Why is D4 not primitive on the vertices of a square?

In summary, for the second part, there is a mistake in assuming that every transposition in ##S_4## is also in ##D_8##. This is not true and can be seen from the example above. Therefore, the conclusion that ##D_8## is primitive on the set of vertices of a square is incorrect.
  • #1
fishturtle1
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Homework Statement
Let ##G## be a transitive permutation group on the finite set ##A##. A block is a nonempty subset of ##B## of ##A## such that for all ##\rho \in G## either ##\rho(B) = B## or ##\rho(B) \cap B = \emptyset##. (here ##\rho(B) = \lbrace \rho(b) : b \in B \rbrace##).

c) A (transitive) group ##G## acts on a set ##A## is set to be primitive if the only blocks in ##A## are the trivial ones: the sets of size ##1## and ##A## itself. Show that ##S_4## is primitive on ##A = \lbrace 1, 2, 3, 4, \rbrace##. Show that ##D_8## is not primitive as a permutation group of on the four vertices of a square.
Relevant Equations
.
Proof: Let ##B = \lbrace a \rbrace \subseteq A## and ##\rho \in S_4##. We have two cases, ##\rho(a) = a## in which case ##\rho(B) = B##, or ##\rho(a) \neq a## in which case ##\rho(B) \cap B = \emptyset##. Its clear that ##\rho(A) = A##. So these sets are indeed blocks. Now let ##C## be any subset of ##A## such that ##2 \le \vert C \vert \le 3##. Then there is ##x, y \in C, z \not\in C## and ##\gamma \in S_4## such that ##\gamma(x) = z## and ##\gamma(y) = y##. This implies ##\gamma(C) \neq C## and ##\gamma(C) \cap C \neq \emptyset##. So ##C## is not a block. We can conclude ##S_4## is primitive on ##A##. []

For the second part we need to show ##D_8## is not primitive on the set of vertices of a square. We label the vertices ##1, 2, 3, 4##. Let ##S \subseteq A## such that ##\vert S \vert = 2## or ##3##. If ##\vert S \vert = 3## and ##\sigma_1 = (1234)##, then ##\sigma_1(S) \cap S \neq \emptyset## and ##\sigma_1(S) \neq S##. So, suppose ##\vert S \vert = 2##. Then ##S = \lbrace a, b \rbrace## and there is ##c \not\in S##. Let ##\sigma_2 = (ac)##. Then ##\sigma_2(S) = \lbrace c, b \rbrace##. It follows ##S## is not a block. Since ##D_8 \subseteq S_4##, it follows the only blocks of ##D_8## acting on ##A## are the trivial ones, and so ##D_8## is primitive on ##A##.

Where did I go wrong in the second part?
 
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  • #2
fishturtle1 said:
Homework Statement:: Let ##G## be a transitive permutation group on the finite set ##A##. A block is a nonempty subset of ##B## of ##A## such that for all ##\rho \in G## either ##\rho(B) = B## or ##\rho(B) \cap B = \emptyset##. (here ##\rho(B) = \lbrace \rho(b) : b \in B \rbrace##).

c) A (transitive) group ##G## acts on a set ##A## is set to be primitive if the only blocks in ##A## are the trivial ones: the sets of size ##1## and ##A## itself. Show that ##S_4## is primitive on ##A = \lbrace 1, 2, 3, 4, \rbrace##. Show that ##D_8## is not primitive as a permutation group of on the four vertices of a square.
Homework Equations:: .

Proof: Let ##B = \lbrace a \rbrace \subseteq A## and ##\rho \in S_4##. We have two cases, ##\rho(a) = a## in which case ##\rho(B) = B##, or ##\rho(a) \neq a## in which case ##\rho(B) \cap B = \emptyset##. Its clear that ##\rho(A) = A##. So these sets are indeed blocks. Now let ##C## be any subset of ##A## such that ##2 \le \vert C \vert \le 3##. Then there is ##x, y \in C, z \not\in C## and ##\gamma \in S_4## such that ##\gamma(x) = z## and ##\gamma(y) = y##. This implies ##\gamma(C) \neq C## and ##\gamma(C) \cap C \neq \emptyset##. So ##C## is not a block. We can conclude ##S_4## is primitive on ##A##. []

For the second part we need to show ##D_8## is not primitive on the set of vertices of a square. We label the vertices ##1, 2, 3, 4##. Let ##S \subseteq A## such that ##\vert S \vert = 2## or ##3##. If ##\vert S \vert = 3## and ##\sigma_1 = (1234)##, then ##\sigma_1(S) \cap S \neq \emptyset## and ##\sigma_1(S) \neq S##. So, suppose ##\vert S \vert = 2##. Then ##S = \lbrace a, b \rbrace## and there is ##c \not\in S##. Let ##\sigma_2 = (ac)##. Then ##\sigma_2(S) = \lbrace c, b \rbrace##. It follows ##S## is not a block. Since ##D_8 \subseteq S_4##, it follows the only blocks of ##D_8## acting on ##A## are the trivial ones, and so ##D_8## is primitive on ##A##.

Where did I go wrong in the second part?

A possible problem I see is the following:

How are you sure that ##\sigma_2\in D_8##? Not every transposition of ##S_4## is in ##D_8## (because ##S_4## is generated by all its transpositions).
 
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  • #3
Math_QED said:
A possible problem I see is the following:

How are you sure that ##\sigma_2\in D_8##? Not every transposition of ##S_4## is in ##D_8## (because ##S_4## is generated by all its transpositions).
Thanks!

Consider the square with vertices ##a = (0,1), b = (1, 1), c = (1,0), d = (0,0)##. Let ##S = \lbrace a, c \rbrace##. Observe,
##(abcd)\cdot S = (abcd)^3\cdot S = (ab)(dc)\cdot S = (ad)(bc)\cdot S = \lbrace b, d \rbrace##
and
##1\cdot S = (abcd)^2 \cdot S = (ac) \cdot S = bd\cdot S = S##.
This shows that for all ##\sigma \in D_4##, we have ##\sigma(S) = S## or ##\sigma(S) \cap S = \emptyset##. This means that ##S## is a nontrivial block. So ##D_4## is not primitive on ##\lbrace a, b, c, d \rbrace##. []
 

Related to Why is D4 not primitive on the vertices of a square?

1. What is a group action?

A group action is a mathematical concept that describes how a group (a set of elements with a defined operation) can act on a set. This means that each element in the group can be associated with a specific transformation or action on the set. Group actions have applications in various fields, including geometry, physics, and computer science.

2. What is the significance of group actions?

Group actions are significant because they provide a way to study symmetry and invariance in mathematical structures. They also allow for the classification of objects based on their symmetries, which can lead to a better understanding of their properties and relationships.

3. Can you give an example of a group action?

One example of a group action is the rotation of a square in the Cartesian plane. The group in this case is the set of all possible rotations (0, 90, 180, and 270 degrees) and the set being acted upon is the square. Each rotation in the group corresponds to a specific transformation of the square, such as flipping it or rotating it 90 degrees clockwise.

4. What are the different types of group actions?

There are three main types of group actions: left group actions, right group actions, and conjugation actions. In a left group action, the group elements act on the set from the left, while in a right group action, they act from the right. Conjugation actions involve transforming an element in the group by an element from the group itself.

5. How are group actions related to blocks?

Group actions and blocks are related through the concept of orbits. The orbit of an element in the set is the set of all elements that can be reached by applying group actions to it. Blocks, on the other hand, are subsets of the set that remain unchanged under the group action. The number of blocks in a set can give insight into the structure of the group and its actions.

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