Math 400
Due Friday 2/24
Homework 6 - Material from Chapters 6-7
Reminder: Justify all answers on computational/example problems. If you work with
anyone or use any resource other than class notes and the text of the textbook, give credit.
No internet resources are allowed (including Google or Chegg or material posted from any
other university).
1. Find an isomorphism from Z to the group of all even integers (under addition), and
prove that it is an isomorphism. This shows that a group can be isomorphic to a proper
subgroup of itself!
2. Show that ϕ(x) = ln x is an isomorphism from R+ under multiplication to R under
addition.
3. Prove that S4 is not isomorphic to D12 .
1 n n ∈ Z under matrix multiplication. (I’m
4. Consider the group of matrices
0 1 telling you it’s a group – you don’t need to prove it.) What familiar group is this
isomorphic to? Explain why you chose your answer, but you don’t need a full proof of
the isomorphism.
5. Suppose ϕ is an isomorphism from some group G to Z, a ∈ G, and ϕ(a) = 4. What is
ϕ(a3 )?
6. Prove that Z under addition is NOT isomorphic to Q under addition.
7. Suppose ϕ is an isomorphism from D4 to itself such that ϕ(R90 ) = R270 and ϕ(V ) = V .
Determine ϕ(D) and ϕ(H).
8. Inner automorphisms:
(a) Let G be a group and x ∈ G a fixed element. Define a function ϕx : G → G by
ϕx (a) = xax−1 for all a ∈ G. Prove that ϕ is an isomorphism. (It is called the
inner automorphism of G induced by x.)
(b) Under what condition(s) is φx the trivial isomorphism (that is, the identity function)?
9. Let H be the subgroup of A4 given by H = {ε, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. (You
don’t need to prove that H is a subgroup.) Find all the left cosets of H in A4 .
10. Let H be the same as in the previous problem. How many left cosets of H are there
in S4 ? (Do this without finding the cosets.)
11. Let H be all the multiples of 3, which is a subgroup of Z. Find all the left cosets of H
in Z.
12. Suppose a has order 12. Find all the left cosets of ha4 i in hai.