#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
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1. Find P2 * P1 and P1 * P2. Are they the same?
A1. 
P1 := randperm(5);
                     P1 := [3, 4, 5, 1, 2]

P2 := randperm(5);
                     P2 := [2, 3, 5, 4, 1]

MulPers(P1, P2);
                        [5, 4, 1, 2, 3]

MulPers(P2, P1);
                        [4, 5, 2, 1, 3]

P1*P2 is not equal to P2*P1

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2.Explain why in your own words if order of pi is k then |GG({pi})| = k.
A2. 
If pi has 5 elements then the number of subgroup is usually k
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3. Prove that if H is a subgroup of a group G then |G| / |H| is an integer.
A3.
The number of subgroups is always divisible by the order of the group
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4. Who's theorem is it ?
A4. Lagranche's theorem states: |G| = n|H|, where G is a group and H is the subgroup

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5. Generate random permutation of length 9 and by hand find its cycle representation. 
Check it with PtoC(pi):
A5.
randperm(9);
                  [4, 1, 3, 9, 7, 6, 8, 2, 5]


# [1, 2, 3, 4, 5, 6, 7, 8, 9]
# 
# [4, 1, 3, 9, 7, 6, 8, 2, 5]
#
# Cyclic permutation: [1, 4, 9, 5, 7, 8, 2] (3, 6)
# [1, 4, 9, 5, 7, 8, 2, 3, 6]
# [4, 9, 5, 7, 8, 2, 1, 3, 6]

PtoC(B);
               [[3], [6], [9, 5, 7, 8, 2, 1, 4]]


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6. Check the range of FunT over all permutations of length 7.
A6. permute(7) has 5040 permutations and they all have the same range. Below is the range for 
the last permutation which is the same range for all the permutations. 

with(Statistics);
A := Range(seq(FunT(permute(7)[i]), i = 1 .. 5040)[5040]);
                            A := 6.



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