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Math Help - Quotient groups of cyclic groups

  1. #1
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    Quotient groups of cyclic groups

    "G is a cyclic group with normal subgroup N. Show that G/N is cyclic."

    Not sure what to do here. I know that all subgroups of a cyclic group are also cyclic. But the quotient group is confusing me here.
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  2. #2
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    Re: Quotient groups of cyclic groups

    Hey phys251.

    Check out this:

    Cyclic group - Wikipedia, the free encyclopedia

    and look at the property regarding prime numbers for the order of the groups
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  3. #3
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    Re: Quotient groups of cyclic groups

    OK thanks, I'll look into that. I just need to get more familiar with the definition of quotient groups.
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  4. #4
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    Re: Quotient groups of cyclic groups

    First of all, note that cyclic groups are of necessity abelian. So ANY subgroup is automatically normal.

    Since G is cyclic, there is some element g of G that generates G. I claim gN generates G/N.

    For suppose xN is ANY element of G/N. Since x is some element of G, we have x = g^k, for some integer k (since G is cyclic, with generator g).

    Thus: xN = g^kN = (gN)^k, which shows that G/N is generated by gN.
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