MAT301 Assignment 4
Show work in all problems.
Please make sure your submission is legible
and mark pages for each submitted question. You will lose points otherwise.
(1) Let ϕ : Z40 → Z40 be an isomorphism such that ϕ(7¯) = 23.
Find ϕ(15).
(2) Let ϕ : D8 → D8 be an automorphism. List all the possibilities for
ϕ(R45◦). Justify your answer.
(3) Let G = (C∗, ·) (recall that C∗ = C \ {0}). Let H be the set of
matrices of the form
(
a 0
0 b
)
where a, b ∈ C∗. The operation on H
is matrix multiplication. You do not need to show that G and H
are groups.
Are G and H isomorphic? Justify your answer.
Hint: Use Theorem 6.2 from Gallian.
(4) Is U(16) isomorphic to Z8? Justify your answer.
(5) Let ϕ : D4 → D4 be an automorphism. Suppose ϕ(H) = D. Find
ϕ(V ). Justify your answer.
Hint: Use Theorem 6.3 from Gallian.
(6) Let G be a group of order 21. Suppose G has exactly one subgroup
of order 3 and exactly one subgroup of order 7. Prove that G is
cyclic.
(7) Compute 22147 mod 13. Show work.
(8) Let G be a finite abelian group. Suppose G contains at least 3
elements of order 3. Prove that |G| is divisible by 9.
(9) Write all the left cosets of H = 〈(123)〉 in A4. Is it true that gH =
Hg for any g ∈ A4?