I dont know how to do questions 2, 4 (b,c),5,6,7,8

Homework 2 Math 113 Summer 2016.
Due Thursday June 30th
Make sure to write your solutions to the following problems in complete English sentences.
Solutions that are unreadable or incoherent will receive no credit. Provide complete justifications
for all claims that you make. Problems will be of varying difficulty, and do not appear in any
order of difficulty.
1. Let g be an element of order k in a group G .
a) If f : G → H is a homomorphism, prove that the order of f (g ) divides k.
b) If f : G → H is an isomorphism, prove that the order of f (g ) is equal to k.
2. Find all automorphisms of Z/4Z.
3. Does k · x = x + k define an action of Z/nZ on R?
4. Let U3 (Z/2Z) be the group of matrices 1 a b def
U3 (Z/2Z) = 0 1 c | a, b, c ∈ Z/2Z 0 0 1
with matrix multiplication as the binary operation.
a) Show that |U3 (Z/2Z)| = 8.
b) Find an element R of order 4 and an element S of order 2 in U3 (Z/2Z), such that
SRS = R −1 .
c) Write the group D8 as {e, r , r 2 , r 3 , s, sr , sr 2 , sr 3 }, where r is counterclockwise 90◦
rotation of the plane and s is a reflection. Consider the map
f : D8 → U3 (Z/2Z)
which sends s i r j to S i R j . This defines a homomorphism (you do not need to prove
this). Prove that f is actually an isomorphism.
5. (Important, and useful for the following exercises!) Prove that “a homomorphism is determined by what it does to the generators”, in the following sense. Suppose that a group
G is generated by some subset B. Suppose f1 and f2 are two homomorphisms to some
other group H such that f1 (g ) = f2 (g ) for all g in B. Prove that f1 = f2 . [CAUTION:
it does not follow, as is the case in linear algebra, that we can define a homomorphism
simply by specifying where to send the generators; one has to be careful about possible
relations that the generators may satisfy]
6. Prove that there are no nontrivial homomorphisms1 from D10 to Z/5Z. [Hint: use
problem 1]
1 The trivial homomorphism from G to H is the map f (g ) = eH for all g ∈ G . A homomorphism is nontrivial
if it is not this one. 1 7. In the dihedral group D12 (symmetries of a regulator hexagon centered at the origin with
two of its vertices on the x-axis) , describe
the subgroup H consisting of transformations
√
which fix the line L given by y = 3x (meaning they leave L unchanged). Find the
right-coset of this subgroup which takes the x-axis to L. In other words, find an element
g ∈ D12 such that the elements of the right-coset Hg are all those symmetries which
take the x-axis to L.
8. Using Lagrange’s theorem, determine all pairs m, n of positive integers for which there
exists a nontrivial homomorphism from Z/nZ to Z/mZ.
9. Find all possible actions on the group Z/2Z on Z/3Z.
10. Finish the “converse” part of Lemma 6.1.2 in the notes, namely: let α be a homomorphism
G → Perm(S); use α to define an action of G on S. 2

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