Introduction to Group Theory

Orbits, Stabilizers and Symmetry

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Contrary to the header of this section, this part will begin with a discussion f symmetry. The unit circle is the circle centered at (0,0) with radius 1. It has the property that every point along it can be described by $$\left(\sin\theta,\cos\theta\right), 0\leq\theta\leq2\pi$$.

The unit circle has infinitely many symmetries. The symmetries are all the rotations $$\left[\hbox{rot}\left(\theta\right)\right]$$ and reflections $$\left[\hbox{ref}\left(\theta\right)\right]$$. These symmetries form a group, usually referred to as the orthogonal group (for the unit circle, this is usually called O2). Closure is satisfied, because the combination of a rotation and a reflection, rotation and rotation, or reflection and reflection, results in either a rotation or reflection. The identity is satisfied by rot(0). Reflective inverse is trivial, because a reflection is its own inverse! Rotational inverse is easy to see as well, since for each $$\hbox{rot}\left(\theta\right)$$ there is also a $$\hbox{rot}\left(-\theta\right)$$. Since composition holds for this group, and associativity holds for composition, the four axioms necessary for a group are satisfied.

Now that symmetry has been defined, we can move on to orbits. Let $$G$$ be some group acting on a set $$A$$. For some element $$a \in A$$, we can generate a subset by taking all the elements of $$A$$ which can be found by applying some element of $$G$$. Let's take the unit circle and the orthogonal group, for example. The point (1,0) has the entire unit circle as its orbit, because with the reflections and rotations in O2, it can be mapped anywhere on the circle. For a group with finite symmetries, the fewer number of orbits, the greater amount of symmetry.

Using the set $$A$$, and group $$G$$ from the last paragraph, we can define a stabilizer. A stabilizer for any element $$a \in A$$ is an element $$g \in G$$, such that $$g \circ a = a$$ (the $$\circ$$ denotes the application of $$g$$ on $$a$$). $$g$$ is a stabilizer if it leaves $$a$$ unchanged. Using the point (1,0) from above and applying O2, we find the stabilizers as $$\left\{\hbox{rot}\left(2\pi\right),\hbox{ref}\left(0\right)\right\}$$.

This workbook covers the group of symmetries of a square, which is part of a family of groups called the dihedral groups, and is denoted D4. D4 covers rotation in increments of $$\frac{\pi}{2}$$ and reflection in increments of $$\frac{\pi}{4}$$. The identity element is rot(0) or "r1", but isn't shown on the page because it doesn't change anything. On this page, you click the various reflection and rotation buttons, and can see the various permutations gotten from applying D4. This also lets you input cycles. See if you can figure out the orbits and stabilizers of various elements of the permutation. This exercise is very similar to the exercise presented later in the book on Group Actions and Orbits.


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Your cycle, with values separated by commas: