#ATTENDANCE QUIZ FOR LECTURE 10 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p10
#with an attachment called
#p10FirstLast.txt
#(e.g. p10DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 9, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE: 5

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH BY THE ANSWER

1.Find P2*P1. Are they the same?
P1=21534, P2=41235
P2*P1 = [3, 2, 1, 5, 4]
no

2.Explain in your own words why if the order of pi is k then |GG({pi})|=k
GG({pi} returns the subgroups generated by the set of permutations pi. Therefore, the number of subgroups is equal to the order of pi.

3. Prove that if H is a subgroup of a group G, then |G|/|H| is an integer.
The left cosets of H in G are the equivalence classes of a certain equivalence relation on G. A left coset yH has the same number of elements as |H|. Since left cosets are identical or disjoint, each element of G belongs to exactly one left coset. From the definition of index of subgroup, there are [G:H] left cosets. So,|G|=[G:H]|H|.
If G is of finite order, all three numbers are finite and it follows.
Now let G be of infinite order. If [G:H]=n is finite, then |G|=n|H| and so H is of infinite order. If H is of finite order such that |H|=n, then |G|=[G:H]×n and so [G:H] is infinite.

4. Whose theorem is it?
Lagrange

5. Generate a random permutation of length 9 (use randperm(9)) and by hand find its cycle representation.
[3, 8, 4, 9, 1, 5, 6, 7, 2]
[[9, 2, 8, 7, 6, 5, 1, 3, 4]]