Question

Let G = 〈(1 2 3 4 5 6), (1 6)(2 5)(3 4)〉. Let H_{1} :=
〈(1 4)(2 5)(3 6)〉 and H_{2} := 〈(1 6)(2 5)(3 4)〉. Determine
if the subgroups H_{1} and H_{2} are normal
subgroups of G.

Answer #1

1. Let T = {(1, 2), (1, 3), (2, 5), (3, 6), (4, 7)}. T : X ->
Y. X = {1, 2, 3, 4}, Y = {1, 2, 3, 4, 5, 6, 7}
a) Explain why T is or is not a function.
b) What is the domain of T?
c) What is the range of T?
d) Explain why T is or is not one-to one?

6. Let A = 3 −12 4 −1 0 −2 −1 5 −1 . 1 (a) Find all
eigenvalues of A5 (Note: If λ is an eigenvalue of A, then λ n is an
eigenvalue of A n for any integer n.). (b) Determine whether A is
invertible (Check if the constant term of the characteristic
polynomial χA(λ) is non-zero.). (c) If A is invertible, find (i)
A−1 using the Cayley-Hamilton theorem (ii) All the eigenvalues...

4. Let A = {1, 2, 3, 4, 5}. Let L = {(x, y) ∈ A × A : x < y}
and B = {(x, y) ∈ A × A : |x − y| = 1}.
i. Draw graphs representing L and B.
ii. Determine L ◦ B.
iii. Determine B ◦ B.
iv. Is B transitive? Explain.

let the universal set be U = {1, 2, 3, 4, 5, 6, 7, 8, 9} with A
= {1, 2, 3, 5, 7} and B = {3, 4, 6, 7, 8, 9}
a.)Find (A ∩ B) C ∪ B
b.) Find Ac ∪ B.

Let p = (8, 10, 3, 11, 4, 0, 5, 1, 6, 2, 7, 9) and let q = (2,
4, 9, 5, 10, 6, 11, 7, 0, 8, 1, 3) be tone rows. Verify that p =
Tk(R(I(q))) for some k, and find this value of k.

Let ? = ? + 2? + 3? and ? = −4? + 6? + ??, where ? represents a
real number.
Express the cross product ? × ? in terms of ? and in component
form, 〈?, ?, ?〉.
Determine the value of ? for which ? × ? is parallel to the
vector ? = 〈15, 3, −7〉.

Let A = {1, 2, 3, 4, 5, 6}. In each of the following, give an
example of a function f: A -> A with the indicated properties,
or explain why no such function exists.
(a) f is bijective, but is not the identity function f(x) =
x.
(b) f is neither one-to-one nor onto.
(c) f is one-to-one, but not onto.
(d) f is onto, but not one-to-one.

Consider the following matrix.
A =
4
-1
-1
2
6
-3
6
4
1
Let B = adj(A). Find b31,
b32, and b33. (i.e., find
the entries in the third row of the adjoint of A.)

Let X = {1, 2, 3, 4} and Y = {2, 3, 4, 5}. Define f : X → Y by
1, 2, 3, 4 → 4, 2, 5, 3. Check that f is one to one and onto and
find the inverse function f -1.

Let A = {0, 3, 6, 9, 12}, B = {−2, 0, 2, 4, 6, 8, 10, 12}, and C
= {4, 5, 6, 7, 8, 9, 10}.
Determine the following sets:
i. (A ∩ B) − C
ii. (A − B) ⋃ (B − C)

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