automorphism group of a cyclic group

Theorem 1.

The automorphism groupMathworldPlanetmath of the cyclic groupMathworldPlanetmath Z/nZ is (Z/nZ)×, which is of order ϕ(n) (here ϕ is the Euler totient function).


Choose a generatorPlanetmathPlanetmathPlanetmath x for /n. If ρAut(/n), then ρ(x)=xa for some integer a (defined up to multiplesMathworldPlanetmathPlanetmath of n); further, since x generates /n, it is clear that a uniquely determines ρ. Write ρa for this automorphismPlanetmathPlanetmathPlanetmath. Since ρa is an automorphism, xa is also a generator, and thus a and n are relatively prime11 If they were not, say (a,n)=d, then (xa)n/d=(xa/d)n=1 so that xa would not generate.. Clearly, then, every a relatively prime to n induces an automorphism. We can therefore define a surjective map


Φ is also obviously injective, so all that remains is to show that it is a group homomorphism. But for every a,b(/n)×, we have


and thus



  • 1 Dummit, D., Foote, R.M., Abstract Algebra, Third Edition, Wiley, 2004.
Title automorphism group of a cyclic group
Canonical name AutomorphismGroupOfACyclicGroup
Date of creation 2013-03-22 18:42:35
Last modified on 2013-03-22 18:42:35
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 6
Author rm50 (10146)
Entry type Theorem
Classification msc 20A05
Classification msc 20F28