Notes on algebra, probability theory, and linear algebra
A group is a monoid $G$ such that every element has an inverse.
That is, for every $a\in G$, there exists $a^{-1}\in G$ with
\(aa^{-1}=a^{-1}a=e.\)
A group is the algebra of reversible actions.
A monoid describes actions that can be composed and includes a “do nothing” action. A group adds the requirement that every action can be undone.
So:
| Structure | Intuition |
|---|---|
| Semigroup | compose non-empty processes |
| Monoid | compose processes including doing nothing |
| Group | compose reversible processes |
If $a$ is an action, $a^{-1}$ is the action that cancels it.
This means elements of a group do not destroy information.
In a group, the equation
\(ax=b\) has the unique solution
\(x=a^{-1}b.\) Similarly,
\(ya=b\) has the unique solution
\(y=ba^{-1}.\) This is one of the main algebraic powers of inverses: they allow division-like reasoning.
If
\(ax=ay,\) then multiplying on the left by $a^{-1}$ gives
\(x=y.\) Groups always have cancellation.
In a monoid we have
\(a^0,a^1,a^2,\dots\) In a group we also have
\(a^{-1},a^{-2},\dots\) So powers are indexed by all integers:
\(a^n, \qquad n\in\mathbb Z.\)
Without inverses, actions may be irreversible.
We may lose:
For example, a non-injective function can collapse two inputs into one output. Once collapsed, the original information cannot be recovered.
\((\mathbb Z,+)\) is a group. The inverse of $n$ is $-n$.
This models reversible displacement along a line.
\((\mathbb R^\times,\cdot)\) is a group. The inverse of $a\neq 0$ is $1/a$.
This models reversible scaling.
All permutations of a set form a group under composition.
Every permutation is reversible.
Invertible $n\times n$ matrices over a field form a group:
\(GL_n(K).\) These are exactly the linear transformations that lose no information.
\((\mathbb N,+)\) is a monoid but not a group, because adding $5$ cannot be undone inside $\mathbb N$.
Passing from $\mathbb N$ to $\mathbb Z$ can be viewed as adding formal additive inverses.
All functions form a monoid under composition, but not a group. Only bijections are invertible.
All square matrices form a monoid under multiplication, but singular matrices are not invertible.
The group is the point where algebra begins to model symmetry.
A group is not merely “a monoid with more axioms.” It is a shift from general actions to reversible actions.
This leads naturally to the next idea: symmetries of objects.