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Math Help - Sylows Theorems

  1. #1
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    Sylows Theorems

    Hi got this question where I need to use sylows theorems (or one of them) to determine the numbers of sylow 11-subgroups and sylow 109-subgroups for a group G of order 11990 ( =2*5*11*109).

    Where do I start with this. Which of sylows theorems do i use and ultimately is it possible to show that G is not simple.
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  2. #2
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    Quote Originally Posted by moolimanj View Post
    Hi got this question where I need to use sylows theorems (or one of them) to determine the numbers of sylow 11-subgroups and sylow 109-subgroups for a group G of order 11990 ( =2*5*11*109).

    Where do I start with this. Which of sylows theorems do i use and ultimately is it possible to show that G is not simple.
    Let n be number of Sylow 11-subgroups. Then n\equiv 1(\bmod 11) and n|11990. The only possibility is n=1. Thus there is only one Sylow subgroup P. But aPa^{-1} is also a Sylow subgroup since it is unique it means P = aPa^{-1} thus P is normal.
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  3. #3
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    So, do I just show that the group is normal and use the same process for the 109 group?
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  4. #4
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    But isn't 2 x 5 x 109 = 1090 \equiv 1 (mod 11)

    and

    2 x 5 x 11 = 110 \equiv 1 (mod 109)
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  5. #5
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    Yeah - thats interesting. Does that imply that there are more than one sylow subgroup for each?

    PerfectHacker - can you clarify your working/thinking please?
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