addition, $(\mathbb{Z},+)$ : here, zero is the identity, and the inverse of an element $a \in \mathbb{Z}$ is $- a$ .
the set of the positive rational numbers $\mathbb{Q}_+$ under multiplication, $(\mathbb{Q}_+,\cdot)$ : $1$ is the identity, while the inverse of an element $\frac{m}{n} \in \mathbb{Q}_+$ is $\frac{n}{m}$ .
for every $n \in \mathbb{N}$ there exists at least one group with n elements,e.g., $(\mathbb{Z}/n\mathbb{Z},+) = (\mathbb{Z}_n,+).$
the set of complex numbers {1, -1, i,-i} under multiplication, where i is the principal square root of -1, the basis of the imaginary numbers. This group is isomorphic to $\mathbb{Z}_{4}$ under mod addition.
the Klein four group consists of the set of formal symbols ${1, i, j, k}$ with the relations $i^{2} =j^{2}=k^{2}=1, \; ij=k, \; jk=i, \; ki=j.$ All elements of the Klein four group (except the identity 1) have order 2. The Klein four group is isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ under mod addition.
the set of "moves" on a Rubik's cube, where a move is understood to be a finite sequence of twists: here, the identity move is to do nothing, while the inverse of a move is to do the move in reverse, thereby undoing it.
The Symmetric group
The general and special Linear groups.