1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Why a group is not a direct or semi direct product

  1. Oct 4, 2015 #1
    1. The problem statement, all variables and given/known data
    Good day all!

    (p.s I dont know why every time I type latex [ tex ] ...... [ / tex ] a new line is started..sorry for this being so "spread" out)

    So I was wondering if my understanding of this is correct:

    The Question asks: "[tex]\mathbb{Z}_4[/tex] has a subgroup is isomorphic to [tex]\mathbb{Z}_2[/tex] The quotient [tex]\mathbb{Z}_4/ \mathbb{Z}_2[/tex] is also isomorphic to [tex]\mathbb{Z}_2[/tex]Nevertheless, [tex]\mathbb{Z}_4[/tex] is not a direct or semidirect product of two copies of [tex]\mathbb{Z}_2[/tex]. Why?

    2. Relevant equations
    None

    3. The attempt at a solution
    Well: Its pretty easy to see that [tex]\mathbb{Z}_{4}\neq \mathbb{Z}_{2}\oplus \mathbb{Z}_{2} [/tex] becuase 2 is not relatively prime to 2. Thus, Z4 isnt a direct product.

    For the semi direct product: since [tex]\mathbb{Z}_{2}\cap \mathbb{Z}_{2}\neq e[/tex] (where e is the identity), Z4 is not a semi direct produt of two copies of Z2. Is this correct? (and sufficient?)
     
    Last edited: Oct 4, 2015
  2. jcsd
  3. Oct 4, 2015 #2

    Orodruin

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Because this is the definition of that tag. Use [ itex] or double hashes instead if you want to write equations in text.

    ##\mathbb Z_4## is not isomorphic to ##\mathbb Z_2##. You cannot have a bijection between sets of different cardinality.

    Can you think of a property of any element in ##\mathbb Z_4## which none of the products reproduce?
     
  4. Oct 4, 2015 #3
    [1] in Z4 has order 4
     
    Last edited: Oct 4, 2015
  5. Oct 4, 2015 #4
    I miss typed the question. I meant to say Z4 has a subgroup isomorphic to Z2 (obviously subgroup generated by [2])
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Why a group is not a direct or semi direct product
  1. Semi Direct Product (Replies: 1)

Loading...