Problem Set 3 Math 415 Honors, Fall 2014

Problem Set 3
Math 415 Honors, Fall 2014
Due: Tuesday, September 30.
Review Sections 6, 7, 8 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, September 30. Remember to fully justify all your answers,
and provide complete details. Neatness is greatly appreciated.
1. Let p and q be distinct prime numbers. Find the number of generators of Z pq .
2. Find the order of the cyclic subgroup of Z30 generated by 25.
3. Let G be a cyclic group with generator a, and let G0 be a group isomorphic to G. If φ : G → G0
is an isomorphism, show that φ is determined by φ(a). That is, show that if ψ : G → G0 is
also an isomorphism, and ψ(a) = φ(a), then ψ(x) = φ(x) for all x ∈ G. Use this fact to find all
isomorphisms of Z12 with itself.
4. Draw a Cayley digraph for Z8 , taking as a generating set S = {2, 5}.
5. Construct a nonabelian group of order 10 generated by two elements of order 2.
6. Draw the digraphs of the two possible nonisomorphic groups of order 4, taking as small a
generating set as possible in each case.
7. For n ≥ 3, the dyhedral group Dn is the subgroup of S n consisting of permutations that arise by
placing two copies of a regular plane n-gon on top of each other and considering the resulting
permutation of the vertices. Find the order of Dn . Using geometric intuition, show that Dn has
a subgroup of order (1/2)|Dn |.
8. Compute the left regular representation of Z4 .
9. Show that the permutation
1 2 3 4 5
∈ S5
2 4 5 1 3
has order 6. Is the subgroup of S 5 generated by ρ isomorphic to S 3 ?
10. Show that if n ≥ 3, and if σ ∈ S n is such that στ = τσ for all τ ∈ S n , then σ is the identity
permutation. Conclude that S n is nonabelian for all n ≥ 3.
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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.
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