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

A group requires closure, associativity, an identity, and inverses; commutativity is an additional property that defines an abelian (commutative) group, not a group in general.

Question 2

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

Accepted Answer

12

Answer

Z_12 has elements {0, 1, 2, ..., 11}, so its order (number of elements) is exactly 12.

Question 3

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

Accepted Answer

3

Answer

The order of element a in Z_n is n/gcd(a, n), so ord(4) = 12/gcd(4, 12) = 12/4 = 3.

Question 4

Which of the following groups is NOT cyclic?

Accepted Answer

Z_2 × Z_2

Answer

Z_2 × Z_2 (the Klein four-group) has no element of order 4, so no single element generates the entire group.

Question 5

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

Accepted Answer

|H| divides |G|

Answer

Lagrange's Theorem states that the order of any subgroup H of a finite group G must divide the order of G.

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Abstract Algebra and Group Theory