The order of an element in a group is the smallest positive integer n such that the element to the power of n equals the identity element. In the case of the automorphism group Aut(C11), the identity element is the identity function, which maps each element to itself.

The cyclic group C11 has 11 elements, and each automorphism in Aut(C11) is determined by where it sends the generator of C11. Since C11 is of order 11, which is prime, its automorphism group is isomorphic to the group of units of the ring of integers modulo 11, which is Z/10Z. This group has 10 elements, each represented by an integer from 1 to 10.

The element φ5 in Aut(C11) is the automorphism that sends the generator a of C11 to a^5. To find the order of φ5, we need to find the smallest positive integer n such that (a^5)^n = a, in the group C11.

Since 5 is relatively prime to 11, by Euler's theorem, we have a^(5*10) = a, in the group C11. Therefore, the smallest such positive integer n is 10.

So, the order of the element φ5 in Aut(C11) is 10.

Question

The order of an element in a group is the smallest positive integer n such that the element to the power of n equals the identity element. In the case of the automorphism group Aut(C11), the identity element is the identity function, which maps each element to itself.

The cyclic group C11 has 11 elements, and each automorphism in Aut(C11) is determined by where it sends the generator of C11. Since C11 is of order 11, which is prime, its automorphism group is isomorphic to the group of units of the ring of integers modulo 11, which is Z/10Z. This group has 10 elements, each represented by an integer from 1 to 10.

The element φ5 in Aut(C11) is the automorphism that sends the generator a of C11 to a^5. To find the order of φ5, we need to find the smallest positive integer n such that (a^5)^n = a, in the group C11.

Since 5 is relatively prime to 11, by Euler's theorem, we have a^(5*10) = a, in the group C11. Therefore, the smallest such positive integer n is 10.

So, the order of the element φ5 in Aut(C11) is 10.

Knowee AI · Accepted Answer

The order of an element in a group is the smallest positive integer n such that the element to the power of n equals the identity element. In the case of the automorphism group Aut(C11), the identity element is the identity function, which maps each element to itself.

The cyclic group C11 has 11 elements, and each automorphism in Aut(C11) is determined by where it sends the generator of C11. Since C11 is of order 11, which is prime, its automorphism group is isomorphic to the group of units of the ring of integers modulo 11, which is Z/10Z. This group has 10 elements, each represented by an integer from 1 to 10.

The element φ5 in Aut(C11) is the automorphism that sends the generator a of C11 to a^5. To find the order of φ5, we need to find the smallest positive integer n such that (a^5)^n = a, in the group C11.

Since 5 is relatively prime to 11, by Euler's theorem, we have a^(5*10) = a, in the group C11. Therefore, the smallest such positive integer n is 10.

So, the order of the element φ5 in Aut(C11) is 10.

Let 𝐶11 be the cyclic group of order 11, and consider its group of automorphisms Aut(𝐶11). What is the order of the element 𝜙5∈Aut(𝐶11)?

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