MATH 446/546. SECOND TEST. SOLUTIONS.
When proving statements, you are allowed to use homework problems and facts
from the book or from the lectures.
Good luck!
Problem 1. (25 points) Let G be a nite group, V a nite-dimensional irreducible
representation
MATH 446/546. SECOND PRACTICE TEST. SOLUTIONS.
Problem 1. The quaternion group is a group of order 8, G = cfw_1, i, j, k ,
with dening relations
i = jk = kj,
j = ki = ik,
k = ij = ji,
i2 = j 2 = k 2 = 1.
a) Find the number and the dimensions of complex ir
MATH 446/546. SECOND TEST.
When proving statements, you are allowed to use homework problems and facts
from the book or from the lectures.
Good luck!
Problem 1. (25 points) Let G be a nite group, V a nite-dimensional irreducible
representation of G over a
MATH 446/546. SECOND PRACTICE TEST
Problem 1. The quaternion group is a group of order 8, G = cfw_1, i, j, k ,
with dening relations
i = jk = kj,
j = ki = ik,
k = ij = ji,
i2 = j 2 = k 2 = 1.
a) Find the number and the dimensions of complex irreducible re
MATH 446/546. FIRST HOURLY TEST. SOLUTIONS.
Problem 1. Let G be a group of order 5 7 19. Prove that G is cyclic.
Solution. We have from Sylows theorem that n5 = n7 = n19 = 1. Thus, G has
normal subgroups K , L, and M of orders 5 ,7, and 19. It follows tha
MATH 256. SECOND MIDTERM.
All problems are worth 25 points, but some are harder then the others. When
proving statements, you are allowed to use homework problems and facts from the
book or from the lectures.
Good luck!
1
1. Compute e 2 i .
1
2
MATH 256.
Math 446/546. First Hourly Test. Problems from
past exams.
Problem 1. Prove that any group of order 105 is solvable.
Solution. Let np (p = 3, 5, 7) be the number of Sylow p-subgroups of G. It
follows from the Third Sylow Theorem that n5 = 1 or 21 and n7 =
MATH 256. FIRST MIDTERM.
All problems are worth 25 points, but some are harder then the others. When
proving statements, you are allowed to use homework problems and facts from the
book or from the lectures.
Good luck!
1. Consider the initial value proble
.
MATH 446/546. PRACTICE FINAL
Problem 1. Let F be the splitting eld of the polynomial x3 + 2x 1 = 0 over
Z3 .
a) What is the degree of F over Z3 ?
b) What is the Galois group of F over Z3 ?
Solution. It is easy to check that polynomial x3 + 2x 1 is irred
.
MATH 446/546. PRACTICE FINAL
Problem 1. Let F be the splitting eld of the polynomial x3 + 2x 1 = 0 over
Z3 .
a) What is the degree of F over Z3 ?
b) What is the Galois group of F over Z3 ?
Problem 2. Let K be the splitting eld of the polynomial x3 + 2x