Math 100A: Abstract Algebra I (UC San Diego, fall 2022)
Homework for week 10
Due Friday, December 2 at 11:59pm
All homework should be submitted to Gradescope. No extensions will be granted.
At the top of each homework assignment, you must specify all outside resources that you
consulted and all collaborators, or write ¯ne)f none were used. See the syllabus for
clarification.
All references to Artin are to Algebra, second edition. The hardcover, softcover, and eBook
versions contain identical text. Exercises are located at the end of each chapter.
Due to the ongoing strike by UC academic workers, this homework will be graded for
completion only, with no revision cycle.
(1) Chapter 7, exercise 2.9(a)#): Determine the class equation for the following groups:
(a) the quaternion group;
(b) D4 ;
(c) D5 .
(2) Determine the class equation for the group A4 .
(3) Let p be a prime number. Let G be the set of matrices of the form
?
?
1 x y
?0 1 z ?
(x, y, z ? Z/Zp).
0 0 1
(a) Show that G is a group of order p3 which is not commutative.
(b) Determine the class equation in the case p = 3.
(4) Chapter 7, exercise 2.14: The class equation of a group G is 1 + 4 + 5 + 5 + 5.
(a) Does G have a subgroup of order 5? If so, is it a normal subgroup?
(b) Does G have a subgroup of order 4? If so, is it a normal subgroup?
(5) Chapter 7, exercise 3.1: Prove the Fixed Point Theorem (7.3.2).
(6) Chapter 7, exercise 4.1: The icosahedral group operates on the set of five inscribed
cubes in the dodecahedron. Determine the stabilizer of one of the cubes.
(7) Chapter 7, exercise 5.4: Describe the centralizer Z(?) of the permutation ? =
(153)(246) in the symmetric group S7 , and compute the orders of Z(?) and C(?).
(8) Let p be a prime number. Show that
|GL3 (Z/Zp)| = (p3 ? 1)(p3 ? p)(p3 ? p2 )
by counting choices for the first row, then the second row, then the third row. Then
use this to show that the subgroup of upper triangular matrices with 1s on the
diagonal (see above) is a Sylow p-subgroup.
(9) Let G be a group of order 100. Use the Sylow theorems to show that G has a normal
subgroup of order 25, and hence is not simple.
(10) Find a Sylow 2-subgroup of S6 . Optional challenge: generalize to Sn for any n.
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