Understanding Aut(Aut(Aut(C_73))) in Group Theory

  • Thread starter jimmycricket
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In summary, the conversation discusses the calculation of Aut(Aut(Aut(C_{73}))), which is shown to be isomorphic to Aut(C_2\times C_2\times C_2) \times Aut(C_3). The speaker initially thought this was equivalent to GL_3(\mathbb{F}_2), but is corrected and shown that Aut(C_{73}) = U(73) = C_{72}, and Aut(C_{72}) = U(2^3) × U(3^2) = C_2 × C_2 × C_2 × C_2 × C_3. The mistake is then corrected and it is acknowledged that the expression given is indeed correct.
  • #1
jimmycricket
116
2
Im calculating [itex]Aut(Aut(Aut(C_{73})))[/itex] and have got as far as [itex]Aut(Aut(Aut(C_{73})))\cong Aut(C_2\times C_2\times C_2) \times Aut(C_3)[/itex]
I thought this was the answer but I have been told that [itex]Aut(C_2\times C_2\times C_2) \times Aut(C_3)\cong GL_3(\mathbb{F}_2)[/itex]
Can someone explain this to me please.
 
  • #3
From what I get, [itex]Aut(C_{73}) = U(73) = C_{72}[/itex]. [itex]Aut(C_{72}) = U(72) = U(2^3) × U(3^2) = C_2 × C_2 × C_2 × C_2 × C_3 [/itex] (since [itex]|U(3^2)| = phi(9) = 6[/itex] and only abelian group of order 6 is [itex]C_2 × C_3[/itex]. So I think you're missing a factor of [itex]C_2[/itex] in your second Aut( ) expression.
 
  • #4
Sorry I just realized that I made a mistake. [itex] U(2^3) = C_2 × C_2 [/itex] so your expression is actually correct. My apologies.
 

1. What is "Finding Aut(Aut(Aut(C_73)))"?

"Finding Aut(Aut(Aut(C_73)))" is a mathematical concept that refers to finding the automorphism group of the automorphism group of the automorphism group of a group C_73. In simpler terms, it involves identifying all the possible ways to map a group onto itself, and then repeating this process three times.

2. Why is "Finding Aut(Aut(Aut(C_73)))" important?

Understanding the automorphism group of a group can provide valuable insights into its structure and properties. By finding the automorphism group of the automorphism group of the automorphism group of a group, we can gain a deeper understanding of the symmetries and transformations of that group.

3. How is "Finding Aut(Aut(Aut(C_73)))" done?

"Finding Aut(Aut(Aut(C_73)))" involves using mathematical techniques and algorithms to systematically determine all possible automorphisms of a group. This process may involve studying the group's elements, subgroups, and other properties to identify patterns and relationships.

4. What is the significance of "C_73" in "Finding Aut(Aut(Aut(C_73)))"?

"C_73" refers to a specific group, known as the cyclic group of order 73. This group has 73 elements, and each element can be mapped onto itself in different ways, leading to a large number of possible automorphisms. By studying this particular group, we can gain a better understanding of the concept of automorphism groups in general.

5. What are the applications of "Finding Aut(Aut(Aut(C_73)))"?

The concept of finding the automorphism group of a group has applications in various fields, including mathematics, computer science, and physics. It can be used to study the symmetries and transformations of different mathematical structures, such as graphs and networks. In computer science, it can help with data encryption and coding theory. In physics, it can aid in understanding the symmetries of physical systems and their interactions.

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