#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

QUESTION #1:

Find P2 * P1. Is it equal to P1 * P2?

ANSWER:

P2 * P1 = 
1 2 3 4 5
2 1 5 3 4
1 4 5 2 3

They are different.

QUESTION #2;

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

ANSWER:

If the order is k, then the subgroup generated by the element pi is:
{e,pi,...,pi^{k-1}} = GG({pi}).
The size of GG({pi}) is thus equal to k.

QUESTION #3:

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

ANSWER:

Let H be a subgroup of G. If H = {e} or H = G, then it is clear that the quotient is an integer. So, suppose that H is a proper subset of G not equal to {e}. The cosets of H are the sets gH, where g in G. The equivalence relation ~, where a ~ b if a = bh, partitions the set G. Furthermore, the equivalence classes for ~ are equal to gH for each g in G. Consider the set of equivalence classes for ~. Since, |gH| = |H| for all g in G, then the size of the set of equivalence classes is equal to |G|/|H|, which is an integer.

QUESTION #4:

Whose theorem proves this?

ANSWER:

Lagrange's theorem.

QUESTION #5:

Use randperm(9) to generate a permutation. Find the cycle representation.

ANSWER:

Using randperm(9), I got:
[3, 8, 4, 9, 1, 5, 6, 7, 2]

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

The cycle representation is (134928765)

QUESTION #6:

Check that the range of FunT over all permutations of length 7 is the same thing.

ANSWER:

Running, FunT(permute(7)), the program times out.