#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: 6

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

1) Find P2*P1. Are they same?

 1 2 3 4 5 	1 2 3 4 5	1 2 3 4 5
[        ] *   [	 ] =   [	 ]
 4 1 2 3 5	2 1 5 3 4	3 2 1 5 4

No, they are not the same.

2) Explain in your own words why if the order of pi is k
then |GG({pi})|=k

GG keeps generating subgroups of the set by multiplying by the generators until there are no new 
members. The number of elements in this subgroup is the order. Thus it makes sense that the 
cardinality is the order.

3) Prove that if H is a subgroup of a group G then 

|G|/|H| is always an integer

I don't know how to do this, but I found it here.

https://crypto.stanford.edu/pbc/notes/group/lagrange.html

4) Whose theorem is it?
Lagrange's Theorem

5) Generate a random permutation of length 9 (use randperm(9))
and by hand find its cycle representation

1, 2, 3, 4, 5, 6, 7, 8, 9
3, 8, 4, 9, 1, 5, 6, 7, 2

1 3 4 9 2 8 7 6 5 
3 4 9 2 8 7 6 5 1

(1,3,4,9,2,8,7,6,5)

Check it with PtoC(pi):

(9, 2, 8, 7, 6, 5, 1, 3, 4)

6) Check that the range of FunT over all permutations of length 7 
(You can use permute(7)) is the same thing (the set of permutations of length 7)

I wasn't able to get FunT to work