Name:
Problem Set 2
Math 415 Honors, Fall 2014
Due: Tuesday, September 23.
Review Sections 1, 2, 3, 6 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 23. Remember to fully justify all your answers,
and provide complete details. Neatness is greatly appreciated.
1. Let U7 be the group of seventh roots of unity and consider the primitive seventh root of unity
ζ = ei(2π/7) ∈ U7 . Assume that ϕ : U7 → Z7 is a function that satisfies the homomorphism
property, and ϕ(ζ) = 4. Find ϕ(ζ m ) for m = 0, 2, 3, 4, 5, 6, and verify that ϕ is an isomorphism.
2. Prove that if ∗ is an associative and commutative binary operation on a set S , then (a ∗ b) ∗ (c ∗
d) = [(d ∗ c) ∗ a] ∗ b for all a, b, c, d ∈ S .
3. Suppose that ∗ is an associative binary operation on a set S . Let H = {a ∈ S | a ∗ x =
x ∗ a for all x ∈ S }. Show that H is closed under ∗.
4. Suppose that ∗ is an associative and commutative binary operation on a set S . Show that
H = {a ∈ S | a ∗ a = a} is closed under ∗.
5. Let φ : S → S 0 be an isomorphism between (S , ∗) and (S 0 , ∗0 ). Show that the inverse function
φ−1 : S 0 → S is an isomorphism between (S 0 , ∗0 ) and (S , ∗). (Here φ−1 is the inverse of φ with
respect to composition of functions, which exists since φ is a bijection.)
6. Prove that if φ : S → S 0 is an isomorphism between (S , ∗) and (S 0 , ∗0 ), and if ψ : S 0 → S 00 is
an isomorphism between (S 0 , ∗0 ) and (S 00 , ∗00 ) then ψ ◦ φ is an isomorphism between (S , ∗) and
(S 00 , ∗00 ).
7. Show that if (S , ∗) and (S 0 , ∗0 ) are isomorphic, and ∗ is commutative, ∗0 is also commutative.
"
#
a −b
8. (30 points) Let H be the set of all 2 × 2 matrices with real coefficients of the form
for
b a
a, b ∈ R.
a. Show that H is closed under matrix addition and matrix multiplication.
b. Show that (H, +) is isomorphic to (C, +).
c. Show that (H, ·) is isomorphic to (C, ·).
9. Let r, s be integers. Show that {nr + ms | n, m ∈ Z} is a subgroup of Z.
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10. Let a and b be elements of a group G. Show that if ab has finite order in G, ba also has finite
order in G.
11. Show that a group that has only a finite number of subgroups must be a finite group.
12. Find a group that is not cyclic, but such that all of its proper subgroups are cyclic.
13. (0 points) For entertainment, you can read: http://www.americanscientist.org/issues/
pub/group-theory-in-the-bedroom.
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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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