Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calculating an automorphism

  1. Jan 16, 2015 #1
    Im doing a question where I have to calculate of composition of automorphisms of a cyclic p-group and something has got me confused. When constructing decompositions of cyclic groups I have gotten used to grouping the direct products of groups with orders of the same prime to a power e.g [itex]C_{20}\cong C_4\times C_5[/itex].
    In this question however I have gotten to a stage where accorging to my lecturer [itex]Aut(C_8)\times Aut(C_9)\cong C_2\times C_2 \times C_6[/itex] and i don't understand why. I would have expressed it has [itex]C_4\times C_6[/itex] since there are 4 numbers less than 8 that are coprime to 8 (eulers totient function). Can anyone help clear up my confusion?
     
  2. jcsd
  3. Jan 16, 2015 #2

    jbunniii

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    ##Aut(C_8)## is not a cyclic group.

    If we write ##C_8## additively, with elements ##\{0,1,2,3,4,5,6,7\}##, then there are four possible generators: ##1,3,5,7##. Any automorphism ##\phi \in Aut(C_8)## is determined completely by ##\phi(1)##, and there are four possibilities: ##\phi## can map ##1## to any of ##1,3,5,7##. So ##Aut(C_8)## has four elements.

    Expressed in cycle notation, ##Aut(C_8)## consists of these four elements:
    $$\phi_1 = \text{identity}$$
    $$\phi_2 = (0)(1 3)(2 6)(4)(5 7)$$
    $$\phi_3 = (0)(1 5)(2)(3 7)(4)(6)$$
    $$\phi_4 = (0)(1 7)(2 6)(3 5)(4)$$
    The orders of the non-identity elements are all ##2##, so ##Aut(C_8)## cannot be ##C_4##. Since ##C_2 \times C_2## is (up to isomorphism) the only other group of order 4, by process of elimination, ##Aut(C_8)## must be ##C_2 \times C_2##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Calculating an automorphism
  1. Automorphism Group (Replies: 13)

  2. Automorphism Group (Replies: 20)

  3. Automorphisms of Z_3 (Replies: 3)

  4. Automorphism groups (Replies: 7)

  5. Inner Automorphism (Replies: 1)

Loading...