2.Group

definition

Let $<G,\ast >$ be binary algebra.

If the operation $\ast$ satisfies the associative law, then $<G,\ast >$ is a semigroup.

If $<G,\ast >$ is a semigroup, and there's an identity element $e$ with respect to the $\ast$ operation, then $<G,\ast >$ is a monoid, denoted as $<G,\ast ,e>$.

Let $<G,\ast >$ be a monoid, if all elements $a$ in $G$ have inverse $a^{-1}$, then $<G,\ast >$ is a group.

If the operation $\ast$ satisfies commutative law, then group $<G,\ast >$ is an Abelian group or Abel group.

The cardinality of the set $G$ is the order of group, denoted as $|G|$​​. If the order is limited, then it's a finite group, otherwise it's an infinite group.

Groups are a special kind of algebraic system.

Property

In group $<G,\ast >$:

(1) All elements are cancelable;

Invertible elements must be cancelable, and all elements of a group are invertible.

(2) There's no idempotent element in $G$ except identity element $e$​;

Let $a$ is an idempotent element, then $a\ast a=a,a\ast a=a\ast e$, According to the elimination law: $a=e$.

(3) Groups of order greater than 1 cannot have zero elements;

Assume the order of $G$ is grater than 1 and have a zero element $\theta$, find a non-zero element $x$, according to the definition of zero element, $x\ast \theta =\theta ,\theta \ast \theta =\theta$, then $x\ast \theta =\theta \ast \theta$. Using the elimination law: $\theta =x$, Contradict the assumption.

(4) No row or column in a group table has the same element.

Power of element

Let $n$ is a nonnegative integer, define the power of an element a in the group:

When $n>0$, $a^n=a\cdot ...\cdot a$, because groups satisfy the law of closure and association.

When $n=0$, $a^0=e$.

Since $a$ has inverse $a^{-1}$, then we can define negative integer power of $a$:

$a^{-n}=(a^{-1}{)}^n=a^{-1}\cdot ...\cdot a^{-1}$.

In group $G$, for all integer $n,m$, $a^n{a}^m=a^{n+m}$, $(a^n{)}^m=a^{nm}$.

If the commutative law is satisfied, $(ab{)}^n=a^n{b}^n$.