**Question 5: Automorphisms of Klein's 4-group**How many automorphisms does Klein’s 4-group have?**Explanation:**Klein's 4-group, often denoted as $ V_4 $, is a group with four elements, typically $\{ e, a, b, c \}$, where $ e $ is the identity element. It has the structure of a direct product of two cyclic groups of order 2. The group is abelian, with each element other than the identity having order 2.An *automorphism* is a bijective homomorphism from a group to itself. For Klein’s 4-group, determining the number of automorphisms involves finding the number of bijective homomorphisms that map the group onto itself while preserving the group operation.**Detailed Explanation:**- **Structure:** The group has three elements of order 2 and an identity element. Each non-identity element generates a subgroup of order 2.- **Automorphism Group:** The automorphism group $\text{Aut}(V_4)$ is isomorphic to the symmetric group $ S_3 $, which is the group of all permutations of three objects. This results because each automorphism is determined by the image of the non-identity elements, which can be permuted in $ 3! = 6 $ ways.Therefore, Klein’s 4-group has 6 automorphisms.

No of automorphism of Klein's 4-group: K4={e,a,b,c} f1=eabceabc = I f2=eabceacb = bc f3=eabcecba =…

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5. How many automorphisms does Klein's 4-group have?

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

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Chapter2: Second-order Linear Odes

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