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Math Help - Number of elements and subgroups

  1. #1
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    Number of elements and subgroups

    I have a question that I feel I am going about in a roundabout way, and would like some help on. I am preparing for an exam.


    Problem: Let G be a group with |G|=150. Let H be a non-normal subgroup in G with |H|=25.


    (a) How many elements of order 5 does G have?


    (b) How many elements of order 25 does G have?


    My attempt:


    G has 6 subgroups of order 25 because the number of subgroups of order 25 has to divide $150/25=6$ and $6 \cong 1 (\mod 5)$. So there are 6 such subgroups: $ \{H_1,H_2,H_3,H_4,H_5,H_6\}$. Let $G$ act on this set by conjugation. The permutation representation of the group action of conjugation on this set is $\phi:G \rightarrow S_6$, and $|S_6|=720$. Also, $|\ker(\phi)||im(\phi)|=150$. Because $|im(\phi)|$ has at most one factor of $5$, $5$ divides $ |\ker(\phi)|$. Suppose $25$ divides $ |\ker(\phi)|$. Then $ \ker(\phi)$ has a Sylow 5-subgroup $H_i$. So $ H_i \subset \ker(\phi) \implies aH_ia^{-1} \subset a(\ker(\phi))a^{-1} = \ker(\phi)$. By the Second Sylow Theorem, $ H_1 \cup \ldots \cup H_6 \subset \ker(\phi) \implies G = \ker(\phi) \implies \phi$ trivial. So $\ker(\phi)$ has a subgroup $K$ with $|K|=5$. So for some $ H_i, \ker(\phi) \cap H_i=K \implies a(\ker(\phi))a^{-1} \cap aH_ia^{-1} = aKa^{-1}$


    What next? Surely this shouldn't be so long-winded.
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  2. #2
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    Re: Number of elements and subgroups

    Hi,
    How's this for an answer? If G is of order 150 with a non-normal subgroup of 25, then G has 237 elements of order 5!! This is a true statement because there is no group of order 150 with a non-normal subgroup of order 25.

    Let G be a group of order 150. Then the Sylow 5 subgroup of G is normal.
    Proof.
    The order of G is 75*2. So G has a normal subgroup N of order 75 -- see Huppert Endliche Gruppen I, page 30 or prove it yourself.
    If you don't read German, I could supply the proof if necessary.
    Thus by Sylow, N has a normal Sylow 5 subgroup which is then normal in G.

    By the way, everything you say is true. This is typically the way you analyze groups of small order.
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  3. #3
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    Re: Number of elements and subgroups

    Thanks for the reply. I am not able to find the book online, so if you could supply the proof it would be great. I have a lot of other material to study for this exam anyway.

    By the way, I understand that G has a normal subgroup N of order 75 because every subgroup of index 2 is normal.
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  4. #4
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    Re: Number of elements and subgroups

    Hi,
    I hope the attachment answers all your questions. By the way, you're right that any subgroup of index 2 in a group is normal. But there need not be any subgroup of index 2. Example: A5 has order 60, but has no subgroup of order 30 since A5 is simple.

    Number of elements and subgroups-mhfgroups20.png
    Thanks from abscissa
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