# Problem Set 3 Math 415 Honors, Fall 2014 ```Name:
Problem Set 3
Math 415 Honors, Fall 2014
Due: Tuesday, September 30.
Review Sections 6, 7, 8 in your textbook.
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
ρ=
∈ 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.
Signed:
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