[Abstract Algebra] Chapter 3

Question 1

What are generators of a group?

Answer 1

A set of elements where the entire group can be written as a finite product of these elements

Question 2

What is a group presentation?

Answer 2

A way to write a group using relations and generators

Question 3

What is the notation for a group presentation?

Answer 3

<generators, … | relations(x, y, …), …, >

Question 4

What is the group presentation of the Klein Four group?

Answer 4

<a,b|a^2=b^2=1,ab=ba>

Question 5

What is the group presentation of the dihedral group of order 2n?

Answer 5

<r,s | r^n=s^2=1,rs=sr^(-1)>

Question 6

What is a group homomorphism?

Answer 6

A map from groups G to H such that f(g1 x g2) = f(g1)f(g2)

Question 7

Where do group homomorphisms from groups G to H send 1G?

Answer 7

To 1H

Question 8

What are the only elements required to specify a homomorphism from G to H?

Answer 8

Generators of group G

Question 9

What is a kernel of a homomorphism?

Answer 9

Elements in g which return 1H when mapped using the homomorphism to H

Question 10

What is the longhand term for coset?

Answer 10

Left coset

Question 11

What is a coset?

Answer 11

A set in the form gH for all g in G given a subgroup H of G

Question 12

What bijection sends a coset to another coset?

Answer 12

x -> g2(g1^-1)x

Question 13

What is a normal subgroup?

Answer 13

A subgroup that is the kernel of some homomorphism

Question 14

What are the elements of a quotient group?

Answer 14

The cosets of N

Question 15

What is the operation in a quotient group?

Answer 15

An operation such that g1N x g2N = (g1g2)N

Question 16

What condition can be used to check whether a group is normal?

Answer 16

for all g in G, h in H: ghg^(-1) in H

Question 17

What are the domain and range of a projection homomorphism?

Answer 17

G to G/H

Question 18

What is the mapping of a projection homomorphism?

Answer 18

maps g to gH

Question 19

What does the First Isomorphism Theorem state?

Answer 19

For a homomorphic f: G -> H, G/ker f isomorphic to f^(img)(G)