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Thread: Frobeniusgroup

  1. #1
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    Frobeniusgroup

    Let $\displaystyle F_{20}$ be the Frobeniusgroup of order $\displaystyle 20.$

    1) Show that $\displaystyle F_{20}$ has the trivial Center.
    2) Determine for every divisor $\displaystyle d$ of $\displaystyle 20$ the number of elements in $\displaystyle F_{20}$ of order $\displaystyle d.$
    3)Determine the number of 5-Sylowgroups in $\displaystyle F_{20}$ and teh number of 2-Sylowgroups in $\displaystyle F_{20}$ .

    4) Determine the normal subgroups of $\displaystyle F_{20}$ .
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  2. #2
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    The Frobeniusgroup of order 20
    is the set of maps from $\displaystyle F_5$ to $\displaystyle F_5$ of the form
    $\displaystyle x \mapsto ax+b$ where $\displaystyle a,b \in F_5, b \neq 0.$
    I'll answer your questions in
    slightly different order:

    2)
    d=1: 1 element, the identity $\displaystyle x$
    d=2: 5 elements, $\displaystyle -x+b, b \in F_5$
    d=4: 10 elements, $\displaystyle ax+b, \; a\in \{2,3\}, b \in F_5$
    d=5: 4 elements, $\displaystyle x+b, \; b \in F_5, b \neq 0$

    1)
    A presentation for $\displaystyle F_{20}$ is

    $\displaystyle
    \alpha^5 = \beta^2 = \gamma^4 = 1, \gamma^2 = \beta, \alpha^\beta=\alpha^{-1}, \alpha^\gamma=\alpha^2
    $

    We take
    $\displaystyle
    \alpha = x+1, \beta = -x, \gamma = 2x
    $
    The elements of order 5 and two together lie in the
    Dihedral group of generated by $\displaystyle \alpha$ and $\displaystyle \beta$ (of order 10).
    This group has trivial center. Hence $\displaystyle F_{20}$ has trivial center.

    3) There is only one Sylow 5 subgroup, the group
    generated by $\displaystyle \alpha = x+1$ (of order 5).
    There are $\displaystyle 5$ Sylow 2 subgroups (of order 4).
    They are the groups generated by $\displaystyle \gamma = 2x$ and its conjugates
    $\displaystyle 2x+b \; (b \neq 0)$.

    4) There are two normal subgroups, both mentioned already:
    The Sylow 5 subgroup generated by $\displaystyle \alpha$,
    and the Dihedral group of order 10 generated by $\displaystyle \alpha$ and $\displaystyle \beta.$

    Best,

    ZD
    Last edited by ZeroDivisor; Apr 26th 2009 at 06:17 AM. Reason: Latex formatting
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