#5 pls help 3.Find Aut(K) where K is the Klein four-group.What about Aut (Z₂ × Z₂)?4. Show that the group of automorphism of D4 is of order 8.5. Show that Aut(Z₂ × Z₁) consists of eight elements sending(1,0) to (1,0) or (1,2) and (0, 1) to (0, 1), (0, 3), (1, 1), or (1, 3).Show that Aut (Z₂ × Z3) ≈ Aut (Z₂) × Aut (Z3).6.~In the following problems 7 and 8, you may invoke Theorem2.1, Chapter 8.7.Let A be a noncyclic finite abelian group. Prove that Aut (4) isnot abelian.8.Let G be a finite group such that |Aut(G)| = p. Prove |G| ≤ 3.(Hint: Note G is abelian)

The group Aut(ℤ2×ℤ4) consists of all automorphisms of the direct product ℤ2×ℤ4. An automorphism of a…

Answered: 5. Show that Aut (Z₂ × Z₁) consists of…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 3E: Exercises 3. Find an isomorphism from the additive group to the multiplicative group of units...

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5. Show that Aut (Z₂ × Z₁) consists of eight elements sending (1,0) to (1,0) or (1,2) and (0, 1) to (0, 1), (0, 3), (1, 1), or (1, 3).