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Transitive groups that are not primitive
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I'm trying to seek some examples of transitive groups that are not primitive. I found out this article Generating Symmetric Groups where is shown as example the group $G subset S_6$ induced by rotations of the cube.
Following the line of the above example I thought also the group induced by rotations of the square does our business, and thus the whole $D_4$ (rotations and reflections of the square) group is a good example.
Can anybody know other toy-examples?
abstract-algebra group-theory
asked Aug 2 at 13:41
Davide Motta
1286
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up vote
3
down vote
favorite
I'm trying to seek some examples of transitive groups that are not primitive. I found out this article Generating Symmetric Groups where is shown as example the group $G subset S_6$ induced by rotations of the cube.
Following the line of the above example I thought also the group induced by rotations of the square does our business, and thus the whole $D_4$ (rotations and reflections of the square) group is a good example.
Can anybody know other toy-examples?
abstract-algebra group-theory
asked Aug 2 at 13:41
Davide Motta
1286
3
By definition, subgroup of $S_4$ that preserves $1,2,3,4$ seems to be the minimal example.
– MigMit
Aug 2 at 13:45
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I'm trying to seek some examples of transitive groups that are not primitive. I found out this article Generating Symmetric Groups where is shown as example the group $G subset S_6$ induced by rotations of the cube.
Following the line of the above example I thought also the group induced by rotations of the square does our business, and thus the whole $D_4$ (rotations and reflections of the square) group is a good example.
Can anybody know other toy-examples?
abstract-algebra group-theory
asked Aug 2 at 13:41
Davide Motta
1286
I'm trying to seek some examples of transitive groups that are not primitive. I found out this article Generating Symmetric Groups where is shown as example the group $G subset S_6$ induced by rotations of the cube.
Following the line of the above example I thought also the group induced by rotations of the square does our business, and thus the whole $D_4$ (rotations and reflections of the square) group is a good example.
Can anybody know other toy-examples?
abstract-algebra group-theory
asked Aug 2 at 13:41
Davide Motta
1286
asked Aug 2 at 13:41
Davide Motta
1286
asked Aug 2 at 13:41
Davide Motta
1286
asked Aug 2 at 13:41
Davide Motta
1286
1286
3
By definition, subgroup of $S_4$ that preserves $1,2,3,4$ seems to be the minimal example.
– MigMit
Aug 2 at 13:45
add a comment |Â
3
By definition, subgroup of $S_4$ that preserves $1,2,3,4$ seems to be the minimal example.
– MigMit
Aug 2 at 13:45
3
3
By definition, subgroup of $S_4$ that preserves $1,2,3,4$ seems to be the minimal example.
– MigMit
Aug 2 at 13:45
By definition, subgroup of $S_4$ that preserves $1,2,3,4$ seems to be the minimal example.
– MigMit
Aug 2 at 13:45
add a comment |Â
1 Answer
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Any subgroup of $Sym(n)$ of the form $Sym(k)wr Sym(l)$ where $n=kl$ fits the criteria (this comes from the O'Nan-Scott Theorem). This construction is known as the wreath product. Concretely this is the stabilizer of a partition into $k$ parts of size $l$.
Here is a fully worked example in $Sym(6)$:
Consider two blocks of size $3$: $1,2,3, 4,5,6$, take all permutations that preserve these subsets in $Sym(6)$, i.e $H=Sym(1,2,3)$ and $K=Sym(4,5,6)$. Now find a $gin Sym(6)$ which switches the two blocks, for example $g=(14)(25)(36)$. Now let $G=langle H,K,grangle$. Then $G$ acts transitively and imprimitively on 6 points. Note that $Gcong Sym(2)wr Sym(3)$.
The minimal example alluded to in the comments has generating set $langle(12),(34),(13)(24)rangle$. That is, it is exactly the set of permutations that preserve 2 sets of 2. A non trivial block structure is then given almost by definition: $1sim2$ and $3sim4$.
answered Aug 2 at 15:49
Sam Hughes
1037
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
Any subgroup of $Sym(n)$ of the form $Sym(k)wr Sym(l)$ where $n=kl$ fits the criteria (this comes from the O'Nan-Scott Theorem). This construction is known as the wreath product. Concretely this is the stabilizer of a partition into $k$ parts of size $l$.
