数学群论代写 Groups and symmetrices

MAT301 Week 5 Exercises

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 

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

 数学群论代写

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

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