Name:
Problem Set 1
Math 415 Honors, Fall 2014
Due: Tuesday, September 16.
Review Sections 4, 5 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 16. Remember to fully justify all your answers,
and provide complete details. Neatness is greatly appreciated.
1. Carefully read the entire course website. Send an email to your instructor containing:
• Math415H in the subject line;
• a review of the course website (What did you like? What would you change? List any
typos you discovered, etc.); and
• an acknowledgement that you understand the policies and procedures for this course.
2. Fix a positive integer n, and define ∗n on Zn to be multiplication modulo n. Show that if n = 10,
the set {1, 3, 7, 9} is a group under ∗10 . Give the table for this group. Do not forget to check the
group operation is associative.
3. Let S = {e, a, b, c}; show that there are only two nonisomorphic group structures on S . To do
this, fix e as the identity of the group, and construct all possible multiplication tables for S
that correspond to group structures. As a first step, analyze the possible values of a · a. Do not
forget to check that the group structures you propose are associative.
4. Let G be a group. For a ∈ G and n ∈ Z, define



a · a · · · a (n factors)
n>0




n
a =
e
n=0




a−1 · a−1 · · · a−1 (−n factors) n < 0
Show that if m, n ∈ Z, and a ∈ G, an am = an+m .
5. Let G be a finite group, and let a ∈ G. Show that there exists n ∈ Z>0 such that an = e.
6. Determine whether the set of all n×n upper triangular matrices with real coefficients and determinant 1 is a subgroup of SLn (R) = {n × n matrices with real coefficients and determinant 1}.
(The operation is matrix multiplication.)
7. Determine whether the set of all n × n matrices with real coefficients and determinant 2 is a
subgroup of GLn (R), the group of all invertible n × n matrices with real coefficients, under
matrix multiplication.
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0 −2
8. Describe all elements in the cyclic group of GL2 (R) generated by
.
−2 0
9. Write the addition table for Z6 , compute all cyclic subgroups of this group, and draw the
corresponding subgroup diagram.
10. Prove that a nonempty subset H of a group G us a subgroup if and only if, for all a, b ∈ H,
ab−1 ∈ H.
11. Let G be an abelian group, written multiplicatively (so the group operation is ·, and the identity
element is e). Show that H = {x ∈ G | x2 = e} is a subgroup of G.
12. Prove that every cyclic group is abelian.
13. Prove that a group with no proper nontrivial subgroups is cyclic.
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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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