CH 3 - Normal Subgroups, Quotients and Homomorphisms

Question 1

Define two elements x and y being conjugate.

What can we say about conjugacy classes in the symmetric group?

Answer 1

Definition 3.1. Let G be a group. Two elements x and y are conjugate in G if there exists g ∈ G
such that y = g^−1xg. We write y = x^g. Conjugacy is an equivalence relation.
.The conjugacy class of x in G is denoted x^G.

Two permutations σ and τ in Sn are conjugate if and only if they have the same disjoint cycle
structure.

Question 2

Define a normal subgroup.

What about normal subgroups in abelian groups?

Answer 2

In an abelian group, every subgroup is normal

Question 3

State and prove the lemma on normal subgroups and conjugacy classes.

Define the quotient group of G by N.

Answer 3

Question 4

Define a group homomorphism.

What happens to identity elements in a homomorphism? What happens to inverse elements?

Answer 4

In a homomorphism from G to H, *f(1_G) = 1_H
* f(x^-1) = (f(x))^-1 for all x

Question 5

Define the kernel of a homomorphism.

State a lemma on the kernel and image of a homomorphism.

State a lemma on injectivity and the kernel.

Answer 5

Question 6

State the First Isomorphism Theorem.

Answer 6

Question 7

State the Correspondence Theorem and represent it with a subgroup diagram.

Answer 7

Question 8

State the Second Isomorphism Theorem.

Less likely to be in an exam.

Answer 8

Question 9

State the Third Isomorphism Theorem.

Answer 9