Groups
A group is a set $G$ and a binary operation $\cdot$ such that

For all $x,y\in G$, $x\cdot y \in G$ (closure).

There exists an identity element $1\in G$ with $x\cdot 1 = 1\cdot x = 1$ for all $x \in G$ (identity).

For all $x,y,z \in G$ we have $(x y)z = x(y z)$ (associativity).

For all $x \in G$ there exists an element $x^{1}$ with $x x^{1} = x^{1} x = 1$ (inverse).
If we only have closure and associativity, then we call $G$ a semigroup. If we have closure, associativity and an identity element, we call $G$ a monoid.
If $x y = y x$ for some $x,y \in G$ then we say $x,y$ commute (or are commutative, or permutable). If $x y = y x$ for all $x,y \in G$ then we say $G$ is abelian (or commutative).
Theorem: The following are alternative axioms for defining finite groups:

Closure.

Associativity.

Right and left cancellation, namely $a x = b x \implies a = b$ and $y a = y b \implies a = b$.
We shall restrict our attention to finite groups for now.
A homomorphism between two groups $G, H$ is a map $f:G\rightarrow H$ with $f(x) f(y) = f(x y)$ for all $x, y \in G$. If $f$ is bijective then we call $f$ an isomorphism.
The order of an element $g$ in a group $G$ is the smallest positive integer $k$ such that $g^k = 1$. This must always exist in a finite group.
Theorem: If $x \in G$ has order $h$, then $x^m = 1$ if and only if $h  m$.
Theorem: If $x \in G$ has order $m n$, where $m,n$ are coprime, then $x$ can be uniquely expressed in the form $x = u v$ where $u$ has order $m$ and $v$ has order $n$.
Proof: Find $a,b$ with $a m + b n = 1$, and pick $u = x^{b n}, v = x^{a m}$. Uniqueness is straightforward.∎
A subset $H$ of $G$ that also satisfy the group axioms is called a subgroup of $G$. Every group $G$ contains two trivial or improper subgroups, $G$ itself and the group consisting of the identity element alone. All other subgroups are called proper subgroups.
Theorem: A nonempty subset $H$ of $G$ is a subgroup if and only if it is closed under multiplication.
A nonempty subset $H \subset G$ is a subgroup if and only if $H^2 \subset H$
Lemma: For a subgroup $H$, for all $h \in H$ we have $h H = H = H h$.
Corollary: For any set $S \subset H$ we have $S H = H = H S$.
We can now strengthen a previous statement. A nonempty subset $H \subset G$ is a subgroup if and only if $H^2 = H$
Theorem: Let $g\in G$. Then for a subgroup $H$, we have $g^{1} H g$ is also a subgroup of $G$ isomorphic to $H$.