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11.4: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

1

Prove that det(AB)=det(A)det(B) for A,BGL2(R). This shows that the determinant is a homomorphism from GL2(R) to R.

2

Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?

  1. ϕ:RGL2(R) defined by

    ϕ(a)=(100a)

  2. ϕ:RGL2(R) defined by

    ϕ(a)=(10a1)

  3. ϕ:GL2(R)R defined by

    ϕ((abcd))=a+d

  4. ϕ:GL2(R)R defined by

    ϕ((abcd))=adbc

  5. ϕ:M2(R)R defined by

    ϕ((abcd))=b,

    where M2(R) is the additive group of 2×2 matrices with entries in R.

3

Let A be an m×n matrix. Show that matrix multiplication, xAx, defines a homomorphism ϕ:RnRm.

4

Let ϕ:ZZ be given by ϕ(n)=7n. Prove that ϕ is a group homomorphism. Find the kernel and the image of ϕ.

5

Describe all of the homomorphisms from Z24 to Z18.

6

Describe all of the homomorphisms from Z to Z12.

7

In the group Z24, let H=4 and N=6.

  1. List the elements in HN (we usually write H+N for these additive groups) and HN.
  2. List the cosets in HN/N, showing the elements in each coset.
  3. List the cosets in H/(HN), showing the elements in each coset.
  4. Give the correspondence between HN/N and H/(HN) described in the proof of the Second Isomorphism Theorem.

8

If G is an abelian group and nN, show that ϕ:GG defined by ggn is a group homomorphism.

9

If ϕ:GH is a group homomorphism and G is abelian, prove that ϕ(G) is also abelian.

10

If ϕ:GH is a group homomorphism and G is cyclic, prove that ϕ(G) is also cyclic.

11

Show that a homomorphism defined on a cyclic group is completely determined by its action on the generator of the group.

12

If a group G has exactly one subgroup H of order k, prove that H is normal in G.

13

Prove or disprove: Q/ZQ.

14

Let G be a finite group and N a normal subgroup of G. If H is a subgroup of G/N, prove that ϕ1(H) is a subgroup in G of order |H||N|, where ϕ:GG/N is the canonical homomorphism.

15

Let G1 and G2 be groups, and let H1 and H2 be normal subgroups of G1 and G2 respectively. Let ϕ:G1G2 be a homomorphism. Show that ϕ induces a homomorphism ¯ϕ:(G1/H1)(G2/H2) if ϕ(H1)H2.

16

If H and K are normal subgroups of G and HK={e}, prove that G is isomorphic to a subgroup of G/H×G/K.

17

Let ϕ:G1G2 be a surjective group homomorphism. Let H1 be a normal subgroup of G1 and suppose that ϕ(H1)=H2. Prove or disprove that G1/H1G2/H2.

18

Let ϕ:GH be a group homomorphism. Show that ϕ is one-to-one if and only if ϕ1(e)={e}.

19

Given a homomorphism ϕ:GH define a relation on G by ab if ϕ(a)=ϕ(b) for a,bG. Show this relation is an equivalence relation and describe the equivalence classes.


This page titled 11.4: Exercises is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Thomas W. Judson (Abstract Algebra: Theory and Applications) via source content that was edited to the style and standards of the LibreTexts platform.

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