MATH 128A San Jose State University Abstract Algebra Question Problem Set

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Math 128A, problem set 05
CORRECTED MON OCT 05
Outline due: Wed Sep 30
Due: WED OCT 07
Last revision due: Mon Nov 23
Problems to be done, but not turned in: (Ch. 5) 21–67 odd; (Ch. 6) 1, 3, 5.
Problems to be turned in:
1. (Ch. 5) 26.
2. Let
H = {α ∈ S8 | α(i) ∈ {1, 2, 3} for all i ∈ {1, 2, 3}} .
Prove that H is a subgroup of S8 . (You may want to use the Pigeonhole Principle, in
the following form: If X is a finite set, then f : X → X is one-to-one if and only if it
is onto.)
3. This problem proves an analogue of Theorem 5.4.
(a) Find two 3-cycles α, β ∈ S4 such that αβ = (1 2)(3 4).
(b) Prove that if β is a cycle of length k ≥ 3, then there exists a 3-cycle α such that
αβ is a cycle of length k − 2. (To make the notation less awkward, just do this
for the k-cycle (1 2 · · · k).)
(c) Prove that if β is a cycle of length k ≥ 3, then there exist 3-cycles α1 , . . . , αr
such that αr · · · α1 β is either the identity or a 2-cycle.
(d) We have the following theorem:
Theorem (*): If β is an even permutation, then then there exist
3-cycles α1 , . . . , αs such that αs · · · α1 β = .
To make the notation less complicated, prove Theorem (*) in the case where
β = (1 2 3 4 5)(6 7 8 9)(10 11 12 13 14 15 16)(17 18),
and explain how you know your method works in general.
(e) Now suppose Theorem (*) holds in general, and not just in the special case
you proved. Prove that every even permutation is a product of 3-cycles (not
necessarily disjoint).
4. Revised version: Let H be a subgroup of Sn that contains at least one odd permutation
α ∈ H, let K = An ∩ H be the subgroup of H that contains the even permutations in
H, and let O be the set of all odd permutations of H.
(a) Prove that if x ∈ K, then αx ∈ O.
(b) Now define a map f : K → O by the formula f (x) = αx. Prove that f is a
bijection.
(c) Prove that |H| = 2 |K|. (It follows that the order of H is even.)
Old version:
(a) Let G be a group, and let a be an element of G. Prove that the map f : G → G
defined by f (x) = ax is a bijection (one-to-one and onto).
(b) Let H be a subgroup of Sn that contains at least one odd permutation. Prove
that the order of H is even.
5. (Ch. 6) 6.
6. (a) Prove that the map ϕ3 : U (11) → U (11) defined by ϕ3 (x) = x3 is an automorphism of U (11).
(b) For which n is the map ϕn : U (11) → U (11) defined by ϕn (x) = xn an automorphism of U (11)? Find a pattern. (You do not need to prove your answer to 6b,
but show some work or evidence.)
of S, that are isomorphic to H.
Exercises
Being a mathematician is a bit like being a manic depressive: you spend your
life alternating between giddy elation and black despair.
Steven G. Krantz, A Primer of Mathematical Writing
1. Find an isomorphism from the group of integers under addition to
the group of even integers under addition.
2. Find Aut(Z).
3. Let R+ be the group of positive real numbers under multiplication
.
Show that the mapping 0(x) = Vx is an automorphism of R+.
4. Show that U(8) is not isomorphic to U(10).
5. Show that_U(8) is isomorphic to U(12).
6. Prove that isomorphism is an equivalence relation. That is, for any
groups G, H, and K
GG;
GzHimplies H = G
G ~ H and H = K implies G ~ K.
7. Prove that S, is not isomorphic to Di2.
8. Show that the mapping a → logio a is an isomorphism from R*
under multiplication to Runder addition,
9. In the notation of Theorem 6.1, prove that T, is the identity and
-1 = TE
that (T)-1 =
8-1
tive notation.
10. Given that o is a isomorphism from a group G under addition to a
group G under addition, convert property 2 of Theorem 6.2 to addi-
11. Let G be a group under multiplication, G be a group under addition
ā
o (a) =
find an expression for o(a’b-2) in terms of a and 7.
12. Let G be a group
. Prove
that the mapping alg) = 8-1 for all g in G
is an automorphism if and only if G is Abelian.
13. If g and h are elements from a groun
21. Complete
even if and only if
22. What cycle is (a, a, …)?
exercise is referred to in Chapter 25.)
n
23. Show that if H is a subgroup of S, then either every member of His
an even permutation or exactly half of the members are even. (This
24. Suppose that H is a subgroup of S of odd order. Prove that H is a
subgroup of A,
25. Give two reasons why the set of odd permutations in S, is not a sub.
group.
26. Let a and ß belong to S. Prove that a-18-‘aß is an eveh
permutation.
27. How many elements are there of order 2 in Ag that have the disjoint
cycle form (a,an)(aza)(aza)(aas)?
28. How many elements of order 5 are in S,?
29. How many elements of order 4 does Shave? How many elements
of order 2 does S. have?
30. Prove that (1234) is not the product of 3-cycles. Generalize.
31. Let B E S, and suppose B4 = (2143567). Find ß. What are the pos-
sibilities for ß if ß ES,?
32. Let ß = (123)(145). Write ß99 in disjoint cycle form.
33. Let (a,a,a,a,) and (azac) be disjoint cycles in S… Show that there is
no element x in S10 such that x? = (a,azaza)(azao).
34. If a and B are distinct 2-cycles, what are the possibilities for laßi?
35. Let G be a group of permutations on a set X. Let a E X and define
stab(a) = {a E Gla(a) = a}. We call stab(a) the stabilizer of a in
G (since it consists of all members of G that leave a fixed). Prove
that stab(a) is a subgroup of G. (This subgroup was introduced by
Galois in 1832.) This exercise is referred to in Chapter 7.
36. Let ß = (1,3,5,7,9,8,6)(2,4,10). What is the smallest nositive inte
ger n for which on = R-52