Results 1 to 3 of 3

Math Help - Isomorphisms

  1. #1
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,054
    Thanks
    368
    Awards
    1

    Isomorphisms

    Have you ever thought you knew something well and talked about it for years, only to suddenly find out one day you were wrong? (Shaking my head in despair!) I thought I knew without a shadow of a doubt what an isomorphism was...

    The problem: Let G be a group and Aut G the set of all automorphisms of G.
    a) Show Aut G is a group. (Easy. I did this.)
    b) Show Aut \, \mathbb{Z} \cong \mathbb{Z}_2 and Aut \, \mathbb{Z}_6 \cong \mathbb{Z}_2.

    Part b) is what's bothering me. Let's take the second case first since it typifies the problem I'm having.

    The automorphism is the set of all bijective maps \{f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_6\}. Now, we may think of the general element of this set essentially as an element of the set of permutations of {0,1,2,3,4,5}. This means the set Aut \, \mathbb{Z}_6 has 6! elements. The set \mathbb{Z}_2 has 2 elements.

    If Aut \, \mathbb{Z}_6 is isomorphic to \mathbb{Z}_2 then there must exist at least one bijection F:Aut \, \mathbb{Z}_6 \rightarrow \mathbb{Z}_2. But the two sets have a different number of members, so we can't define a bijection between them!

    Needless to say my argument must have a flaw in it somewhere. But where?

    I am having a similar problem with the first part using Aut \, \mathbb{Z}. I presume whatever I got wrong above is similar to what I've got wrong with this one.

    (This is what happens when a Physicist tries to teach himself Mathematics! )

    -Dan
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by topsquark
    Have you ever thought you knew something well and talked about it for years, only to suddenly find out one day you were wrong? (Shaking my head in despair!) I thought I knew without a shadow of a doubt what an isomorphism was...

    The problem: Let G be a group and Aut G the set of all automorphisms of G.
    a) Show Aut G is a group. (Easy. I did this.)
    b) Show Aut \, \mathbb{Z} \cong \mathbb{Z}_2 and Aut \, \mathbb{Z}_6 \cong \mathbb{Z}_2.

    Part b) is what's bothering me. Let's take the second case first since it typifies the problem I'm having.

    The automorphism is the set of all bijective maps \{f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_6\}. Now, we may think of the general element of this set essentially as an element of the set of permutations of {0,1,2,3,4,5}. This means the set Aut \, \mathbb{Z}_6 has 6! elements. The set \mathbb{Z}_2 has 2 elements.
    Aut \, \mathbb{Z}_6 is the set of all bijective maps \{f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_6\}
    which preserve the structure of  \mathbb{Z}_6. So it is not equivalent to
    the set of all permutations of {0,1,2,3,4,5}, as such a
    permutation need not preserve the required structure
    (persumably the field structure, or maybe the additve
    group structure?).

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,054
    Thanks
    368
    Awards
    1
    Quote Originally Posted by CaptainBlack
    Aut \, \mathbb{Z}_6 is the set of all bijective maps \{f: \mathbb{Z}_6 \rightarrow \mathbb{Z}_6\}
    which preserve the structure of  \mathbb{Z}_6. So it is not equivalent to
    the set of all permutations of {0,1,2,3,4,5}, as such a
    permutation need not preserve the required structure
    (persumably the field structure, or maybe the additve
    group structure?).

    RonL
    THERE we go! I knew I had to be forgetting something! Thanks!

    -Dan
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Isomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: October 28th 2010, 09:03 PM
  2. isomorphisms......
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 27th 2010, 10:11 PM
  3. Isomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 23rd 2009, 06:05 PM
  4. Z2 + Z2 + Z3 isomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 2nd 2008, 06:34 PM
  5. isomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 30th 2008, 06:20 PM

Search Tags


/mathhelpforum @mathhelpforum