Find explict automorphisms generating Aut(𝐶37 ) as a direct product of cyclic groups We know that the automorphism group of the cyclic group 𝐶37=⟨𝑥∣𝑥37=1⟩ is isomorphic to a direct product of cyclic groups. If 𝜙𝑟:𝐶37→𝐶37 is the homomorphism 𝑥↦𝑥𝑟 find a minimal list of generators for Aut(𝐶37)=⟨𝜙𝑟1,…,𝜙𝑟𝑡⟩ . (Your answer should be a set of one or more integers. For example if Aut(𝐶37)=⟨𝜙3,𝜙7⟩ then enter {3,7} .)
Question
Find explict automorphisms generating Aut(𝐶37 ) as a direct product of cyclic groups
We know that the automorphism group of the cyclic group 𝐶37=⟨𝑥∣𝑥37=1⟩ is isomorphic to a direct product of cyclic groups.
If 𝜙𝑟:𝐶37→𝐶37 is the homomorphism 𝑥↦𝑥𝑟 find a minimal list of generators for Aut(𝐶37)=⟨𝜙𝑟1,…,𝜙𝑟𝑡⟩ .
(Your answer should be a set of one or more integers. For example if Aut(𝐶37)=⟨𝜙3,𝜙7⟩ then enter {3,7} .)
Solution
The automorphism group of a cyclic group of order n, Aut(C_n), is isomorphic to the group of units modulo n, (Z/nZ)^*.
The group of units modulo 37, (Z/37Z)^*, is a cyclic group of order 36.
The generators of a cyclic group of order 36 are the integers k such that gcd(k, 36) = 1.
Therefore, the generators of Aut(C_37) are the integers r such that the homomorphism φ_r : C_37 → C_37 defined by x ↦ x^r is an automorphism.
The integers r that satisfy this condition are precisely the integers r such that gcd(r, 37) = 1.
Since 37 is a prime number, the integers r that satisfy this condition are all the integers from 1 to 36.
Therefore, a minimal list of generators for Aut(C_37) is {1, 2, 3, ..., 36}.
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