Test on Group Theory: Core Concepts

Question 1: If x and y are elements of a group G, is x considered conjugate to y if there exists an element g in G such that x equals gyg⁻¹?

A. Yes

B. No

Explanation: x is conjugate to y if there exists an element g in G such that y equals gxg⁻¹.

Question 2: In a group G, for two elements x and y to be conjugate, what must be true about the relationship between them?

A. There must exist an element g in G such that y = gxg⁻¹.

B. The elements x and y must be identical, i.e., x = y.

C. There must exist an element g in G such that x = gyg⁻¹.

D. The elements x and y must commute, i.e., xy = yx.

Explanation: The study materials define conjugacy: 'Let x, y ∈ G. We say that x is conjugate to y if there exists g ∈ G such that y = gxg⁻¹.'

Question 3: When proving conjugacy is an equivalence relation, the symmetric property demonstrates that if y = gxg⁻¹, then x = gyg⁻¹.

A. Yes

B. No

Explanation: The study materials state that for the symmetric property, if y = gxg⁻¹, then x = g⁻¹yg, not x = gyg⁻¹.

Question 4: The permutations (12), (13), and (23) form a single conjugacy class in S3.

A. Yes

B. No

Explanation: The study materials explicitly state that the conjugacy classes of S3 are { e }, { (12), (13), (23) }, and { (123), (132) }. Therefore, (12), (13), and (23) indeed form one of the conjugacy classes.

Question 5: Based on the provided study materials, which of the following statements about conjugacy in the symmetric group S n is correct?

A. Two permutations in S n are conjugate if and only if they have the same cycle structure.

B. The identity element in S n is conjugate to all other elements in S n .

C. Permutations with different cycle structures can still be conjugate if the group G is large enough.

D. The transpositions (12) and (123) are conjugate in S 3 because they are both permutations.

Explanation: The study materials state, "In the symmetric group S n , two permutations are conjugate if and only if they have the same cycle structure." This directly supports option 0. Option 1 is incorrect because the identity is conjugate only to itself, as stated. Option 2 contradicts the rule that conjugacy is determined by cycle structure. Option 3 is incorrect because (12) is a 2-cycle and (123) is a 3-cycle, meaning they have different cycle structures and therefore are not conjugate in S 3, as illustrated by the example conjugacy classes of S 3.