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Math Help - subgroups and cosets

  1. #1
    Junior Member
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    subgroups and cosets

    Show that the nonzero elements in Z_n form a group under the product [a][b]=[ab] if and only if n is a prime. I know that they definitely do not form a group when n is not prime but I am not sure how to prove otherwise. Please show steps. Thank you!
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  2. #2
    Super Member Gamma's Avatar
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    Iowa City, IA
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    Here is a start

    What you are trying to show is that \mathbb{Z}_p^{\times} is a group.

    Multiplication is well defined and that is pretty easy to show. just show that (a+pz)(b+py)=ab+apy+bpz+p^2yz = ab + p( ay + bz + pyz) \equiv ab (mod p)
    so it is closed under multiplication

    You just gotta show it has an inverse. But this too is easy, if 0 \not = a\in \mathbb{Z}_p, then you know p and a are relatively prime. Otherwise p would not be prime as a and p would share a divisor.

    but this means there exist integers x and y so that
    ax+py=1 but this tells you when you reduce mod p
    ax \equiv 1 (mod p)
    so x reduced mod p is the inverse of a. Thus you have shown it to be a group
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  3. #3
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    Thank you sooo much! Very helpful!
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