Name:
Problem Set 4
Math 415 Honors, Fall 2014
Due: Tuesday, October 7.
Review Sections 9, 10 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Tuesday, October 7. Remember to fully justify all your answers, and
provide complete details. Neatness is greatly appreciated.
1. Write the the product of cycles (1, 2)(4, 7, 8)(2, 1)(7, 2, 8, 1, 5) ∈ S 8 using two-line notation.
2. Express the permutations
!
1 2 3 4 5 6 7 8
σ=
,
8 2 6 3 7 4 5 1
1 2 3 4 5 6 7 8
τ=
3 1 4 7 2 5 8 6
!
as products of disjoint cycles and as products of transpositions.
3. Fix n ≥ 3. Show that any permutation σ ∈ S n can be written as a product of at most n − 1
transpositions.
4. Show that S n is generated by {(1, 2), (1, 2, 3, . . . , n)}.
Hint: What is (1, 2, 3, . . . , n)(1, 2)(1, 2, . . . , n)n−1 ? What is (1, 2, 3, . . . , n)2 (1, 2)(1, 2, . . . , n)n−2 ?
5. Fix a positive integer n ≥ 2, and fix σ ∈ S n an odd permutation. Show that every odd
permutation in S n is a product of σ and some element of An .
6. Find all cosets of the subgroup h18i of Z36 .
7. Let H and K be subgroups of G. Define ∼ on G by a ∼ b if and only if a = hbk for some
h ∈ H and k ∈ K. Prove that ∼ is an equivalence relation on G and describe the elements of
the equivalence class containing a ∈ G. (These equivalence classes are called double cosets.)
8. Let G be a group of order pq, where p and q are prime numbers. Show that every proper
subgroup of G is cyclic.
9. Show that a group with at least two elements but with no proper nontrivial subgroups must be
finite and of prime order.
10. Let G be a finite group of order n. Show that an = e for all a ∈ G.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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