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What is the action of a primitive permutation group of type SD?

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1

$begingroup$

My goal is to show that $Alt(5) wr Sym(3)$ is a primitive permutation group of type SD.

Let $G$ be a primitive permutation group, $N$ a minimal normal subgroup, and $H$ a point stabilizer (so $NH = G$). $G$ is of type SD if

1. 
   $N$ is non-abelian and the unique minimal normal subgroup (so $N cong T^k$, where $T$ is a non-abelian simple group)


2. 
   $N cap H$ is the diagonal subgroup $(t, t, ..., t): t in T$
   

It follows (as claimed here) that $N$ acts on the cosets of $N cap H$ in $N$, where $(t_1, ..., t_k)$ takes the coset of $(s_1, ..., s_k)$ to the coset of $(t_k^-1s_1t_1, ..., 1)$, so we identify $Omega$ (the underlying set) with $T^k-1$. And we identify $G$ with a subgroup of $Aut(T^k) cong Aut(T) wr Sym(k)$.

In the case $N = Alt(5)^3$, $G =Alt(5) wr Sym(3)$ is in the form I want. The only thing I am struggling to understand is the action of this group.

1. Is the action of $NH$ on the cosets of $H$ isomorphic to the action of $N$ on the cosets of $N cap H$?




2. If the underlying set $Omega$ is isomorphic to $Alt(5)^2$, how can I explicitly describe the action of $Alt(5) wr Sym(3)$?

I am not sure how to answer the $2$nd question without assuming the first. For the first question, (since we do not assume $H$ is normal), I defined a natural bijection between $NH/H$ and $N / N cap H$, but I could not show that this bijection respects the action.

abstract-algebra group-theory group-actions

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asked Mar 22 at 8:09

vxnturevxnture

40410

$endgroup$

add a comment | 

1

$begingroup$

My goal is to show that $Alt(5) wr Sym(3)$ is a primitive permutation group of type SD.

Let $G$ be a primitive permutation group, $N$ a minimal normal subgroup, and $H$ a point stabilizer (so $NH = G$). $G$ is of type SD if

1. 
   $N$ is non-abelian and the unique minimal normal subgroup (so $N cong T^k$, where $T$ is a non-abelian simple group)


2. 
   $N cap H$ is the diagonal subgroup $(t, t, ..., t): t in T$
   

It follows (as claimed here) that $N$ acts on the cosets of $N cap H$ in $N$, where $(t_1, ..., t_k)$ takes the coset of $(s_1, ..., s_k)$ to the coset of $(t_k^-1s_1t_1, ..., 1)$, so we identify $Omega$ (the underlying set) with $T^k-1$. And we identify $G$ with a subgroup of $Aut(T^k) cong Aut(T) wr Sym(k)$.

In the case $N = Alt(5)^3$, $G =Alt(5) wr Sym(3)$ is in the form I want. The only thing I am struggling to understand is the action of this group.

1. Is the action of $NH$ on the cosets of $H$ isomorphic to the action of $N$ on the cosets of $N cap H$?




2. If the underlying set $Omega$ is isomorphic to $Alt(5)^2$, how can I explicitly describe the action of $Alt(5) wr Sym(3)$?

I am not sure how to answer the $2$nd question without assuming the first. For the first question, (since we do not assume $H$ is normal), I defined a natural bijection between $NH/H$ and $N / N cap H$, but I could not show that this bijection respects the action.

abstract-algebra group-theory group-actions

share|cite|improve this question

asked Mar 22 at 8:09

vxnturevxnture

40410

$endgroup$

add a comment | 

$begingroup$

My goal is to show that $Alt(5) wr Sym(3)$ is a primitive permutation group of type SD.

Let $G$ be a primitive permutation group, $N$ a minimal normal subgroup, and $H$ a point stabilizer (so $NH = G$). $G$ is of type SD if

1. 
   $N$ is non-abelian and the unique minimal normal subgroup (so $N cong T^k$, where $T$ is a non-abelian simple group)


2. 
   $N cap H$ is the diagonal subgroup $(t, t, ..., t): t in T$
   

It follows (as claimed here) that $N$ acts on the cosets of $N cap H$ in $N$, where $(t_1, ..., t_k)$ takes the coset of $(s_1, ..., s_k)$ to the coset of $(t_k^-1s_1t_1, ..., 1)$, so we identify $Omega$ (the underlying set) with $T^k-1$. And we identify $G$ with a subgroup of $Aut(T^k) cong Aut(T) wr Sym(k)$.

In the case $N = Alt(5)^3$, $G =Alt(5) wr Sym(3)$ is in the form I want. The only thing I am struggling to understand is the action of this group.

1. Is the action of $NH$ on the cosets of $H$ isomorphic to the action of $N$ on the cosets of $N cap H$?




2. If the underlying set $Omega$ is isomorphic to $Alt(5)^2$, how can I explicitly describe the action of $Alt(5) wr Sym(3)$?

I am not sure how to answer the $2$nd question without assuming the first. For the first question, (since we do not assume $H$ is normal), I defined a natural bijection between $NH/H$ and $N / N cap H$, but I could not show that this bijection respects the action.

abstract-algebra group-theory group-actions

share|cite|improve this question

asked Mar 22 at 8:09

vxnturevxnture

40410

$endgroup$

share|cite|improve this question

asked Mar 22 at 8:09

vxnturevxnture

40410

share|cite|improve this question

asked Mar 22 at 8:09

vxnturevxnture

40410

asked Mar 22 at 8:09

vxnturevxnture

40410

asked Mar 22 at 8:09

vxnturevxnture

40410