#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 as P1*P2? [20:26]

A1. MulPers(P2, P1) = [3,2,1,5,4]



Q2. explain in own words why if the order of pi is k then |GG({pi})|=k? 

A2. Because assume order of pi is k. Then it means pi^k = pi. (k is the smallest positive integer that make this happen). So this means that after k pow, everything will have to start repeating all over again. Since the result of |GG({pi})| is a set, then it only contain unique elements which can only generated by pi^1 -> pi^(k - 1). 



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

A3. https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)



Q4. whose theorem is it? 

A4. Lagrange's Theorem



Q5. Generate a random permutation of length 9 [randperm(9)], and by hand find its cycle representation and check it with PtoC

A5. randperm(9) = [3, 8, 4, 9, 1, 5, 6, 7, 2]

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)

PtoC([3,8,4,9,1,5,6,7,2]) = [[9, 2, 8, 7, 6, 5, 1, 3, 4]] which is the same. 



Q6. Check that the range of FunT over all permutaiton of length 7 (can use permute(7))

A6. range has 5040 elements in it, should be the same as permute(7). 



The code is as follows: 

r := proc (permutations) local i, ans; 

ans := {}; 

for i to nops(permutations) do 

	ans := {FunT(permutations[i]), op(ans)} 

end do; 

return ans 

end proc



nops(r(permute(7))) = 5040