Group Theory - rFronteddu/general_wiki GitHub Wiki

• Loosely speaking, a group is a set of actions satisfying some mild properties:
  • deterministic
  • reversibility
  • closure (if you do a sequence of moves that is still a move)
• A generating set for a group is a sub-collection of actions that together can produce all actions in the group - like a spanning set in a vector space.
• Given a generating set, the individual actions are called generators.

The set of all possible ways to scramble a Rubik's cube is an example of a group. Two actions are the same if they have the same "net effect", e.g., twisting a face 1 time vs. twisting a face 5 times. Note that the group is the set of actions one can perform, not the set of configurations of the cube. However, there is a bijection between these two sets (if you take any scrambled configuration and a fixed solved state there is one way to go from one to the other).

The Rubik's cube group has $4.3*10^9$ actions but we it can be generated by a set of size 6.

A Cayley graphs is a mapping of a group.

In general, a Cayley diagram consists of nodes that are connected by colored (or labeled) arrows, where:

• an arrow of a particular color represents a specific generator;
• each action of the group is represented by a unique node (sometimes we will label nodes by the corresponding action).
• Equivalently, each action is represented by a (non-unique) path starting from the solved state.

An arrow corresponding to the generator g from node x to node y means that node y is the result of applying the action $g \in G$ to node x

If an action $h \in G$ is its own inverse (that is, $h^2 = e$), then we have a 2-way arrow. This happens with horizontal and vertical flips. For clarity, our convention is to drop the tips on all 2-way arrows.

Consider a clear glass rectangle and label it as follows:

We will call the above configuration the solved state of the puzzle (note that you could use colors instead of numbers).

• The idea of the game is to scramble the puzzle and then find a way to return the rectangle to its solved state.
• We are allows two moves: horizontal flip, and vertical flip. We allow these moves because they preserve the footprint of the rectangle. We could also do nothing or rotate the rectangle.
• The moves of the puzzle form a group, they satisfy the 4 rules mentioned earlier.
• For our convenience, let's say that when we flip the rectangle, the numbers automatically become right-side-up
• By using only sequences of horizontal and vertical flips, we can obtain only four configurations. We can draw this roadmap:

• There are two generators (red and blue arrows) and the order doesn't matter.
• Let G denote the rectangle group, This is a set of four actions. The identity action e, a horizontal flip h, a vertical flip v, and a 180 rotation r. G = {e,h,v,r}
• We need two actions to generate G. In our diagram, each generator is represented by a different type (color) of arrow. We write G = <h,v> " G is generated from h and v."
• The map shows us how to get from any one configuration to any other. There is more than one way to follow the arrows.
• Since the order of the actions is irrelevant, we call such a group abelian.
• Every action in G is its own inverse. Note that the Rubik's cube group does not have this property nor the one before.
• How we chose to layout the map is irrelevant, the important thing is that the connections between the various states are preserved.

Let's map a group we will call 2-Light Switch:

• Consider two light switches side by side that both start in the off position (this is our "solved state").
• We are allowed 2 actions: flip L switch and flip R switch

Noticed how the Cayley diagrams for the Rectangle Puzzle G = {e, v, h, r} and the 2-Light Switch Group G'={e,L,R,B} are essentially the same. Although these groups are superficially different, the Cayley diagrams help us see that they have the same structure (the two groups are isomorphic).

Any group with the same Cayley diagram as the Rectangle Puzzle and the 2-Light Switch Group is called the Klein 4-group, denoted by $V_4$ for vierergruppe, "four-group" in German.

The triangle puzzle group, often denoted $D_3$ has 6 actions:

• The identity action: e
• A (Clockwise) 120 rotation: r
• A (clockwise) 240 rotation $r^2$
• A horizontal flip: f
• Rotate + horizontal flip: rf
• Rotate twice + horizontal flip: $r^2$ f

One set of generators: $D_3$ = <r, f>

Note that multiple paths can lead us to the same node. These give us relations in our group.

Note that this group is non-abelian: rf != fr

• at every node of a Cayley graph, there is exactly one out-going edge of each color.

• Suppose an action g has the property that gx = x for some other action x. Then g is the identify action, i.e., gh=h=hg for all other action h.

Something is symmetrical when it looks the same from more than one point of view. Previous examples deal with arrangements of similar things.

Calyley's Theorem: Every group can be viewed as a collection of ways to rearrange some set of things.

Groups relate to symmetry because an object's symmetries can be described using arrangements of the object's parts.

Algorithm to construct a group that describes a physical object's symmetry.

1. Identify all the parts of the object that are similar and give each such part a different number.
2. Consider the actions that may rearrange the numbered parts, but leave the object in the same physical space.
3. (Optional) To visualize the group, explore and map it as shown in earlier puzzles.

• The physical space that an object occupies is its footprint.