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What are $~\mathbb{Z}~$ and $~\mathbb{Z}/m\mathbb{Z}~$ really?

We make $\mathbb{Z}$ from $\mathbb{N}$ to have the benefit of undoing addition. Usually we think of subtraction as undoing addition but subtraction is just addition of negative numbers.

To get to $\mathbb{Z}$, we take each $n \in \mathbb{N}$ and we “create” a new element $-n$ that is characterized by the property \(n + (-n) = 0 = (-n) + n\) We’ve seen how to do this via a quotient of $\mathbb{N} \times \mathbb{N}$ by an equivalence relation and via inductive types in Lean.

The structure of $(\mathbb{Z},+)$ is a particular instance of a general notion of a group.

Definition. A group is a triple of data $(G,\cdot,e)$ consisting of

  • a set $G$,
  • a binary operation $\cdot : G \times G \to G$, and
  • an element $e \in G$

which together satisfy the following conditions:

  • $\cdot$ is associative: $a \cdot (b \cdot c) = (a \cdot b) \cdot c$
  • evaluating $\cdot$ on $e$ is the identity function (on either side): $a \cdot e = e \cdot a = a$
  • and there exists inverses: for each $a \in G$ there is some $b \in G$ with $a \cdot b = b \cdot a = e$.

Often when $\cdot$ and $e$ are clear from the context we just use the underlying set $G$ to refer to a group. The element $e \in G$ is called the identity element of $G$.

Example. $\mathbb{Z}$ is a group under addition where $\mathbb{N}$ is not a group under addition. The defect with $\mathbb{N}$ is the lack of additive inverses which is rectified by passage to $\mathbb{Z}$.

Example. $\mathbb{Z}/m\mathbb{Z}$ is group for all $m$ under modular addition. For associativity, we can use that for $\mathbb{Z}$: \([n] + ([m] + [k]) = [n] + [m+k] = [n+(m+k)] \\ = [(n+m)+k] = [n+m] + [k] = ([n] + [m]) + [k]\) The identity element of $\mathbb{Z}/m\mathbb{Z}$ is the equivalence class of $0$.

We can see that $[-n]$ is the inverse to $[n]$ since \([n] + [-n] = [n+(-n)] = [0] = [(-n)+n] = [-n] + [n]\) But in $\mathbb{Z}/m\mathbb{Z}$ there are more efficient choices for representative of the equivalence class of the inverse. For example, assume we start with $0 \leq n < m$. Then \([n] + [m-n] = [m] = [0] = [m] = [m-n] + [n]\) so $m-n$ is another representative for $[-n]$ and it satisfies $0 < m - n \leq m$.

Example. The integers are not a group under multiplication because for a general $n \in \mathbb{Z}$, there is no $m$ with $nm = 1$.

Theorem. Let $X$ be a set. Then the set of all self-bijections of $X$ is a group under composition.

Proof. (Expand to view)

We have seen that function composition is associative. The identity element is the identity function $\operatorname{id}_X$. Composition with the identity function returns the original function. We have already seen that $f: X \to X$ is a bijection if and only if it has a inverse $g: X \to X$, ie a function $g$ such that $g \circ f = f \circ g = \operatorname{id}$. So we have inverses.

Definition. Let $X$ be a set. The symmetric group on $X$, denoted $S_X$, is set of all self-bijections of $X$ under composition.

Example. Let’s take the symmetric group of the set $\lbrace 0,1,2,3\rbrace$. We can represent an element $f: \lbrace 0,1,2,3 \rbrace \to \lbrace 0,1,2,3 \rbrace$ as a list \(f \mapsto f(0) f(1) f(2) f(3)\) without losing any information. Thus, $1032$ would be the function that swaps $0$ and $1$ and swaps $2$ and $3$.

The symmetric group behaves a way that might be unexpected from the examples above. Let’s compose $0321$ and $1023$ in the two orders: \(0321 \circ 1023 = 3021 \\ 1023 \circ 0321 = 1320\) Note that $3021 \neq 1320$. In the symmetric group, the order of applying the group operation matters whereas in $\mathbb{Z}$ or $\mathbb{Z}/m\mathbb{Z}$ it doesn’t.

Definition. We say a group $G$ is a commutative or abelian if $a \cdot b = b \cdot a$ for all $a,b \in G$.

If $X$ has more than two elements, then $S_X$ is never abelian.

The symmetric group is, as the name suggests, the group of symmetries of the underlying set. More generally, given a set with extra mathematical structure, it is very natural to consider the set of self-bijections that perserve the structure. These also form groups.

Example. Let’s consider functions $f : \mathbb{Z} \to \mathbb{Z}$ which preserve addition and satisfy $f(0) = 0$. Then, for positive $n$, since $n = 1 + \cdots + 1$, we have \(f(n) = f(1 + \cdots + 1) = f(1) + \cdots + f(1) = nf(1)\) For $-n$, we have \(f(n) + f(-n) = f(n + (-n)) = f(0) = 0\) so $f(-n) = -f(n)$ and $f$ is completely determined by the value $f(1)$.

What can $f(1)$ be for $f$ to be a bijection? If $f(1) = 0$, then $f(n) = 0$ for all $n$. Not so good.

For $f$ to be surjective, we need to be able to write every integer as a multiple of $f(1)$. In other words, $f(1)$ has to divide $n$ for all $n \in \mathbb{Z}$. This forces $f(1) = \pm 1$.

So the set of symmetries $\mathbb{Z}$ under addition is a two element set given by the identity function and the negative function $n \mapsto -n$. This is rather small compared to the symmetric group of $\mathbb{Z}$ the set which is infinite.

There are the same number of symmetries of $(\mathbb{Z},+)$ as there are elements of the group $\mathbb{Z}/2\mathbb{Z}$: two. Are they the same group?

On its face, the answer is no but there is a natural way to loosen equality.

Definition. Let $G$ and $H$ be groups. An homomorphism of groups is a function $\phi : G \to H$ such

  • $\phi(e_G) = e_H$ and
  • $\phi(a\cdot b) = \phi(a) \cdot \phi(b)$

An isomorphism is a bijective homomomorphism. If $G = H$, we call an isomorphism an automorphism.

Above we computed the automorphism group of $(\mathbb{Z},+)$. It is in fact isomorphic to $\mathbb{Z}/2\mathbb{Z}$ using the function \([0] \mapsto \operatorname{id}_{\mathbb{Z}} ~,~ [1] \mapsto (n \mapsto -n)\) You should use the definitions above to check that this indeed is a homomorphism.

Understanding the structure of groups is a fundamental pursuit throughout all of math.