Math 100A: Abstract Algebra I (UC San Diego, fall 2022)
Homework for week 7
Due Friday, November 11 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.
(1) Chapter 6, exercise 7.1: Let G = D4 be the dihedral group of symmetries of the
square.
(a) What is the stabilizer of a vertex? of an edge?
(b) G operates on the set of two elements consisting of the diagonal lines. What is
the stabilizer of a diagonal?
(2) Chapter 6, exercise 7.11: Prove that the only subgroup of order 12 of the symmetric
group S4 is the alternating group A4 .
(3) Let G be the group Sn . Let S be the 3-fold Cartesian product of {1, . . . , n}. Describe
the orbits of the action of G on S given by
g ? (s1 , s2 , s3 ) = (g(s1 ), g(s2 ), g(s3 )).
(4) Let G be the set of symmetries of a cube (not necessarily preserving orientation).
Determine the stabilizer of a vertex of G.
(5) Let G be an arbitrary group.
(a) Prove that the formula
g ? a = ga
defines an action of G on itself which is faithful : for any g ? G, if g ? a = a for
all a ? G, then g = 1.
(b) Prove that the formula
g ? a = gag ?1
defines an action of G on itself.
(c) Find a group G for which the action defined in (b) is not faithful.
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