1. ## Automorphism

1) What is Aut(G) and how do you show that the automorphism is isomorphic to a group i.e. Aut(C5) =~ C4, and what is Aut(C9) is isomorphic to which cyclic group, is it group C8?

2) Describe Aut(Cn) explicitly for n = 24
How do you show that Aut(C12) =~ C2 * C2 and Aut(C14) =~ C6 ?. Find the correspoding isomorphism for Aut(C24)

Another question is: Does it mean that, C2 * C2 = C4
C4 * C4 = C16
C5 * C5 = C25
and so on ... so in general Cn * Cn = C(n*n). I dont know

Many thanks

2. Originally Posted by knguyen2005
1) What is Aut(G) and how do you show that the automorphism is isomorphic to a group i.e. Aut(C5) =~ C4, and what is Aut(C9) is isomorphic to which cyclic group, is it group C8?

2) Describe Aut(Cn) explicitly for n = 24
How do you show that Aut(C12) =~ C2 * C2 and Aut(C14) =~ C6 ?. Find the correspoding isomorphism for Aut(C24)

Another question is: Does it mean that, C2 * C2 = C4
C4 * C4 = C16
C5 * C5 = C25
and so on ... so in general Cn * Cn = C(n*n). I dont know

Many thanks

The automorphism of the cyclic group Z/nZ is $\displaystyle (Z/nZ)^{\times}$, which is of order $\displaystyle \phi(n)$ (link)

$\displaystyle \phi(9)=6$, thus Aut(C9) is not isomorphic to C(8).
$\displaystyle \phi(mn) = \phi(m)\phi(n)$, when gcd(m,n)=1.
$\displaystyle \phi(3*8)=\phi(3)\phi(8) = 2 *4 =8$, so the order of Aut(C24) should be 8.

A cyclic group is an abelian group by definition.

$\displaystyle C_{m} \times C_{n} = C_{mn}$, when gcd(m,n)=1. Thus $\displaystyle C_{4} \times C_{4} \neq C_{16}$

Note: Someone might give you a more thorough answer for this. This is my attempt to this problem.

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### automorphisam of c4

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