Groups

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

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

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

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

4. 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:

1. Closure.

2. Associativity.

3. 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 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 $$H$$.

Ben Lynn blynn@cs.stanford.edu 💡