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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

    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


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    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])
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