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 \(H\).