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September 10th, 2017, 12:15 PM   #1
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action of Frobenius group

Let $p$ be a prime number. Show that the Frobenius group $F_{p(p-1)}$ acts on the set $\mathbb{F}_p$ by:

$$\begin{pmatrix}a&b \\\ 0&1\end{pmatrix}x:=ax+b$$
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September 10th, 2017, 01:11 PM   #2
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$\displaystyle \left(\begin{pmatrix}a_1&b_1 \\\ 0&1\end{pmatrix}\begin{pmatrix}a_2&b_2 \\\ 0&1\end{pmatrix}\right)x = \begin{pmatrix}a_1a_2&a_1b_2+b_1 \\\ 0&1\end{pmatrix}x=
a_1a_2x+a_1b_2+b_1.$

$\displaystyle \left(\begin{pmatrix}a_1&b_1 \\\ 0&1\end{pmatrix}\begin{pmatrix}a_2&b_2 \\\ 0&1\end{pmatrix}\right)x=a_1(a_2x+b_2)+b_1.$
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September 10th, 2017, 01:49 PM   #3
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And how does that answer the quetion? Can you explain to me more? Thanks in advance.

Last edited by skipjack; September 11th, 2017 at 02:56 PM.
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September 10th, 2017, 03:36 PM   #4
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The set of polynomials $\displaystyle ax+b$, where $\displaystyle a,b\in\mathbb{F}_p$ and $\displaystyle a\neq 0$, is a group (the operation is composition $\displaystyle \circ$). The group acts on $\displaystyle \mathbb{F}_p$:

$\displaystyle ax+b:\ t\mapsto at+b, \ \ \ t\in \mathbb{F}_p$.

And $\displaystyle (ax+b)(\mathbb{F}_p)=\mathbb{F}_p$. It permutes the elements of $\displaystyle \mathbb{F}_p$.

One has isomorphism $\displaystyle \varphi$ from the group of matrices $\displaystyle \begin{pmatrix}a&b \\\ 0&1\end{pmatrix}$ to the group of $\displaystyle ax+b$:

$\displaystyle \varphi:\ \begin{pmatrix}a&b \\\ 0&1\end{pmatrix} \mapsto ax+b$.

$\displaystyle \varphi\left(\begin{pmatrix}a_1&b_1 \\\ 0&1\end{pmatrix}\begin{pmatrix}a_2&b_2 \\\ 0&1\end{pmatrix}\right)=a_1a_2x+a_1b_2+b_1=a_1(a_2 x+b_2)+b_1= (a_1x+b_1)\circ(a_2x+b_2)=
$
$\displaystyle
=\varphi\left(\begin{pmatrix}a_1&b_1 \\\ 0&1\end{pmatrix}\right)
\circ\varphi\left(\begin{pmatrix}a_2&b_2 \\\ 0&1\end{pmatrix}\right).
$
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September 10th, 2017, 08:09 PM   #5
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Thanks for the explanation. and is it possible to show that this action is sharply 2-transitive i.e. if $x_1,x_2\in\mathbb{F}_P$ are distinct and $x_1,x_2\in\mathbb{F}_P$ are distinct then there is exactly one $g\in F_{p(p-1)}$ such that $gx_{1}=y_1$ and $gx_{2}=y_2$ ?

Last edited by skipjack; September 11th, 2017 at 02:56 PM.
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September 11th, 2017, 05:10 AM   #6
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Let $\displaystyle g=\begin{pmatrix}a&b \\\ 0&1\end{pmatrix}$.
If $\displaystyle gx_{1}=y_1$ and $\displaystyle gx_2=y_2$ then a,b have to be the solution of the system
$\displaystyle \begin{cases}
x_1\cdot a + b = y_1,\\
x_2\cdot a +b = y_2.
\end{cases}$
This system has unique solution. And $\displaystyle a\neq 0$, otherwise we would have $\displaystyle y_1=y_2$.

