BMath Bachelor of Mathematics Abstract Algebra and Group Theory

Question 1

Which of the following is NOT a required property for a set G with binary operation * to be a group?

Accepted Answer

Commutativity of *

Answer

Closure under *

Answer

Associativity of *

Answer

Existence of an identity element

Question 2

What is the order of the group Z_12 (integers modulo 12 under addition)?

Accepted Answer

12

Answer

6

Answer

11

Answer

24

Question 3

What is the order of the element 4 in the group Z_12?

Accepted Answer

3

Answer

4

Answer

6

Answer

12

Question 4

Which of the following groups is NOT cyclic?

Accepted Answer

Z_2 × Z_2

Answer

Z_7

Answer

Z_4

Answer

Z_6

Question 5

Lagrange's Theorem states that for a finite group G with subgroup H:

Accepted Answer

|H| divides |G|

Answer

|G| divides |H|

Answer

|H| = |G|/2 always

Answer

|G| = |H|²

Question 6

A subgroup N of a group G is called normal if which condition holds?

Accepted Answer

gNg⁻¹ = N for all g in G

Answer

N is cyclic

Answer

N has prime order

Answer

N equals the center Z(G)

Question 7

What is the order of the symmetric group S_3?

Accepted Answer

6

Answer

3

Answer

9

Answer

12