Here is a fully worked example in $Sym(6)$:
Consider two blocks of size $3$: $1,2,3, 4,5,6$, take all permutations that preserve these subsets in $Sym(6)$, i.e $H=Sym(1,2,3)$ and $K=Sym(4,5,6)$. Now find a $gin Sym(6)$ which switches the two blocks, for example $g=(14)(25)(36)$. Now let $G=langle H,K,grangle$. Then $G$ acts transitively and imprimitively on 6 points. Note that $Gcong Sym(2)wr Sym(3)$.
The minimal example alluded to in the comments has generating set $langle(12),(34),(13)(24)rangle$. That is, it is exactly the set of permutations that preserve 2 sets of 2. A non trivial block structure is then given almost by definition: $1sim2$ and $3sim4$.
answered Aug 2 at 15:49
Sam Hughes
1037
add a comment |Â
up vote
5
down vote
accepted
Any subgroup of $Sym(n)$ of the form $Sym(k)wr Sym(l)$ where $n=kl$ fits the criteria (this comes from the O'Nan-Scott Theorem). This construction is known as the wreath product. Concretely this is the stabilizer of a partition into $k$ parts of size $l$.
Here is a fully worked example in $Sym(6)$:
Consider two blocks of size $3$: $1,2,3, 4,5,6$, take all permutations that preserve these subsets in $Sym(6)$, i.e $H=Sym(1,2,3)$ and $K=Sym(4,5,6)$. Now find a $gin Sym(6)$ which switches the two blocks, for example $g=(14)(25)(36)$. Now let $G=langle H,K,grangle$. Then $G$ acts transitively and imprimitively on 6 points. Note that $Gcong Sym(2)wr Sym(3)$.
The minimal example alluded to in the comments has generating set $langle(12),(34),(13)(24)rangle$. That is, it is exactly the set of permutations that preserve 2 sets of 2. A non trivial block structure is then given almost by definition: $1sim2$ and $3sim4$.
answered Aug 2 at 15:49
Sam Hughes
1037
add a comment |Â
up vote
5
down vote
accepted
up vote
5
down vote
accepted
Any subgroup of $Sym(n)$ of the form $Sym(k)wr Sym(l)$ where $n=kl$ fits the criteria (this comes from the O'Nan-Scott Theorem). This construction is known as the wreath product. Concretely this is the stabilizer of a partition into $k$ parts of size $l$.
Here is a fully worked example in $Sym(6)$:
Consider two blocks of size $3$: $1,2,3, 4,5,6$, take all permutations that preserve these subsets in $Sym(6)$, i.e $H=Sym(1,2,3)$ and $K=Sym(4,5,6)$. Now find a $gin Sym(6)$ which switches the two blocks, for example $g=(14)(25)(36)$. Now let $G=langle H,K,grangle$. Then $G$ acts transitively and imprimitively on 6 points. Note that $Gcong Sym(2)wr Sym(3)$.
The minimal example alluded to in the comments has generating set $langle(12),(34),(13)(24)rangle$. That is, it is exactly the set of permutations that preserve 2 sets of 2. A non trivial block structure is then given almost by definition: $1sim2$ and $3sim4$.
answered Aug 2 at 15:49
Sam Hughes
1037
Any subgroup of $Sym(n)$ of the form $Sym(k)wr Sym(l)$ where $n=kl$ fits the criteria (this comes from the O'Nan-Scott Theorem). This construction is known as the wreath product. Concretely this is the stabilizer of a partition into $k$ parts of size $l$.
Here is a fully worked example in $Sym(6)$:
Consider two blocks of size $3$: $1,2,3, 4,5,6$, take all permutations that preserve these subsets in $Sym(6)$, i.e $H=Sym(1,2,3)$ and $K=Sym(4,5,6)$. Now find a $gin Sym(6)$ which switches the two blocks, for example $g=(14)(25)(36)$. Now let $G=langle H,K,grangle$. Then $G$ acts transitively and imprimitively on 6 points. Note that $Gcong Sym(2)wr Sym(3)$.
The minimal example alluded to in the comments has generating set $langle(12),(34),(13)(24)rangle$. That is, it is exactly the set of permutations that preserve 2 sets of 2. A non trivial block structure is then given almost by definition: $1sim2$ and $3sim4$.
answered Aug 2 at 15:49
Sam Hughes
1037
edited Aug 2 at 16:47
answered Aug 2 at 15:49
Sam Hughes
1037
answered Aug 2 at 15:49
Sam Hughes
1037
answered Aug 2 at 15:49
Sam Hughes
1037
1037
add a comment |Â
add a comment |Â
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3
By definition, subgroup of $S_4$ that preserves $1,2,3,4$ seems to be the minimal example.
– MigMit
Aug 2 at 13:45