Complex numbers.
1. Definition and notations
A complex number z is a formal expression of the form z = x + yi where
x, y R. Denote by
C = cfw_x + yi | x, y R
the set of all complex numbers. For a complex number z = x + yi C,
where x, y R put
Re(z ) := x,
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Homework #4, Math 427,
problem 6 corrected 9/18.
1
Prof. Eugene Lerman
Due Wednesday, Sep 23, 2015 in class
Let G be a group a, b G. Prove that
(ab)1 = b1 a1 .
Hint: What do you need to check?
2
(i)
Let G be a group, a G.
Show that the map
ca (g) := aga1
Homework #5, Math 427,
Due Wednesday, Sep 30, 2015 in class
Prof. Eugene Lerman
1 Prove that any subgroup H of the integers Z is generated by one element. Hints: if H = cfw_0 not much to prove. If H = cfw_0 show that
S := cfw_n H | n > 0 is non-empty and
a?
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Homework #2, Math 427,
Due Wednesday, Sep 9, 2014 in class
Prof. Eugene Lerman
1 Prove that for two integers a, b if ab 1 mod n then gcd(a, n) = 1.
Prove that, conversely, if gcd(a, n) = 1 then there is b Z so that ab 1
mod n.
2
(a) Prove that for any x Z
Homework #3, Math 427,
Prof. Eugene Lerman
Due Wednesday, Sep 16, 2015 in class
(problem 6 corrected 9/14: a has to be nonzero for the collection of ane
functions to form a group)
1 Let f : G H be a homomorphism between two groups and let K be
a subgroup
Math 427 Midterm Exam 1 (Solutions)
PRINT YOUR NAME:
Problem 1.
Consider GL(2, R) as group with respect to matrix multiplication. Consider the subset
H=cfw_
ab
| a, b, c, d Z, and ad bc = 0 GL(2, R).
cd
Determine whether or not H is a subgroup of GL(2, R)
Handout on cyclic subgroups and subgroups generated by a set
1. Cyclic subgroups.
Proposition-Denition. Let G be a group and g G.
Then the subset g := cfw_g n |n Z G is a subgroup of G. This subgroup
is called the cyclic subgroup of G generated by g .
Den
Math 427
Midterm Exam 2 (Solutions)
Due Wednesday, Oct 30, 2013
Problem 1.
Let G be a group and let H G be a subgroup of G.
(1) Suppose that [G : H ] = m < . Prove that for every g G there
exists i cfw_1, . . . , m such that g i H .
(2) Give an example of
Math 427 Extra Credit set 2. Due Wed, Dec 4
Problem 1.
Recall that if R is a ring, a R and k Z, then ka R is the element
dened as
0 if k = 0
a + a + + a if k > 0
ka :=
k times
(a) + (a) + + (a) if k < 0
|k| times
Also, if R is a ring, an element a R is ca
Math 427 Midterm Exam 3 (Solutions)
Problem 1. [12 points]
Consider the ideal I = (2x2 , x3 )
ideal.
Z[x]. Prove that I is not a principal
Solution.
Suppose, on the contrary, that I is a principal ideal, so that there exists
f Z[x] such that I = (f ) Z[x]
Math 427 Extra Credit set 1. Due Monday, October 21
Problem 1.
If G is a group, the center Z (G) is dened as
Z (G) = cfw_a G| for every b G we have ab = ba.
(a) [1 exam point] Prove that for every group G the center Z (G) is
a normal subgroup of G.
(b) [3
Math 427 Extra Credit set 2 (Selected solutons)
Problem 1.
Recall that if R is a ring, a R and k Z, then ka R is the element
dened as
0 if k = 0
a + a + + a if k > 0
ka :=
k times
(a) + (a) + + (a) if k < 0
|k| times
Also, if R is a ring, an element a R i
Math 427 H/wk 2, due Fri, Sep 13.
PRINT YOUR NAME:
1. Compute the following permutation S7 in a two-row notation
and then represent as a product of disjoint cycles:
= (4 1 2)(3 4 7 6 1)(5 1 3 4 2)
Then do the same for 1 .
2. For S7 from Problem 1, comput
Math 427
H/wk 8
Due Friday, Oct 25, 2013
Problem 1.
Let (R, +, ) be a ring and let I, J be ideals in R.
(1) Prove that I + J := cfw_a + b|a I, b J is an ideal in R.
(2) Prove that I J is an ideal in R.
(3) Prove that I J := cfw_a1 b1 + + an bn |n 1, ai I
Math 427 H/wk 3, due Fri, Sep 20.
PRINT YOUR NAME:
1. Give a careful proof, using induction on n, that if (G, ) is a group
and a, b G are such that ab = ba, then for every integer n 1 we have
abn = bn a.
2. Consider the subset H S4 dened as
H = cfw_1, (1