# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## 107: The Order of the Product of 2 Group Elements

Where $G$ is a group, and $a, b \in G$, it is always the case that

$|ab| = |ba|$.

That is, we can speak of ‘the’ order of the product of 2 group elements without worrying about which product it is.

## 106: Group Isomorphisms

In group theory, a group isomorphism between 2 groups $G, H$ does not simply refer to an ‘isomorphism’ between the underlying sets of $G, H$, but to a special type of homomorphism between them: one which is bijective. In a sense, isomorphic groups are the same group, just with different symbols representing their elements and operations — something which is not necessarily true if they’re homomorphic or if just their underlying sets are isomorphic.

In particular, if $\varphi : G \mapsto H$ is a group isomorphism, then

• $|G| = |H|$
• $G$ is abelian $\leftrightarrow H$ is abelian
• for all $x \in G, |x| = |\varphi (x)|$.

## 104: Bijectivity of Left- or Right-Multiplication by Group Elements

Let $G$ be a group and $x$ be any element of that group. Then the functions

$f: a \mapsto xa$

$g: a \mapsto ax$

are bijective.

#### Proof:

Consider the function $f$. By the left-cancellation property of groups, if $ab = ac$, then $b = c$. This means that if $b \neq c$, then $ab \neq ac$, and it follows that $f$ is injective. Since the domain and codomain of $f$ are the same ($G$), this means that $f$ is in fact bijective.

The proof for bijectivity of $g$ is identical, but uses the right-cancellation property instead.

The intuitive significance of this is that left- or right-multiplication by a given element always sends distinct elements to distinct elements, which is pretty neat.

## 101: Non-Commutativity of Symmetric Groups

In general, the symmetric group $S_n$ (the set of permutations of $n$ distinct objects under permutation composition) is non-abelian. The somewhat trivial exception is when $n \leq 2$.

## 92: Order (Groups)

The order of a group $G$, often denoted $|G|$, is the cardinality of its underlying set. If the order is finite, then $G$ is called a finite group, and likewise for infinite order.

An example of a group with finite order would be the dihedral group $D_n$ for some $n$. An example of one with infinite order would be the group of $\mathbb{Z}$ together with addition.

## 85: The Group as a Category

The group can be viewed as equivalent to a specific kind of category, specifically the category $C$ with only a single element and all of whose morphisms are isomorphisms. In this equivalence, the elements of the group correspond to the morphisms of $C$, the group operation to $\circ$ (composition), and the group identity to the identity morphism on the single element of $C$. Since all morphisms in $C$ are isomorphisms, each ‘element’ has an inverse, and $\circ$ is associative, analogous to the associativity of the group operation.