TRANSFORMATION GROUPS OF THE PLANE
In class I dened the dihedral group to be a certain class of symmetries of the
plane. This is not exactly in the spirit of the textbook, but thats life. Anyway,
there was a request for more information about general symm
MATH H113: HOMEWORK 9 SOLUTIONS
Section 7.4
Problem 8. If (a) = (b) then a (b) so a = ub for some b R. Similarly b (a)
so b = va for some v R. If a = 0 then b = va = 0 = (1)0 = 1a, so b equals a unit
times a. If a = 0 then note a = ub = uva so a(1 uv) = 0
MATH H113: HOMEWORK 5 SOLUTIONS
Section 3.3
Problem 4. Everything to check follows from the corresponding facts about C
and D. Since C A, we have that C contains 1A , is closed under multiplication and
inverses, and is closed under conjugation by elements
MATH H113: HOMEWORK 3 SOLUTIONS
Section 1.2
Problem 4. Let z = rk D2n where n = 2k. Then z is a rotation by 2k/n =
so z is not the identity, but z 2 = rn = 1. Therefore |z| = 2. Multiplying both sides
of z 2 = 1 by z 1 gives that z = z 1 (i.e. rk = rk )
MATH H113: HOMEWORK 1 SOLUTIONS
Section 1.1
Problem 6
(d) The set is not closed under addition, so it cannot be a group. As an example,
2 and -3/2 both have absoluted value greater than or equal to 1, but their sum
2 + (3/2) = 1/2 does not.
(e) The set in
MATH H113: HOMEWORK 7 SOLUTIONS
Section 3.4
Problem 1. Let G be an abelian simple group. By denition, simple groups
are nontrivial so there exists some g G cfw_1. Since G is abelian, the subgroup
H = g is normal in G. Since G is simple, this implies H = c
MATH H113: MIDTERM PRACTICE PROBLEMS
Problem 1. The following problem outlines a proof that 1 and Z2 are (up to
isomorphism) the only nite groups G for which Aut(G) is trivial. The missing step
(that direct products of Z2 s are the only nite groups in whi
MATH H113: MIDTERM SOLUTIONS
Problem 1 (3+3+4+3 points).
Consider Z2 = Z Z which is a group under compentwise addition. Dene H Z2
by
H = cfw_(i, j) Z2 : 5 | (i + 3j)
(a) Prove that H is a subgroup of Z2 .
First, 5|0 = 0 + 3(0) so (0, 0) H. Suppose (a, b)
MATH H113: HOMEWORK 11 SOLUTIONS
Section 9.1
Problem 4. Dene : Q[x, y] Q[y] and : Q[x, y] Q by
(f )(y) = f (0, y)
(f ) = f (0, 0).
We have seen that evaluations of polynomials at a point give a homomorphism to
the ground ring. This proves that is a homomo
Math H113 Syllabus
Course name: Honors Introduction to Abstract Algebra
Instructor
Max Glick
895 Evans Hall
maxglick@berkeley.edu
Course information
Meeting time: Tuesday/Thursday, 3:30-5, in 6 Evans Hall
Textbook: Abstract Algebra, Dummit and Foote, 3rd
MATH H113: FINAL EXAM PRACTICE PROBLEMS
Problem 1. The following problem lls in a gap in the proof of Eisensteins
criterion from class.
(a) Let S be an integral domain. Suppose f (x), g(x) S[x] satisfy
xn = f (x)g(x)
for some n 0. Prove that f (x) = uxk a