Group

definitiongroup-theory

Definition

For a set $G$ and an operation $\ast$ on $G$, we say $(G,\cdot )$ is a group if

1. $\ast$ is associative on $G$. In other words, for $x,y,z\in G$, $x\ast (y\ast z)=(x\ast y)\ast z$.
2. $\ast$ has an identity element $e$ in $G$ ( $e\cdot g=g\cdot e=g$ ).
3. Every element in $G$ has an inverse in $G$ ( $g\cdot g^{-1}=g^{-1}\cdot g=e$ ).

Examples

1. $(\mathbb{Z},+)$ is an abelian group.
2. $({\mathbb{R}}^{\ast },\cdot )$ is a group.
3. $(G{L}_n(\mathbb{R}),\cdot )$ is a group (where $G{L}_n(\mathbb{R})$ is the set of invertible $n\times n$ matrices).

Notation Convention

For a general group, we say group $(G,\ast )$, identity element is $e$, and the inverse element of $g$ is $g^{-1}$ .

If the operation on $G$ is a multiplication operation, we say group $(G,\cdot )$, with identity element $e=1$, and the inverse element of $g$ is $g^{-1}$.

If the operation on $G$ is an addition operation, we say group $(G,+)$, with identity element $e=0$, and the inverse element of $g$ is $-g$.

Basic Properties of Groups

1. The inverse of any element is unique in a group.