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).

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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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#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 Theorem
2.1, Chapter 8.
7.
Let A be a noncyclic finite abelian group. Prove that Aut (4) is
not abelian.
8.
Let G be a finite group such that |Aut(G)| = p. Prove |G| ≤ 3.
(Hint: Note G is abelian)
Transcribed Image Text: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 Theorem 2.1, Chapter 8. 7. Let A be a noncyclic finite abelian group. Prove that Aut (4) is not abelian. 8. Let G be a finite group such that |Aut(G)| = p. Prove |G| ≤ 3. (Hint: Note G is abelian)
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