#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:

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

#Q1 Find P2*P1, are they the same?
     1 2 3 4 5     1 2 3 4 5     1 2 3 4 5
#A: [         ] * [         ] = [          ], which is not the same as p1*p2.
     4 1 2 3 5     2 1 5 3 4     3 2 1 5 4

#Q2 Explain in your own words why if the order of pi is k, then |GG({pi})|=k

#A: It seems that GG() is based oof of the number of orders, as it permutates and iterates through the Set to generate the outcome of GG({pi}). Since there are k terms in the set, |GG({pi})|=k.

#Q3 Prove that if H is a subgroup of a group G, then |G|/|H| is an interger.

#A: Since we can take answers from online, one proof for lagranges Theorem is "For any element x of G, Hx = {h • x | h is in H} defines a right coset of H. By the cancellation law each h in H will give a different product when multiplied on the left onto x. Thus each element of H will create a corresponding unique element of Hx. Thus Hx will have the same number of elements as H. " (taken from http://dogschool.tripod.com/lagrange.html).

#Q4 Whose theorem is it?

#A: Lagranges Theorem

#Q5 Generate a random permuation of length 9 (use randperm(9)) and by hand find its cycle representation and check it with PtoC(pi) (1:05:00)

                      [1,2,3,4,5,6,7,8,9]
#A: S:=randperm(9) -> [8,5,6,4,3,9,7,2,1]
#:  {[8,2,5,3,6,9,1],[4],[7]}
#:  PtoC(S) = {[4],[7],[9,1,8,2,5,3,6]} checks out

#Q6 Check that the range of FunT over all permutations of length 7 (you can use permute(7)) is the samething (the set of permutations of length 7) (1:24:00)

#A: FunT(permute(7)) never halts in maplecode, uncheckable.