Math 5890: Modern Algebra
HW 3
6 February 2013
p 35 #19 The map : Q Q dened by (x) = 3x 1 is one to one and onto Q. Give the denition of a binary operation
on Q such that is an isomorphism mapping
a. Q, onto Q, .
Solution Dene by a b = (ab + a + b 2)/3. T
Math 5890: Modern Algebra
HW 11
1 March 2013
p 142 #1 The order of Z6 / 3 is 3.
p 142 #2 The order of (Z4 Z12 )/ 2 2 is 4.
p 142 #3 The order of (Z4 Z2 )/ (2, 1) is 4.
p 142 #4 The order of (Z3 Z5 )/cfw_0 Z5 is 3.
p 142 #5 The order of (Z2 Z4 )/ (1, 1) is
Math 5890: Modern Algebra
HW 10
27 February 2013
p 133 #16 If : S3 Z2 if given as in example 13.3, Ker() = cfw_(1), (1, 2, 3), (1, 3, 2).
p 133 #17 If : Z Z7 is dened by (1) = 4, then Ker() = cfw_7n | n Z, (25) = 2.
p 133 #18 If : Z Z10 is dened by (1) =
Math 5890: Modern Algebra
HW 9
22 February 2013
p 110 #6 The order of (3, 10, 9) in Z4 Z12 Z15 is 60.
p 110 #7 The order of (3, 6, 12, 16) in Z4 Z12 Z20 Z24 is 60.
p 110 #11 The subgroups of Z2 Z4 are cfw_0 cfw_0, cfw_0 cfw_0, 2, cfw_0 Z4 , Z2 cfw_0, Z2 c
Math 5890: Modern Algebra
HW 8
20 February 2013
p 102 #27 Let H be a subgroup of group G and let g G. Dene a one-to-one map of H onto Hg.
Proof: Dene : H Hg by (h) = hg and dene : Hg H by (a) = ag 1 .
Since and are inverses of each other, is one-to-one an
Math 5890: Modern Algebra
HW 7
18 February 2013
p 94 #1 The orbits are cfw_1, 5, 2, cfw_3, cfw_4, 6
p 94 #2 The orbits are cfw_1, 5, 8, 7, cfw_2, 6, 3, cfw_4
p 94 #3 The orbits are cfw_1, 2, 3, 5, 4, cfw_6, cfw_7, 8
p 94 #10 (1, 8)(3, 6, 4)(5, 7)
p 94 #11
Math 5890: Modern Algebra
HW 5
11 February 2013
p 58 #43 Show that if H and K are subgroups of an abelian group G, then
cfw_hk | h H and k K
is a subgroup of G.
Proof: Let G be an abelian group and H and K subgroups of G.
Let HK = cfw_hk | h H and k K. No
Math 5890: Modern Algebra
Solutions
Examination II
3 April 2013
1. (10 points) Complete the following denition:
Let G be a group and p be a prime. A subgroup H of G is called a p-Sylow subgroup of G if it is a
maximal p-subgroup of G.
Note: A subgroup is
Math 5890: Modern Algebra
HW 4
8 February 2013
p 48 #31 If is a binary operation on a set S, an element x of S is an idempotent for if x x = x.
Prove that a group has exactly one idempotent element.
Proof: Clearly the identity element e is an idempotent.
Math 5890: Modern Algebra
Final Examination
15 May 2013
Please show all your work on the pages provided. Your responses should be complete sentences and not just
phrases. You will be graded on what you write, not on what I might think (or hope) that you i
Mathematics 5890: Modern Algebra
Examination I
25 February 2013
Solutions
1. (19 points.) Let G be a group of order 35. If H1 , H2 are subgroups of G of order 5, show that either
H1 H2 = cfw_e or H1 = H2 . (Note: e is the identity of G.)
Solution H1 H2 is
Math 5890: Modern Algebra
HW 13
4 March 2013
p 152 #24 Let z0 U = cfw_z C | |z| = 1. Show that z0 U = cfw_z0 z | z U is a subgroup of U and compute
U/z0 U .
1
1
Proof: Note that 1 z0 U since 1 = z0 z0 and z0 U . If x, y z0 U there exist , U such
that x =