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Isomorphic groups

  1. Dec 1, 2009 #1
    Hi guys just a quick question on how I would go about showing the units of Z21 is isomorphic toC2*C6(cyclic groups).I have done out the multiplicative table but they seem to be different to me. What else can I do???
     
  2. jcsd
  3. Dec 1, 2009 #2
    Start by proving that [tex]\mathbb{Z}_{21} \simeq \mathbb{Z}_3 \times \mathbb{Z}_7[/tex]. What are the groups of units of [tex]\mathbb{Z}_3[/tex] and [tex]\mathbb{Z}_7[/tex]?

    (Hint: it's important that GCD(3,7)=1.)
     
  4. Dec 2, 2009 #3
    Sorry Rochfor,
    I actually cant prove that, I know it should be true as gcd(3,7)=1.
    A little more help please?
    Thanks.
     
  5. Dec 2, 2009 #4
    Try proving that the group on the right is cyclic.
     
  6. Dec 2, 2009 #5
    Jeez i cant even do that.
    Im having a terrible day with this.
     
  7. Dec 2, 2009 #6
    So we want to show that every element of [tex]\mathbb{Z}_3 \times \mathbb{Z}_7[/tex] is of the form [tex]n \cdot ( [1]_3, [1]_7 )[/tex]. So for [tex]x, y \in \mathbb{Z}[/tex], we want [tex]( [x]_3, [y]_7 ) = n \cdot ( [1]_3, [1]_7 ) = ( [n]_3, [n]_7 )[/tex]. So we need to find a number n so that [tex]x \equiv n \mod 3[/tex] and [tex]y \equiv n \mod 7[/tex]. The http://mathworld.wolfram.com/ChineseRemainderTheorem.html" [Broken] is your friend.
     
    Last edited by a moderator: May 4, 2017
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