MTH 461 University of Miami Modern Algebra Questions

MTH 461: Survey of Modern Algebra, Spring 2021Homework 6
Homework 5
1. Determine whether each map defined is a homomorphism or not. If it is a homomorphism, prove it, and also determine whether it is an isomorphism.
(a) The maps φ1 , φ2 , φ3 ∶ pRˆ , ˆq Ñ GL2 pRq given by the following, for each a P Rˆ :
ˆ
˙
ˆ
˙
ˆ
˙
1 0
2a 0
1 a
φ1 paq “
,
φ2 paq “
,
φ3 paq “
0 a
0 a
0 1
(b) The map φ ∶ pZn , `q Ñ pCˆ , ˆq given by φpk pmod nqq “ e2πik{n . (Here i “
?
´1.)
(c) The map Cˆ Ñ GL2 pRq given by sending z “ a ` bi P Cˆ to:
ˆ
˙
a b
φpa ` biq “
´b a
2. A while back on HW 1 you showed that the set Z equipped with the binary operation
a ˚ b “ a ` b ` 1 defines a group. Show that this group pZ, ˚q is isomorphic to the usual
group of integers with addition, pZ, `q.
3. Let φ ∶ G Ñ G1 be a group homomorphism. Define
impφq “ ta1 P G1 ∶ a1 “ φpaq for some a P Gu Ă G1
This set is called the image of φ. Sometimes people write impφq “ φpGq.
(a) Show that impφq is a subgroup of G1 .
(b) Prove or provide a counterexample: impφq is a normal subgroup of G1 .
4. Let G and G1 be isomorphic groups. Prove the following statements.
(a) If G is abelian, then G1 is abelian.
(b) If G is cyclic, then G1 is cyclic.
(c) Show that Q is not isomorphic to Z.
5. (a) Find two non-isomorphic groups of order 4.
(b) Show that A4 and Z2 ˆ S3 , although both of order 12, are not isomorphic.
(c) Show that S3 ˆ Z4 and S4 , although both of order 24, are not isomorphic.
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MTH 461: Survey of Modern Algebra, Spring 2021
Homework 1
Homework 1
1. Verify the axioms of a group for the general linear group GL2 pRq.
2. For each of the following examples, either show that it is a group, or explain why it
fails to be a group. If the example is a group, also determine whether it is abelian.
(a) The set of natural numbers N “ t0, 1, 2, . . .u with the operation of addition.
(b) The integers Z with the operation a ˝ b “ a ´ b.
(c) The integers Z with the operation a ˝ b “ a ` b ` 1.
(d) The set of positive integers with the operation of multiplication.
(e) The following set of 2 ˆ 2 matrices with matrix multiplication:
”
ˆ
˙
*
a b
A“
∶ a, b, c, d P Z, ad ´ bc “ 1, a, d odd, b, c even
c d
3. For which subsets of integers S Ă Z does the set S with the operation of multiplication
define a group? Explain your reasoning.
4. Consider the set G “ te, r, b, g, y, ou with operation defined in Lecture 1. In this exercise
you will verify that the operation defined by the Cayley table makes G a group.
(a) Explain why Axiom 2 holds.
(b) Write down the inverse of each element in G. Conclude Axiom 3 holds.
(c) Verify Axiom 1, associativity. For example, check r ˝ pb ˝ gq “ pr ˝ bq ˝ g using the
Cayley table. Write down at least 3 other examples verifying this axiom. (On
your own you can verify that all other possibilities satisfy the axiom.)
5. Let G be an arbitrary group. Given the equations ax2 “ b and x3 “ e, solve for x.
6. For each of the following examples, show that the subset is a subgroup.
(a) The subset t5k ∶ k P Zu of the group pZ, `q.
(b) The subset t3k ∶ k P Zu of the group pQˆ , ˆq.
?
(c) The subset ta ` b 2 ∶ a, b P Q, a, b not both 0u of the group pRˆ , ˆq.
7. Suppose a group G has the property that a2 “ e for all a P G. Show that G is abelian.
8. Show that the intersection of two subgroups of a group is again a subgroup.
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