帮写MATH 3175 GROUP THEORY QUIZ 4代做Prolog

日期：2024-03-17 10:41

QUIZ 4 (20 MIN, OPEN BOOK)

MATH 3175 GROUP THEORY

(Each problem is worth 10 points.)

(1) In each example below, show that the group G1 is not isomorphic to G2.

(a) G1 = (S4, ◦) and G2 = (, ·);

(b) G1 = (, ·), G2 = (Z8, +).

(c) G1 = (Q, +) and G2 = (Z × Z, +).

(2) Suppose that H and K are subgroups of a group G.

(a) Prove that the intersection H ∩ K is a subgroup of G as well.

(b) Suppose that HK = KH. Prove that HK is a subgroup of G.

(Recall: HK = {hk | h ∈ H, k ∈ K}.)