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)?
Question
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)?
Solution
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.
Similar Questions
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}.)
Let G be a group and f: G→ G be a mapping defined by () = −1 ∀ x ∈ G. Then f is an automorphism if and only if i) G is commutative ii) G is non-commutative iii) G is a finite cyclic group iv) G≠ {e}, e is the identity element of G
Find all possible automorphism of group of integers under addition modulo 8.
Find the order of each element in Z/10Z. Hence find all subgroups of Z/10Z. [Hint: everysubgroup of a cyclic group is cyclic.]
For the group G = <Z19*, × > a. Find the order of the group. b. Find the order of each element in the group. c. Find the number of primitive roots in the group. d. Find the primitive roots in the group. e. Show that the group is cyclic.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.