The answer is in the pic. If any doubt still remained, let me know in the comment section.
If this solution helped, please don't forget to upvote to encourage us. We need your support. Thanks ☺☺☺ :)
the following questions are relative,please solve them, thanks! 4. Let G be a group. An isomorphism : G G is called an automorphism of G. (a) Prove that the set, Aut(G), of all automorphisms of G forms a group under composition. (b) Let g E G. Show that the map ф9: G-+ G given by c%(z)-gZg", įs an automorphism. These are called the inner automorphisms of G (c) Show that the set of all g E G such that Og-Pe...
[8 pts) Let G be a group and the center of G is defined as Z(G) = {x € G | xg = gx for all g € G}. In Homework 3, we have showed that the center Z(G) is a subgroup of G. Let H be a subgroup of G. Prove that the set HZ(G) = {hz|he H,2 E Z(G)} is a subgroup of G.
I. Functions and Isomorphisms. Let G be a group and let a EG be any non-identity element (so a #e). Define a function f : GG so that, for any r EG, f(x) = (xa)-1 (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer. (c) Is f an isomorphism? Prove your answer.
Let G be a group, and let a ∈ G. Let φa : G −→ G be defined by φa(g) = aga−1 for all g ∈ G. (a) Prove that φa is an automorphism of G. (b) Let b ∈ G. What is the image of the element ba under the automorphism φa? (c) Why does this imply that |ab| = |ba| for all elements a, b ∈ G? 9. (5 points each) Let G be a group, and let...
22 Must the center of a group be Abelian? 23. Let G be an Abelian group with identity e and let n be some integer Prove that the set of all élements of G that satisfy the equation* - e is a subgroup of G. Give an example of a group G in which the set of all elements of G that satisfy the equation :2 -e does not form a subgroup of G. (This exercise is referred to in...
2. The center of a group G is the set (a) Prove that Z(G) is a subgroup of G, and that it is normal in G (b) Compute the center of the following groups: GG, Di D, Qs, At, Sa, and Dax Qs 2. The center of a group G is the set (a) Prove that Z(G) is a subgroup of G, and that it is normal in G (b) Compute the center of the following groups: GG, Di D,...
15. The center Z of a group G is defined by Z xe G). Prove that Z is a subgroup of G. Can you recognize Z as C(T) for some subgroup Tof G? eGzxxz all
(5 points each) Let G be a group, and let a € G. Let da: G+ G be defined by @a(g) = aga-l for all g E G. (a) Prove that Pa is an automorphism of G. (b) Let b E G. What is the image of the element ba under the automorphism ..? (c) Why does this imply that |ab| = |ba| for all elements a, b E G?
2. Assume the group G acts on the set S. For E S, define Then G is a subgroup of G , which is called the stabilizer of r. The set is called the orbit of r (a) Consider the map ф' G S, defined by фг (g) :-9-x. Prove that there is one map (and only one) : G/G, S such that Vz ยู่'z q (where q: G -G/G, is the quotient map). (b) Prove that is injective. (Hint:...
4. If G is a group, then it acts on itself by conjugation: If we let X = G (to make the ideas clearer), then the action is Gx X = (g, x) H+ 5-1xg E G. Equivalence classes of G under this action are usually called conjugacy classes. (a) If geG, what does it mean for x E X to be fixed by g under this action? (b) If x E X , what is the isotropy subgroup Gx...