Groups

Question 1

What is C_n?

Answer 1

The cyclic group, rotational symmetry for

n-gon with directed sides

Question 2

What is D_n?

Answer 2

Dihedral group, rotation and reflection for n-gon with no direction on sides.

Question 3

What is the order of C_n?

Answer 3

n

Question 4

What is the order of D_n?

Answer 4

2n

Question 5

What is the order of S_n?

Answer 5

n!

Question 6

What is the order of Z_n?

Answer 6

n

Question 7

What is S_n?

Answer 7

Permutations of n objects.

Question 8

What is Z_n?

Answer 8

Integers under addition mod n

Question 9

What is the condition for a group to be Abelian?

Answer 9

Composition law is commutative

Question 10

How many binary operators act on a Field?

Answer 10

2, addition and multiplication.

Question 11

How many binary operators act on a group?

Answer 11

1, multiplication

Question 12

List the 6 field axioms.

Answer 12

1. Closure
2. Commutativity
3. Associativity
4. Distributivity
5. Identities exist
6. Inverses exist

Question 13

List the 4 group axioms

Answer 13

1. Closure
2. Associativity
3. Unique Identity
4. Inverses

Question 14

List all conjugacy classes of D3

Answer 14

(e) - trivial class
(c, c^2)
(m, mc, mc^2)

Question 15

Define a subgroup of G

Answer 15

A subset of G which itself follows the composition law of G and obeys all group axioms.

Question 16

Define the coset of g of G with a given subgroup H = {h1,h2,…,hn}

Answer 16

Coset = gH = {gh1, gh2, gh3,…,ghn}

Question 17

What are the two options for the coverage of two coset of G?

Answer 17

Completely overlapping or completely disjoint

Question 18

What are the three axioms of an equivalence relation ~?

Answer 18

1. Reflexive a~a
2. Symmetric a~b implies b~a
3. Transitive a~b and b~c implies a~c

Question 19

State the equivalence relation axioms for a chosen for example equal gender.

Answer 19

Reflexive: a~a
Mark is the same gender as Mark.

Symmetric: a~b => b~a
If Dave is the same gender as Mark, then Mark is the same gender as Dave.

Transitive: a~b & b~c => a~c
If Dave is the same gender as Mark, and Steve is the same gender as Dave, then Steve is the same gender as Mark.

Question 20

Define the normal subgroup H of G.

Answer 20

gHg^1 = H for all g in G

Or gH = Hg is sufficient.

I.e all the conjugates of h are also contained in H

Question 21

The product of two cosets g1H, g2H =? Where H is a normal subgroup of G

Answer 21

(g1H)(g2H) = g1g2H

Because H is normal subgroup:

g1Hg2H = g1HHg2 = g1Hg2 = g1g2H