11.4: Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
Prove that det(AB)=det(A)det(B) for A,B∈GL2(R). This shows that the determinant is a homomorphism from GL2(R) to R∗.
Which of the following maps are homomorphisms? If the map is a homomorphism, what is the kernel?
- ϕ:R∗→GL2(R) defined by
ϕ(a)=(100a)
- ϕ:R→GL2(R) defined by
ϕ(a)=(10a1)
- ϕ:GL2(R)→R defined by
ϕ((abcd))=a+d
- ϕ:GL2(R)→R∗ defined by
ϕ((abcd))=ad−bc
- ϕ:M2(R)→R defined by
ϕ((abcd))=b,
where M2(R) is the additive group of 2×2 matrices with entries in R.
Let A be an m×n matrix. Show that matrix multiplication, x↦Ax, defines a homomorphism ϕ:Rn→Rm.
Let ϕ:Z→Z be given by ϕ(n)=7n. Prove that ϕ is a group homomorphism. Find the kernel and the image of ϕ.
Describe all of the homomorphisms from Z24 to Z18.
Describe all of the homomorphisms from Z to Z12.
In the group Z24, let H=⟨4⟩ and N=⟨6⟩.
- List the elements in HN (we usually write H+N for these additive groups) and H∩N.
- List the cosets in HN/N, showing the elements in each coset.
- List the cosets in H/(H∩N), showing the elements in each coset.
- Give the correspondence between HN/N and H/(H∩N) described in the proof of the Second Isomorphism Theorem.
If G is an abelian group and n∈N, show that ϕ:G→G defined by g↦gn is a group homomorphism.
If ϕ:G→H is a group homomorphism and G is abelian, prove that ϕ(G) is also abelian.
If ϕ:G→H is a group homomorphism and G is cyclic, prove that ϕ(G) is also cyclic.
Show that a homomorphism defined on a cyclic group is completely determined by its action on the generator of the group.
If a group G has exactly one subgroup H of order k, prove that H is normal in G.
Prove or disprove: Q/Z≅Q.
Let G be a finite group and N a normal subgroup of G. If H is a subgroup of G/N, prove that ϕ−1(H) is a subgroup in G of order |H|⋅|N|, where ϕ:G→G/N is the canonical homomorphism.
Let G1 and G2 be groups, and let H1 and H2 be normal subgroups of G1 and G2 respectively. Let ϕ:G1→G2 be a homomorphism. Show that ϕ induces a homomorphism ¯ϕ:(G1/H1)→(G2/H2) if ϕ(H1)⊂H2.
If H and K are normal subgroups of G and H∩K={e}, prove that G is isomorphic to a subgroup of G/H×G/K.
Let ϕ:G1→G2 be a surjective group homomorphism. Let H1 be a normal subgroup of G1 and suppose that ϕ(H1)=H2. Prove or disprove that G1/H1≅G2/H2.
Let ϕ:G→H be a group homomorphism. Show that ϕ is one-to-one if and only if ϕ−1(e)={e}.
Given a homomorphism ϕ:G→H define a relation ∼ on G by a∼b if ϕ(a)=ϕ(b) for a,b∈G. Show this relation is an equivalence relation and describe the equivalence classes.