Download M 431 Final Exam Fall 2011

1
M 431 Final Exam Fall 2011
Name________________________________________
You may use a graphing calculator and your notes, but no other references are allowed.
Please show your work where indicated, write neatly, and use a pencil. Use the reverse
side if needed. Circle one item per MC question (circling two items means zero credit).
1. Define non-negative powers of a permutation f on A as follows:
f0 = iA, f1 = f, and fn+1 = fn o f for n > 0.
Now, let f and g be any 2 permutations on a nonempty set A and,
using induction, show that (f-1 o g o f)n = f-1 o gn o f holds for all n > 0.
2. What property of the integers states that any nonempty set, S, of positive
integers has a least element (i.e. an m ∈ S such that m ≤ x for all x ∈ S)?
3. Find the general solution of the congruence 10x  14 (modulo 21):
4. Identify the set of units in Z12, i.e. the congruence classes mod 12 forming a
group under multiplication modulo 12. Also, find the inverses of these units:
5. Let : G → G', a mapping from one group G to another group G', be a homomorphism.
Let eG be the identity in G and eG' the identity in G'. Consider the following statements:
1.
2.
3.
4.
 takes the identity of G to the identity of G', i.e. (eG) = eG'.
For x ∈ G, (x−1) = ((x))−1, i.e.  maps inverses to inverses.
The kernel of  is a normal subgroup of G.
The image of  is a normal subgroup of G'.
Then
(a)
(b)
(c)
(d)
statement 4 may be false, but the others all hold.
statements 1 & 2 are true, but not 3 & 4.
each of these statements is true.
only statements 1& 3 are true.
2
6. In showing the uniqueness of inverses in a group,
which of the following properties is needed?
(a)
(b)
(c)
(d)
the commutative property
the distributive property
the associative property
none of these
7. Consider the relation R defined on the set Z of integers:
xRy if and only if 7 divides 3x – 10y
(a) Show that this is an equivalence relation:
1. R is reflexive:
2. R is symmetric:
3. R is transitive:
(b) Describe the distinct equivalence classes of R:
8. An element x in a group is called idempotent if x2 = x. Show that
the identity element e is the only idempotent element in a group:
9. A subset H of the group G is a subgroup of G
if and only if the following conditions hold:
1. H is nonempty, and
2. For any x & y ∈ H, x y−1 ∈ H
True
False
10. In a multiplicative group G, given that we have defined yn for y ∈ G and n a
positive integer, how should we define xn for x ∈ G if n is a negative integer?
3
11. If every proper subgroup of a group G is cyclic, then G itself must be cyclic.
True
False
12. If K is a subgroup of H & H is a subgroup of G, then K is a subgroup of G.
True
False
13. Show the mapping  : C\{0} → H defined by (a + bi) =
is an
isomorphism from the multiplicative group of nonzero complex numbers
C\{0} to the multiplicative group H = {
: a & b real with a2 + b2 ≠ 0}.
Fill in the blanks:
14. If is an epimorphism from the group G to the group G' with kernel K,
then G' is isomorphic to G/ _____.
15. Every field is an integral domain, but when is an integral domain a field?
If the integral domain in question is _____.