Let $\displaystyle x_3\in \mathbb{F}_p$ and $\displaystyle x_3\neq x_1,x_2$.
$\displaystyle g$ necessarily sends x_3 to $\displaystyle ax_3+b$, not to an arbitrary element one could choose.
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September 11th, 2017, 12:00 PM   #7
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In fact, I have a quiz next Friday and I tried to identify the center $Z(F_{p(p-1)})$ and $F'_{p(p-1)}=[F_{p(p-1)},F_{p(p-1)}]$ with familiar groups but I didn't manage to do that. Can you please help me? I will be grateful if you could help me.

Last edited by skipjack; September 11th, 2017 at 02:53 PM.
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September 12th, 2017, 03:09 AM   #8
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Let $\displaystyle H=\left\{ \begin{pmatrix}1&b \\\ 0&1\end{pmatrix}\ \bigg| \ b \in \mathbb{F}_p \right\}$.

$\displaystyle H$ is a normal subgroup in $\displaystyle F_{p(p-1)}$. The order of $\displaystyle H$ is equal to $\displaystyle p$.

Find the commutator of $\displaystyle \begin{pmatrix} a&b \\\ 0&1\end{pmatrix},\begin{pmatrix}c&d \\\ 0&1\end{pmatrix}\in F_{p(p-1)}$:

$\displaystyle \begin{pmatrix} a&b \\\ 0&1\end{pmatrix}\begin{pmatrix}c&d \\\ 0&1\end{pmatrix}\begin{pmatrix} \frac{1}{a} & -\frac{a}{b} \\\ 0&1\end{pmatrix}\begin{pmatrix}\frac{1}{c}& -\frac{d}{c} \\\ 0&1\end{pmatrix} = \begin{pmatrix} ac& ad+b \\\ 0&1\end{pmatrix}\begin{pmatrix}\frac{1}{ac}& -\frac{d}{ac}-\frac{b}{a} \\\ 0&1\end{pmatrix}=\begin{pmatrix} 1& \ldots \\\ 0&1\end{pmatrix}$.

So, $\displaystyle H$ contains every commutator of $\displaystyle F_{p(p-1)}$. If a group is not commutative, its commutator subgroup is not trivial.

The commutator subgroup $\displaystyle [F_{p(p-1)},F_{p(p-1)}]$ is a subgroup of $\displaystyle H$. The order of $\displaystyle [F_{p(p-1)},F_{p(p-1)}]$ is a divisor of the order of $\displaystyle H$ which is prime. So, if $\displaystyle p\gt 2$ the order of the commutator group is equal to $\displaystyle p$, and
$\displaystyle [F_{p(p-1)},F_{p(p-1)}]=H$.

If $\displaystyle p=2$ then $\displaystyle F_{p(p-1)}$ is commutative and its commutator subgroup is trivial.

Last edited by ABVictor; September 12th, 2017 at 03:14 AM.
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September 12th, 2017, 07:01 AM   #9
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and can us identify $F_{p(p-1)}/F'_{p(p-1)}$ with a familiar group?
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September 12th, 2017, 11:33 AM   #10
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The quotient group $\displaystyle F_{p(p-1)}/F'_{p(p-1)}$ consists of (p-1) congruence classes $\displaystyle aH,\ a\in \mathbb{F}_p, \ H=F'_{p(p-1)}$.

The classes are $\displaystyle \begin{pmatrix} a&0 \\\ 0&1\end{pmatrix}H=\left\{ \begin{pmatrix} a&b \\\ 0&1\end{pmatrix} \ \bigg| \ b\in\mathbb{F}_p \right\}$.

The operation in the quotient group:
$\displaystyle \begin{pmatrix} a_1&0 \\\ 0&1\end{pmatrix}H \cdot \begin{pmatrix} a_2&0 \\\ 0&1\end{pmatrix}H = \left(\begin{pmatrix} a_1&0 \\\ 0&1\end{pmatrix}\cdot\begin{pmatrix} a_2&0 \\\ 0&1\end{pmatrix}\right)H=\begin{pmatrix} a_1a_2&0 \\\ 0&1\end{pmatrix}H $

The quotient group is isomorphic to the multiplicative group of field $\displaystyle \mathbb{F}_p$:

$\displaystyle \varphi_2: a \mapsto \begin{pmatrix} a&0 \\\ 0&1\end{pmatrix}H, \ \ a\in \mathbb{F}_p, \ a\neq 0. $
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