Shreyas’ Notes

# MATH 356

Def: $b \mid a$ ("$b$ divides $a$") if $a \div b \in \mathbb{Z}$

Division algorithm: $a, b \in \mathbb{Z}$, $b > 0$. Then, there exist $q, r \in \mathbb{Z}$ where $0 \leq r < b$ such that $a = qb + r$. Additionally, $q, r$ are unique.

Proposition: If $n \mid a$ and $n \mid b$, then $N \mid a + b$.

Proposition: If $n \mid a$ and $c \in Z$, then $n \mid ac$.

Definition: $d$ is a common divisor of $a$ and $b$ if $d \mid a$ and $d \mid b$.

Definition: Given $a, b \in Z$ not both 0, we say $d$ is a greatest common divisor of $a$ and $b$ if:

• $d > 0$
• $d$ is a common divisor of $a$ and $b$
• $c$ is any common divisor of $a$ and $b$ such that $c \mid d$

Proposition: Given $a$ and $b$ not both 0, there is exactly one gcd of $a$ and $b$.

## Groups §

A group is a set $G$ together with a binary operation $*$ on $G$ satisfying:

1. associativity: for all $x, y, z \in G$, $(x * y) * z = x * (y * z)$
2. identity: there exists a (unique) $e \in G$ such that $e * x = x * e = x$ for all $x \in G$
3. inverse: for all $x \in G$, there exists a (unique) $y \in G$ such that $x * y = y * x = e$

If, further, we have the property that $x * y = y * x$ for all $x, y \in G$, $G$ is abelian.

${GL}_n(\mathbb{R}) = \{A \in M_{n \times n}(\mathbb{R}) : A \textrm{ is invertible}\}$