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 GG a set and :G×GG\circ: G \times G \to G an operation, (G,)(G, \circ) is a group if

  1. There exists an element eGe \in G such that for all gGg \in G, ge=g=egg \circ e = g = e \circ g.
  2. For all gGg \in G, there exists an element g1Gg^{-1} \in G such that gg1=e=g1gg \circ g^{-1} = e = g^{-1} \circ g.
  3. For all a,b,cGa, b, c \in G, (ab)c=a(bc)(a \circ b) \circ c = a \circ (b \circ c).

We usually denote (g,h)\circ(g, h) by ghg \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 aa and bb are identity elements, a=ba = b)

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

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

  1. The identity element of GG is contained in HH.
  2. HH is closed under \circ: For all a,bHa, b \in H, abHa \circ b \in H.
  3. HH is closed under inversion: For all aHa \in H, a1Ha^{-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=ab=abab = a \cdot b = a \circ b and we will just denote groups with for example GG instead of (G,)(G, \cdot).

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

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

Definition 4: A homomorphism φ:GH\varphi: G \to H is an isomorphism if it is injective and bijective. Then we call GG and HH 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={1,3,13,32,132,33,133,}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 (Z,+)(\mathbb Z, +).