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Math Help - Help using lagrange....

  1. #1
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    Help using lagrange....

    Hi I have this problem, and I cant seem to get it. It is not for a grade just for study practice. Let me know if you can do it. Thanks...

    Let G be a group, |G|= p^{t}m for a p a prime and (p,m) =1 and H \le G with |H|= p^{t}. If K \le G with |K|= p^{s} and HK = KH, then show K \le H.

    Need to show K is a subgroup of H. Thanks again.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ElieWiesel View Post
    Hi I have this problem, and I cant seem to get it. It is not for a grade just for study practice. Let me know if you can do it. Thanks...

    Let G be a group, |G|= p^{t}m for a p a prime and (p,m) =1 and H \le G with |H|= p^{t}. If K \le G with |K|= p^{s} and HK = KH, then show K \le H.

    Need to show K is a subgroup of H. Thanks again.
    What are you having trouble with exactly? What points aren't clear? Do you see why it is significant that (p,m)=1?
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  3. #3
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    Okay well maybe if I can explain what I have so far, maybe It will help.

    HK= HK implies that HK \le G

    I also know that the |HK| = |H||K|/|H \cap K|

    I also know that |G| = |H| [G:H] which seems to imply that [G:H] = m by lagrange

    Also I know that |HK| divides |G| by lagrange

    Im thinking I have all the pieces put I dont know how to put them together right.
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