Group Theory
A group is a set of elements together with an operation that combines any two of its elements to form a third element also in the set while satisfying four conditions called the group, namely closure, associativity, identity and inverse.
In mathematical term, a group is a set G together with a binary operation
(a,b) (a*b): GXGG
Then the following conditions should be satisfied
G1: (associativity) for all a, b, c G
(a*b)*c = a*(b*c)
G2: (existence of a neutral element) there exists an element e G such that
a*e = a = e*a
for all a G;
G3: (existence of inverses) for each a G, there exists and a’ G such that
a *a’ = e = a’*a
For example the set of integer with addition operation is a group. Addition of any two integers form another integer belong to Z. Addition of three integers satisfies associative property. An identity element exists in set of integer, 0 Z. Each element of Z has it inverse and there result is equal to identity element 0.
Symmetry Group
Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged, and the operation of combining two such transformations by performing one after the other. Various physical systems, such as crystal and the hydrogen atom, space shuttle, airplanes and many other engineering works use symmetry groups modeling.
An object or situation is symmetry if it looks the same from more than one point of view. The computer CD, flowers, art work, human dancing and crystal all things have commonality of symmetry group.
The picture shown below of a square with number 1, 2, 3 and 4 on its corner is an example of symmetry group. If we flip the box horizontally, then flip again vertically, again horizontally and again vertically the square box come to the same position. We look from different point of views when we were flipping the square box it looks same and after two horizontals and verticals flips the box came to the same position. It proves that after set of transformation the object remain unchanged.
Let see another visual example of symmetry group we use propeller in this example. We mark the number on each blade of propeller as 1, 2 and 3. We have three persons looking on from different position as shown in figure. Each person see their own respective view and when we rotate the propeller counter clock wise at 120 degree angle the person one see number 2 blade, person 2 sees number 3 blade and person three sees number 1 blade. But their view of seeing the propeller is unchanged,after three rotation of 120 degree angle the object will come to same position, unchanged. We also can rotate the persons at 120 degree and their view of propeller will not be changed. After 3 rotation of 120 degree the persons will come to the same position. Mean after certain transformation the view of the object is not changed and it comes to position of 360 degree.
But the above scenario will not be true if we have one rotation of 70 and next rotation of 90. It means we need a fixed set of rotation to see the unchanged forms of the blades and propeller. The fixed set of movement Cayley’s diagram as shown in figure below the rotation in clock wise for each step is 120 degree.
If we replace the propeller with node we will have mathematical form of the symmetry graph, where arrow describes the symmetry operation on the nodes.
Let see another example of symmetry group, I will use star fish in this example. Star fish have five arms looks similar from different point of view. We rotate star fish in counter clock wise in 72 degree angle, which is 1/5 of 360 degree after 5 rotationsthe star fish come to its original position, unchanged as shown in the outer circle in fig below. Star fish has more symmetry than the propeller, we can flip it over and it still possesses symmetry property. Let’s flip all starfish from the outer diagram and move them inside the bigger circle connected with blue line. The blue line shows the two way operation. After flipping all the star fishes we find another five symmetry star fishes. Inner circle is anti-clock wise transformation of 5 rotation of 72 degree angle. The object star fish will not change. This proves the symmetry group definition.
Now let’s make the graph look like more mathematical model. Let replace the star fish with nodes. We will have a symmetry graph as shown in image below. We studied graph in our textbook here is another example.
Bibliography
Miller, Willard Jr. (1972). Symmetry Groups and Their Applications, New York: Academic Press. OCLC589081
Burns, G.; Glazer, A.M. (1990). Space Groups for Scientists and Engineers (2nd ed.). Boston: Academic Press, Inc. ISBN 0-12-145761-3
Visual Group Theory, Nathan Carter, Presentation on You tube.