$\textbf{Group}$
A group is a set $G$ together with a binary operation $\cdot$ that fulfill the following requirements:
1) Closure: For all $a,b\in G$, the result $a\cdot b \in G$.
2) Associativity: For all $a,b$ and $c$ in $G$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3) Identity element: There exists an element $e \in G$ such that $e \cdot a=a\cdot e=a$ for all $a \in G$.
4) Inverse element: For each $a\in G$ there exists $b \in G$ such that $a\cdot b=b\cdot a=e$.
Notes:
- The element $e$ in 3) is uniquely defined and called the identity element.
- The element $b$ in 4) is uniquely determined by $a$ and called the inverse of $a$ and is commonly denoted by $a^{-1}$.
$\textbf{Group}$
A group is a set $G$ together with a binary operation $\cdot$ that fulfill the following requirements:
1) Closure: For all $a,b\in G$, the result $a\cdot b \in G$.
2) Associativity: For all $a,b$ and $c$ in $G$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3) Identity element: There exists an element $e \in G$ such that $e \cdot a=a\cdot e=a$ for all $a \in G$.
4) Inverse element: For each $a\in G$ there exists $b \in G$ such that $a\cdot b=b\cdot a=e$.
Notes:
- The element $e$ in 3) is uniquely defined and called the identity element.
- The element $b$ in 4) is uniquely determined by $a$ and called the inverse of $a$ and is commonly denoted by $a^{-1}$.
copied
๐๐ซ๐จ๐ฎ๐ฉ
A group is a set ๐บ together with a binary operation โ that fulfill the following requirements:
1) Closure: For all ๐,๐โ๐บ, the result ๐โ ๐โ๐บ.
2) Associativity: For all ๐,๐ and ๐ in ๐บ, (๐โ ๐)โ ๐=๐โ (๐โ ๐).
3) Identity element: There exists an element ๐โ๐บ such that ๐โ ๐=๐โ ๐=๐ for all ๐โ๐บ.
4) Inverse element: For each ๐โ๐บ there exists ๐โ๐บ such that ๐โ ๐=๐โ ๐=๐.
Notes:
- The element ๐ in 3) is uniquely defined and called the identity element.
- The element ๐ in 4) is uniquely determined by ๐ and called the inverse of ๐ and is commonly denoted by ๐โปยน.