# 数学群论代写 Groups and symmetrices

## MAT301 Week 5 Exercises

### Exercise 1. Prove that the group of integers is isomorphic to all of its non-trivial subgroups.  数学群论代写

Exercise 2. Let tt be a group and let φ be an automorphism of tt. Prove that the set of elements of tt that are fixed by φ is a subgroup of tt.

Exercise 4. Let tt be an abelian group and let φ : tt tt be the homomorphism defined by φ(g) = g2.Exercise  3.  For R = Z, R, Q, C, what group is the multiplicative group ..1    aΣ : a RΣ isomorphic to?

1. Suppose that tt is finite. Describe when φ is an automorphism of tt.

2. Given an example of an infinite group tt such that φ is injective, but not

Exercise 5. Let tt be a group. For each g tt, let Inn(g) be the inner automorphism of tt determined by

g, that is, Inn(g) : tt tt is defined by Inn(g)(x) = gxg1.

1. Let g tt. When is Inn(g) = idG? When is Inn(tt) = {idG}?

2. Let g1, g2 tt. When is Inn(g1) = Inn(g2)?  数学群论代写

3. Prove that the map Inn : tt Inn(tt) is a homomorphism. What is its kernel?

Exercise 6. Let tt and H be isomorphic groups and let φ, ψ : tt H be isomorphisms. Prove that

{x tt : φ(x) = ψ(x)} is a subgroup of tt.

### Exercise 7. Prove that the dicyclic group Dic5 of order 20 is not isomorphic to the dihedral group D10  of  order 20. 数学群论代写

Exercise 9.  Find an injective homomorphism φ : Q>0   Q>0 that is not surjective. Conclude that Q>0 is isomorphic to a proper subgroup of itself.Exercise  8.  Prove that R×  is not isomorphic to the multiplicative group ..a    0Σ : a, b R×Σ.

Exercise 10. Describe all homomorphisms from the additive group Q to itself. Conclude that there is no injective homomorphism from Q to itself that is not surjective, and therefore Q is not isomorphic to a proper subgroup of itself.

Exercise 11. Prove that Q× is not isomorphic to Q.

Exercise 12. Let φ Aut(R×). Prove that φ(R>0) = R>0 and φ(R<0) = R<0.

Exercise 13. Let φ : tt H and ψ : H K be homomorphisms. How are ker φ and ker(ψ φ) related?  数学群论代写

Exercise 15.
Let tt be a finite group. Prove that tt is abelian if and only if there exists an automorphismExercise 14.  Let φ : tt   H  be a surjective homomorphism.  Prove that φ(Z(tt))        Z(H). Can you find a counterexample for this result if you drop the assumption that φ is surjective?

φ of tt such that

1. φ2 = id; and

2. for all x tt, we have φ(x) = x if and only if x = e.

(Hint:  The “only if” direction is easier and holds even in the case when tt is not finite.  For the “if” direction, prove that if an automorphism φ of tt with the above properties exists, then for all x tt there exists y tt such that x = φ(y)y1, and think about what this tells you about φ(x). How can you prove that for all x tt there exists y tt such that x = φ(y)y1 without explicitly determining y? Make sure you use the hypothesis that tt is finite!)  数学群论代写