Notes - Groups HT23, Group axioms

Can you state the definition of a binary operation $\ast$?

A binary operation $\ast$ on a set $S$ is a map $\ast : S \times S \to S$ with $a \ast b$ denoting the image of $(a, b)$ under $\ast$.

Why is $\div$ not a binary operation on $\mathbb R$?

Because it is not defined for $0 \in \mathbb R$.

Can you state the definition of a binary operation $\ast$ being associative?

For all $a, b, c \in S$,

\[(a \ast b) \ast c = a \ast (b \ast c)\]

Can you state the defintion of a binary operation $\ast$ being commutative?

For all $a, b \in S$,

\[a \ast b = b \ast a\]

Can you state the definition of an element $e \in S$ being an identity for a binary operation $\ast$?

For any $a \in S$,

\[e \ast a = a = a \ast e\]

Can you state what it means for $a \in S$ to have an inverse $b \in S$ with respect to a binary operation $\ast$?

\[a \ast b = e = b \ast a\]

Can you state the group axioms?

A group $(G, \ast)$ consists of a set $G$ and a binary operation $\ast : G \times G \to G$ such that

Can you state what it means for a group $(G, \ast)$ to be Abelian?

For all $a, b \in G$,

\[a \ast b = b \ast a\]

What is the notation for the group of $n \times n$ invertible real matrices?

$n$th general linear group,

\[\text{GL}(n, \mathbb R)\]

What is the direct product of two groups $(G, \ast _ G)$ and $(H, \ast _ H)$?

The set of all ordered pairs equipped with componentwise inversion and multiplication.

How could you describe the group $(\mathbb R^2, +)$ as the direct product of two groups?

\[(\mathbb R, +) \times (\mathbb R, +)\]

What is the order of a group $G$?

The number of elements in $G$.

What is the order $o(g)$ of an element of a group $G$?

The smallest positive integer $n$ such that $g^n = e$.

Prove:

Let $\ast$ be a binary operation on $S$ and $a \in S$. Prove that if an identity $e$ exists, then it is unique.

Todo.

Prove:

Let $\ast$ be a binary operation on $S$ and $a \in S$. Prove that if $a$ has an inverse $a^{-1}$, then it is unique.

Todo.