1. Fundamental Concepts

Question 1

A group (G,*) is a set and * is a binary operation on G which satisfies the following properties

Answer 1

1. Associativity: (x ∗ y) ∗ z = x ∗ (y ∗ z) ∀x, y, z ∈ G.
2. Identity: There exists an element e ∈ G such that
   e ∗ x = x = x ∗ e ∀x ∈ G.
3. Inverse element: ∀x ∈ G ∃y ∈ G such that x ∗ y = e = y ∗ x.

Question 2

|S 4| =

Answer 2

4! = 24

Question 3

Subgroup test - Let (G,*) be a group with identity element e. A subset H # G is a subgroup in and only if which following properties hold?

Answer 3

1) e∈H,
2) x,y∈H =⇒ x∗y∈H (‘H isclosedunderthebinaryoperation’), 3) x ∈ H =⇒ x−1 ∈ H (‘H is closed under taking inverses’).

Question 4

Let G be a group and H ≤ G. The index of H in G. What is it denoted by and what does it say about for H in G?

Answer 4

[G : H], is the number of left cosets of H in G.

Question 5

Let φ : G → H be a homomorphism of groups. Show that φ(eG) = eH.

Answer 5

Proposition 2.28. (Properties of homomorphisms)
φ(eG)eH = φ(eG) = φ(eGeG) = φ(eG)φ(eG).

Question 6

Let G and H be groups and φ : G → H be an isomorphism. What properties hold?

Answer 6

1) The inverse map φ−1 : H → G is an isomorphism;
2) |G| = |H|;
3) G is abelian ⇐⇒ H is abelian.

Question 7

When is a group (G,*) called abelian?

Answer 7

when x * y = y * x for x,y e G

Question 8

Let (G,∗G) and (H,∗H) be groups. A homomorphism from G to H is a map φ:G→H satisfying

Answer 8

Definition 1.9 : φ(x∗G y)=φ(x)∗H φ(y) ∀x,y∈G.