# Group theory 101.0

2020-01-18 (2020-01-09)

This is an introduction to group theory, covering the concepts of groups, subgroups and (homo)morphisms.

Definition 1: For $G$ a set and $\circ: G \times G \to G$ an operation, $(G, \circ)$ is a group if

1. There exists an element $e \in G$ such that for all $g \in G$, $g \circ e = g = e \circ g$.
2. For all $g \in G$, there exists an element $g^{-1} \in G$ such that $g \circ g^{-1} = e = g^{-1} \circ g$.
3. For all $a, b, c \in G$, $(a \circ b) \circ c = a \circ (b \circ c)$.

We usually denote $\circ(g, h)$ by $g \circ h$.

Exercise 1: Find/construct groups with 1, 2, 3 and 4 elements.

Exercise 2: Show that the identity element is unique. (Show that if $a$ and $b$ are identity elements, $a = b$)

Exercise 3: Show that every element has a unique inverse.

Definition 2: For a group $(G, \circ)$, a subset $H \subseteq G$ is a subgroup if

1. The identity element of $G$ is contained in $H$.
2. $H$ is closed under $\circ$: For all $a, b \in H$, $a \circ b \in H$.
3. $H$ is closed under inversion: For all $a \in H$, $a^{-1} \in H$.

Exercise 4: Which subgroups do the groups from exercise 1 have?

From now on, we will usually denote the group operation like multiplication: $ab = a \cdot b = a \circ b$ and we will just denote groups with for example $G$ instead of $(G, \cdot)$.

Definition 3: For groups $G$ and $H$, a function $\varphi: G \to H$ is a (homo)morphism if for all $a, b \in G$

$\varphi(ab) = \varphi(a) \varphi(b).$

Definition 4: A homomorphism $\varphi: G \to H$ is an isomorphism if it is injective and bijective. Then we call $G$ and $H$ isomorphic.

An isomorphism "preserves the structure of the group". If two groups they are isomorphic, they are essentially the same.

Exercise 5: Show that there is, up to isomorphism, exactly one group with one element. That is: show that every two groups with one element are isomorphic.

Exercise 6: Show that there is, up to isomorphism, one group with two elements.

Exercise 7: What about groups with 3 or 4 elements?

Exercise 8: Show that the set

$G = \left\{1, 3, \frac 1 3, 3^2, \frac{1}{3^2}, 3^3, \frac{1}{3^3}, \ldots \right \}$

together with the normal multiplication is isomorphic to $(\mathbb Z, +)$.