Groups HT23, Group axioms
Flashcards
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 $\ast$?
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
- $\ast$ is associative
- There exists an identity element
- For each $a \in G$ there exists an inverse
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?
What is the order $o(g)$ of an element of a group $G$?
The smallest positive integer $n$ such that $g^n = e$.
Proofs
Prove:
Let $\ast$ be a binary operation on $S$ and $a \in S$. Prove that if an identity $e$ exists, then it is unique.
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.
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.