Here we consider definition of a group and some simple properties, operations, examples.
Fromal definition#
Group#
A pair $(X, +)$ is called a group if $X$ is a set, $+$ is a function $+:X\times X\to X$ and:
- operations are associative: $a + (b + c) = (a + b) + c$
- X has an identity element 0 such that $\forall x\in X$ $x + 0 = 0 + x = x$
- each element $x$ has an inverse $x^{-1}$: $x+ x^{-1} = x^{-1}+ x = 0$
Semigrups#
Semigroup is a wekaer structure than the group it does not require:
- an identity element
- inverses It does require:
- closure
- associativity
Abellian groups#
A group $G = (X,+)$ is called abellian if the operation $+$ is commutative
Properties#
- identity element is unique
- inverses are unique
- cancelation law: if $a + b = a + c$ then $b = c$
- if $a + x = b$ then $x = a^{-1} + b$
Examples of Groups#
Integers under addition#
$$ (\mathbb{Z}, +) $$
- identity: $0$
- inverse of $a$: $-a$
This is an Abelian group.
Nonzero real numbers under multiplication#
$$ (\mathbb{R} \setminus {0}, \cdot) $$
- identity: $1$
- inverse of $a$: $1/a$
Also an Abelian group.
Integers modulo $n$#
$$ (\mathbb{Z}_n, +) $$
Elements:
$$ {0,1,2,\dots,n-1} $$
Operation: addition modulo $n$.
Example for $n=5$:
$$ 3 + 4 \equiv 2 \pmod{5} $$
Matrix groups#
Invertible matrices form groups under multiplication:
$$ GL(n, \mathbb{R}) $$
These groups are not commutative.
Permutation groups#
A permutation is a rearrangement of elements in a set.
The set of all permutations of $n$ elements forms the symmetric group
$$ S_n $$
The group operation is composition of permutations.
Example: permutations of the set ${1,2,3}$.
One permutation might send:
$$ 1 \rightarrow 2,\quad 2 \rightarrow 3,\quad 3 \rightarrow 1 $$
Applying two permutations one after another gives another permutation.
Properties:
- identity permutation leaves all elements unchanged
- every permutation has an inverse
- generally not commutative
For $n$ elements the group contains
$$ n! $$
permutations.
Example:
$$ |S_3| = 6 $$
Permutation groups are extremely important because many groups can be represented as permutation groups.