Section 17.2 Concepts: Discovery guide
Discovery 17.1.
(a)
Suppose \(\funcdef{\varphi}{G}{S_X}\) is a homomorphism. What element of \(S_X\) must \(\varphi(e)\) be?
(b)
Suppose a group \(G\) acts on a set \(X\text{,}\) and \(x\) is an element of \(X\text{.}\) What must the group-action result \(\grpact{e}{x}\) be?
Discovery 17.2.
(a)
Suppose \(\funcdef{\varphi}{G}{S_X}\) is a homomorphism, and \(g_1,g_2\) are elements of \(G\text{.}\) What does the word homomorphism say about \(\varphi(g_1 g_2)\text{?}\)
(b)
Suppose a group \(G\) acts on a set \(X\text{,}\) \(g_1,g_2\) are elements of \(G\text{,}\) and \(x\) is an element of \(X\text{.}\) Let \(h\) represent the product element \(g_1 g_2\) in \(G\text{.}\) What does Task a say about the group-action result of \(\grpact{h}{x}\text{?}\)
For group \(G\) and set \(X\text{,}\) suppose we have an interpretation of each group element \(g\) in \(G\) as a function \(\funcdef{g}{X}{X}\text{.}\) To verify that this defines a group action of \(G\) on \(X\text{:}\)
Check that each function \(\funcdef{g}{X}{X}\) is a permutation of \(X\) (i.e. a bijective correspondence of \(X\) with itself).
For each pair of group elements \(g_1,g_2\text{,}\) the permutations of \(X\) corresponding to elements \(g_1\text{,}\) \(g_2\text{,}\) and the product element \(g_1 g_2\) satisfy the pattern you found in Task 17.2.b.
Discovery 17.3.
Recall that the set of natural numbers is
Given group element \(p\) in \(S_9\text{,}\) define a function \(\funcdef{p}{\N}{\N}\) as follows. For number \(m\) in \(\N\text{,}\) create output number \(p(m)\) by replacing the digits in \(m\) (in place) according to \(p\) (and leaving digit \(0\) alone). For example, if \(p = \threecycle{1}{3}{8} \twocycle{2}{7} \) then
Verify that this interpretation of group elements in \(S_9\) as functions \(\N \to \N\) defines a group action of \(S_9\) on \(\N\text{.}\)
Discovery 17.4.
In each of the following, determine both the orbit and stabilizer of the given set elements under the described group action. As you go, look for any patterns that might emerge.
NOTE: Keep a record of your results, you will be asked about them later in this Discovery Set.
(a)
The group of \(3 \times 3\) permutation matrices acting on \(\R^n\) by matrix-times-column-vector. (Note that this group is isomorphic to \(S_3\text{.}\))
(i)
Orbit and stabilizer of \(\vec{x} = (1,0,0)\) (as a column vector).
(ii)
Orbit and stabilizer of \(\vec{x} = (1,2,3)\) (as a column vector).
(iii)
Orbit and stabilizer of \(\vec{x} = (1,1,1)\) (as a column vector).
(b)
Group \(D_3\) acting on the set of labelled vertices of a hexagonal plate by rotational symmetries.
With
as usual, let \(r\) represent a rotation by \(2 \pi / 3\) around the \(z\)-axis in a counter-clockwise direction when looking down at the \(x y\)-plane from above, and let \(s\) represent a rotation by \(\pi\) around the \(x\)-axis.
(i)
Orbit and stabilizer of vertex \(1\text{.}\)
(ii)
Orbit and stabilizer of vertex \(2\text{.}\)
(c)
The cyclic subgroup of \(S_5\) generated by the element \(\threecycle{1}{2}{3} \twocycle{4}{5}\text{,}\) acting on the set \(X = \{1,2,3,4,5\}\text{.}\)
(i)
Orbit and stabilizer of \(1\text{.}\)
(ii)
Orbit and stabilizer of \(4\text{.}\)
Discovery 17.5.
Use the Subgroup Test to verify that a stabilizer \(G_x\) is always a subgroup of \(G\text{.}\)
Discovery 17.6.
For \(G = \GL_n(\R)\) acting on \(\R^n\) by matrix-times-column-vector as usual, what linear algebra concept arises when considering what it means for a matrix \(A\) to be in a stabilizer \(G_{\vec{x}}\) of a vector \(\vec{x}\text{?}\)
Discovery 17.7.
Verify that when a group \(G\) acts on a set \(X\text{,}\) the orbits Partition \(X\text{.}\) That is, verify that each element of \(X\) is in one and only one orbit under the action of \(G\text{.}\)
Alternatively, you could verify that the relation on \(X\) where “\(x\) is related to \(y\)” means that \(x\) is in the orbit \(G(y)\) is an Equivalence relation on \(X\text{.}\) (This would imply that the orbits partition \(X\text{,}\) since the orbits would then be the equivalence classes.)
