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)?

The order of an element in a group is the smallest positive integer n such that the element raised to the power of n equals the identity element.

In the case of the cyclic group C11, the identity element is 1.

The automorphism group Aut(C11) consists of all bijective functions from C11 to itself that preserve the group operation.

The element φ5 is such a function, and we want to find the smallest positive integer n such that φ5^n = 1.

Since C11 is a cyclic group of order 11, it is generated by a single element, say g. Then φ5(g) is also a generator of C11.

The order of φ5 is the smallest positive integer n such that (φ5(g))^n = g^n = 1.

Since the order of g in C11 is 11, the order of φ5 must divide 11.

The only divisors of 11 are 1 and 11, so the order of φ5 is either 1 or 11.

If the order of φ5 were 1, then φ5 would be the identity automorphism, which it is not.

Therefore, the order of φ5 must be 11.