MT310 Homework 8
Solutions
Due Wednesday, March 31 by 5:00 pm
Exercise 1. Suppose G is a nonabelian group of order 8. In previous homework, you have shown
that G = AB , where A = a is cyclic of order four and B = b is cyclic, generated by some element
b G
ORDERS OF ELEMENTS IN A GROUP
KEITH CONRAD
1. Introduction
Let G be a group and g G. We say g has finite order if g n = e for some positive integer
n. For example, 1 and i have finite order in C , since (1)2 = 1 and i4 = 1. The powers
of g repeat themselv
Mathematics 310
Robert Gross
Homework 1
Answers
Remember that the Fibonacci numbers are defined with the three equations
F1 = 1
F2 = 1
Fn = Fn1 + Fn2
For example, we have F3 = 2, F4 = 3, and F5 = 5.
1. Let k be a positive integer. Prove that F3k is always
MT310 Homework 4
Solutions
Due Friday, February 26 by 5:00 pm
Exercise 1. The direct product of two groups G and H is the group G H with group operation
(g, h)(g , h ) = (gg , hh ) for g, g G and h, h H . Suppose G = g and H = h are cyclic of orders
m and
MT310 Homework 3
Solutions
Due Friday, February 12 by 5:00 pm
Exercise 1. On a group G dene a relation by: a b if there exists g G such that b = gag 1 . Prove
that this is an equivalence relation.
Proof. Let a, b, c G.
Since a = eae1 , we have a a so the
MT310 Homework 2
Solutions
Due Friday, February 5
Exercise 1. Let G = a be a nite cyclic group of order n, with generator a. Prove that ak generates
G if and only if gcd(k, n) = 1.
Proof. () Suppose that ak generates G. Then a is some power of ak , so we
MT310 Homework 6
Your Name
Due Friday, March 19 by 5:00 pm
Exercise 1. Suppose G has two subgroups H, K with K
K HK and that HK/K H .
G and that H K = cfw_e. Prove that
Proof. In general, if J is a subgroup of G containing H , then since gHg 1 = H for all
MT310 Homework 7
Solutions
Due Friday, March 26 by 5:00 pm
Exercise 1. Suppose G is a nonabelian group of order p3 . Prove that for any a G we have ap Z (G).
Hint: Recall the structure of G/Z (G).
Proof. Let us write Z = Z (G). From previous homework, we
MT310 Homework 11
Solutions
Due Friday, April 30 by 5:00
Exercise 1. Let F be a eld, let V be a vector space over F and let V 0 be the set of nonzero vectors
in V . For u, v V 0 , say that u v if there exists a F such that au = v . Prove that this is an
e
MT310 Homework 9
Solutions
Due Saturday, April 17 by midnight
Exercise 1. Suppose G is a nonabelian group of order p3 . We have seen that the center Z = Z (G)
has order p. Let a G, but a Z .
/
a) Prove that for all b G, we have a1 bab1 Z . (See exam 2, pr
Mathematics Course 111: Algebra I
Part II: Groups
D. R. Wilkins
Academic Year 1996-7
6
Groups
A binary operation on a set G associates to elements x and y of G a third element x y of G. For
example, addition and multiplication are binary operations of the