Discovery 17.8.
In Task b.i of Discovery 17.4, you computed the orbit and stabilizer of vertex \(1\) under the action of \(G = D_3\) on a hexagonal plate. For each vertex \(v\) in that orbit (besides vertex \(1\) itself), use a particular choice of element of \(G = D_3\) that sends \(1\) to \(v\) to turn the stabilizer \(G_1\) into the stabilizer \(G_v\text{.}\)
Discovery 17.9.
Let's try to create a general description of the pattern of Discovery 17.8 in the general context of a group \(G\) acting on a set \(X\text{.}\)
(a)
Suppose \(x,y\) are set elements and \(g\) is a group element so that \(y = g(x)\text{.}\) (So we are assuming that \(y\) is in the orbit of \(x\text{.}\))
(i)
For each group element \(a\) in the stabilizer \(G_x\) (so that \(a(x) = x\)), create a corresponding element in the stabilizer \(G_y\text{.}\)
Since we have assumed \(y = g(x)\text{,}\) you are trying to use \(a\) to create a group element to fill in the question mark in
(ii)
For each group element \(b\) in the stabilizer \(G_y\) (so that \(b(y) = y\)), create a corresponding element in the stabilizer \(G_x\text{.}\)
Since we have assumed \(y = g(x)\text{,}\) the fact that \(b\) is in \(G_y\) means that \(b\bbrac{g(x)} = g(x)\text{.}\) Based on this, you are trying to use \(b\) to create a group element to fill in the question mark in
(b)
Describe the overall algebra pattern of Task a: set elements in the same orbit have stabilizers.
Discovery 17.10.
In Task b.i of Discovery 17.4, you computed the orbit and stabilizer of vertex \(1\) under the action of \(G = D_3\) on a hexagonal plate.
(a)
For each vertex \(v\) in that orbit, determine the complete collection of group elements that send vertex \(1\) to vertex \(v\text{.}\) (This time start with \(v = 1\text{.}\)) What is the pattern?
(b)
Repeat Task a but “starting” at a different vertex in that orbit. (That is, choose one of the other vertices in the orbit \(G_1\) to play the role of vertex \(1\text{.}\)) Do you see the same pattern?
Discovery 17.11.
Let's try to create a description of the pattern of Discovery 17.10 in the general context of a group \(G\) acting on a set \(X\text{.}\)
Suppose \(x,y\) are set elements and \(g\) is a group element so that \(y = g(x)\text{.}\) (So we are assuming that \(y\) is in the orbit of \(x\text{.}\))
(a)
Verify that every group element in the coset \(g G_x\) will send \(x\) to \(y\text{.}\)
(b)
Verify that every group element that sends \(x\) to \(y\) must be in the coset \(g G_x\text{.}\)
Start with the assumption that group element \(h\) sends \(x\) to \(y\text{.}\) Using \(y = g(x)\text{,}\) this means that
Manipulate this equality algebraically so that it becomes
Discovery 17.12.
Let's take Discovery 17.11 even further — for each set element \(x\) in \(X\text{,}\) the pattern of that activity sets up a correspondence between the orbit of \(x\) and the set of cosets of the stabilizer of \(x\text{.}\) That is, we have a correspondence
as follows. For each set element \(y\) in the orbit \(G(x)\text{,}\) we can find a group element \(g_y\) so that \(g_y(x) = y\text{.}\) So match \(y\) with the coset \(g_y G_x\) — which means, from the patterns in Discovery 17.11, that we are matching \(y\) with the complete collection of group elements that send \(x\) to \(y\text{.}\)
(a)
Verify that the correspondence \(y \leftrightarrow g_y G_x \) is well-defined with respect to the choice of coset representative. That is, if we were to choose a different group element \(g_y'\) so that \(g_y'(x) = y\) and use that group element to create the matching coset \(g_y' G_x\text{,}\) would we then have still matched \(y\) with the same coset in \(G / G_x\) as when we initially choose \(g_y\text{?}\)
(b)
Verify that the correspondence \(y \leftrightarrow g_y G_x \) is bijective. That is, verify that each coset \(g G_x\) will be matched up with one and only one set element \(y\) in this way.
Discovery 17.13. Orbit-Stabilizer Theorem.
Assume that \(G\) is a finite group acting on a set \(X\text{.}\)
(a)
Convince yourself that each orbit \(G(x)\) must be a finite set, even if \(X\) itself is infinite.
(b)
For a set element \(x\) in \(X\text{,}\) use Discovery 17.12 to come up with a relationship between the order of \(G\text{,}\) the order of the stabilizer \(G_x\text{,}\) and the size of the orbit \(G(x)\text{.}\)
(c)
When you were asked to “look for any patterns that might emerge” in Discovery 17.4, is this a pattern you noticed? If not, go back at your calculations and see if this pattern holds up.