The factorial is defined as $n! = 1 \cdot 2 \cdot 3 \cdots n$ for $n \geq 1$ though we may define $0!=1$.
We also have binomials:
$$\binom{n}{i} = \frac{n!}{i!(n-i)!}.$$
Fact: If $p$ is prime, then $p$ divides $\binom{p}{i}$ unless $i=0$ or $i=p$.
We can also write out the binomial expansion:
$$(a+b)^n = \sum_{i=0}^n \binom{n}{i} a^ib^{n-i}$$
where $a,b$ are “numbers.” Importantly, the binomial coefficient is counting something, specifically the number of subsets of $1, \ldots, n$ with exactly $i$ elements.
If $S$ is a set, then a function $f : S \to S$ which is a bijection is called a permutation on $S$.
There are two notations commonly used to describe permutations:
- $2$-$1$ notation shows how each element maps to its image in two rows, and it is what the textbook uses:
$$\sigma = \binom{1 \quad 2 \quad 3 \quad 4 \quad 5 \quad 6 \quad 7}{4 \quad 5 \quad 2 \quad 7 \quad 1 \quad 3 \quad 6}.$$
- Cycle notation points each element to the next element in the list, e.g. $\sigma = (*, ?) (\cdot)$ (each set of parentheses is a cycle). Note that in this example $(\cdot)$ is a trivial cycle, so we may leave it out.
The identity permutation $e$ is the function $e(x) = x$.
We can also write the inverse permutation $\sigma^{-1}$ which maps each element to the element that points to it in $\sigma$.
Composition of permutation functions is referred to as product of permutation. It acts as function composition normally would and is notated the same way using $\circ$. Additionally, order of composition matters, and that a composition of bijections is also a bijection. We avoid composing permutations that act on different sets.
Fact: The number of fixed points in $\sigma\circ\tau$ and $\tau\circ\sigma$ is the same.
Permutation composition has several properties:
- Associativity: $(\sigma_1\circ\sigma_2) \circ \sigma_3 = \sigma_1 \circ (\sigma_2\circ\sigma_3)$.
- Identity: $\sigma\circ e = \sigma = e\circ\sigma$.
- Inverse: $\sigma\circ\sigma^{-1} = e = \sigma^{-1}\circ\sigma$.
These properties hold for function composition in general.
A permutation group is a set of permutations that satisfies three properties:
- It contains the identity $e$.
- It is closed under the inverse $^{-1}$, e.g. if $\sigma\in G$, then $\sigma^{-1} \in G$.
- It is closed under product, e.g. if $\sigma,\tau\in G$, then $\sigma\circ\tau\in G$.
Some examples of permutation groups are
- the set of all permutations
- $\{e\}$
- the set of all motions that move all the points on a sponge and maintain its shape
Fact: If a set $S$ is finite, it is enough to verify only closure under product to ensure that a set of permutations on $S$ is a permutation group.
The order of a permutation group is the number of elements in the group. This is according to our textbook, but some people say it is the number of elements in the set $S$.
We can also talk about order of elements. We say $\sigma$ is of order $n$ if $\sigma^n=e$.
The order of $e$ is $1$, and obviously, it is the only permutation with order $1$.
Fact: If $S$ is finite, then any permutation has finite order, meaning repeatedly taking permutations results in a repeating cycle of sets.
An abstract group is a set $G$ with a binary operation $* : G \times G \to G$ with some axioms:
- Associativity: $(a*b)*c = a*(b*c)$.
- Existence of a special element (identity) $e$ or $1$: $(a*e)=a=(e*a)$.
- Existence, for any element $a$, of an element $a^{-1}$ (inverse): $a*a^{-1}=e=a^{-1}*a$.
Example: Any permutation group is an abstract group.