Hi all,
The first midterm will on my zoom room. You need a camera and then I can monitor you online.
The first midterm will be on May 2 (Sunday) 4:30- 5:30 pm. (Another 10 minutes to submit your solution) (California time). (Special time, not the lecture time)
I choose this time, since I think it will work for most of you. If this really conflicts your schedule, please let me know.
I will upload some practical question before the end of this week. And May 3 will have no lecture.
PRACTICAL QUESTIONS
(1) Prove that every subgroup H of cyclic group G is a cyclic group. Find all generators of group Z32.
(2) Show that group Un (n th unit root) and group Zn are isomorphic. (3) Is all groups of order 6 commutative ? If not commutative, give an example. How about
groups of order 5? (Are they always commutative). (4) Let set G be the Cartisian product of G1 ×G2, where G1 and G2 are 2 groups. Define a
binary operation ∗ on G by (a, b) ∗ (c, d) = (ac, bd). Show that (G,∗) is group. If both G1 and G2 are commutative groups, show G is also a commutative group.
(5) Let H := {A ∈ Mn×n|A = AT , det(A) 6= 0} be a set, ∗ be matrix multiplication. Is the set (H,∗) a group? What if ∗ is the matrix addition?
(6) Assume that G is a group such that for all x ∈ G, x∗x = e. Prove that G is an abelian group.
Date: April 23, 2021.
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