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Thread: Sylow's Theorem

  1. #1
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    Sylow's Theorem

    Let $\displaystyle |G|=p^{\alpha}m, \ p\nmid m$. Then $\displaystyle Syl_p(G)\neq\O$ that is $\displaystyle n_p\geq 1$ where $\displaystyle n_p$ number of Sylow p subgroups.

    Let G be a minimal counterexample.

    $\displaystyle p\mid |Z(G)| \ \ \text{or} \ \ p\nmid |Z(G)|$

    Why are me looking at p dividing or not dividing the center of G?
    Last edited by dwsmith; Nov 12th 2011 at 08:18 PM.
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  2. #2
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    Re: Sylow's Theorem

    Quote Originally Posted by dwsmith View Post
    Let $\displaystyle |G|=p^{\alpha}m, \ p\nmid m$. Then $\displaystyle Sly_p(G)\neq\O$ that is $\displaystyle n_p\geq 1$ where $\displaystyle n_p=|Sly_p(G)|$.

    Let G be a minimal counterexample.

    $\displaystyle p\mid |Z(G)| \ \ \text{or} \ \ p\nmid |Z(G)|$

    Why are we looking at p dividing or not dividing the center of G?
    what is your working definition of $\displaystyle Syl_p(G)$? also, is $\displaystyle n_p$ supposed to number of subgroups $\displaystyle Syl_p(G)$, or the cardinality of such a subgroup?

    without having a bit more context, i can only hazard a guess, but my thought is, you are looking at an inductive proof, and will eventually consider G/Z(G).
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  3. #3
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    Re: Sylow's Theorem

    I will just type up the whole proof but my question is still in regards to the center.

    proof:
    Let G be a minimal counterexample. Either $\displaystyle p||Z(G)| \ \text{or} \ p\nmid |Z(G)|$.

    Case 1: $\displaystyle p||Z(G)|$
    The center is abelian. By Cauchy's Theorem (Let G be a finite abelian group and $\displaystyle p||Z(G)|$, then G has an element of order p), there is an element $\displaystyle g\in Z(G)$ of order p.
    Set $\displaystyle <g>=H$. $\displaystyle H\trianglelefteq G$ since $\displaystyle g\in Z(G)$.
    $\displaystyle |G|=|G/H||H|=|G/H|p$ since the order of H is p.
    $\displaystyle |G|=p^{\alpha}m=|G/H|p$
    So $\displaystyle |G/H|<|G|$.
    Since G is a minimal counterexample, G\H has a subgroup of order $\displaystyle p^{\alpha-1}$ (WHY), say $\displaystyle K<G\H$.
    By the 4th Isomorphism Theorem, there exist $\displaystyle N\leq G$ such that $\displaystyle H\leq N$ and $\displaystyle N/H=K$.
    Then $\displaystyle |N/H||H|=|N|$ which is $\displaystyle p^{\alpha-1}p=p^{\alpha}$.
    Therefore, $\displaystyle N\in Syl_p(G)$

    Case 2: $\displaystyle p\nmid |Z(G)|$
    By the Class Equation, $\displaystyle p^{\alpha}m=|G|=|Z(G)|+\sum_{i=1}^r|G:C_G(g_i)|$.
    So $\displaystyle |Z(G)|=|G|-\sum_{i=1}^r|G:C_G(g_i)|$.
    p must not divide $\displaystyle |G:C_G(g_i)|$ for some i (WHY). By Lagrange's Theorem, $\displaystyle |G:C_G(g_i)||C_G(g_i)|=|G|$. Since p divides G and by Euclid's Lemma, $\displaystyle p||C_G(g_i)|$; moreover, $\displaystyle p^{\alpha}||C_G(g_i)|$.
    So $\displaystyle |C_G(g_i)|=p^{\alpha}m'$ where $\displaystyle m'<m$ otherwise $\displaystyle G=C_G(g_i)$ which contradicts the Class Equation.
    By induction, there exist $\displaystyle P\leq C_G(g_i)$ such that $\displaystyle |P|=p^{\alpha}$ so $\displaystyle P\in Syl_p(G)$
    Last edited by dwsmith; Nov 12th 2011 at 08:26 PM.
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  4. #4
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    Re: Sylow's Theorem

    ok, now we have some context.

    the reason why we look at |Z(G)|, is in order to use the class equation.

    specifically, if p does not divide |Z(G)|, then the sum of the centralizer indexes must not be divisible by p, either. because if the sum was, then

    since p divides |G|, and the sum of centralizer indexes, it would divide the difference, which is |Z(G)|, a contradiction.

    in turn, if the sum of centralizer indexes is not divisible by p, at least one centralizer index is not divisible by p.

    but since |G| = [G:C(g)]|C(g)| for any centralizer C(g), all the powers of p in the factorization of |G| are in |C(g)|.

    since C(g) is strictly less than G (since otherwise g is in Z(G)), it has a p-sylow subgroup, which in turn is a subgroup of G.

    however, if p divides Z(G), we can't use the class equation, because it might be that p divides the index of all our centralizers.

    so then we use cauchy's theorem for abelian groups to produce a normal subgroup of G, <g>, of order p, and consider G/<g>.

    since |G/<g>| < |G|, we can assume G/<g> has a p-sylow subgroup (since G is, by assumption, the minimal counter-example).

    this subgroup has order $\displaystyle p^{\alpha-1}$. we then use the 1-1 correspondence of subgroups of G/<g> and subgroups of G

    containing <g> (guaranteed by the 4th isomorphism theorem, which is a consequence of the 1st isomorphism theorem),

    to produce a subgroup of G containing <g> which has order $\displaystyle p^{\alpha-1}$ (this subgroup is N in the proof).

    N/<g> is isomorphic to the p-sylow subgroup of G/<g>, and |N| = |N/<g>||<g>| = $\displaystyle (p^{\alpha-1})(p) = p^\alpha$,

    which is the desired p-sylow subgroup of G.

    in either case, we get a contradiction: by assuming G is a minimal counter-example, we prove G is NOT a counter-example.

    therefore, there is no minimal counter-example, so every finite group G has a p-sylow subgroup.
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