A mathematical group, G

Thirumohoor Kalamegaperumal Temple
The Art of Symmetry in Architecture

A collection of symmetric objects or molecules obeying certain rules form a group. More specifically, the set of symmetry operations performed with reference to the symmetry elements (A, B, C, D, ..,) supporting the rules (listed below) makes a mathematical group, G. Also the symmetry elements,  A, B, C, D, ... belongs to the mathematical group, G.

Rules/ Properties of a Group:

1. The square of symmetry element or the product of any two symmetry elements should be the symmetry elements of the group, G.
2. The commutative law of combination must hold good.
3. The associative law of combination should exist.
4. There should be an identity element that commutes with all other symmetry elements, leaving it unchanged.
5. The product of some symmetry elements nullifies the operation, bringing the orientation to the original.
6. Every symmetry element has its own inverse. The product of the symmetry element with its inverse gives the identity element.

Types of Group:

1. Abelian Group: Here the symmetry elements are commutative i.e., each element commute with every other element in the group.
2. Non-abelian group: Here the symmetry elements are not commutative.
3. Cyclic Group: Group of symmetries deals with only rotations. All cyclic groups are abelian.
4. Dihedral Group: Group of symmetries that deals with rotations and mirror planes or reflections. In addition to this, we should remember for any dihedral group there must be nC2 axes of symmetry perpendicular to Cn, the principal axis. Designated as 'D'

Mathematically, nC₂ ⟂ C_n

Group Multiplication Table: The multiplication operations of symmetry elements in a group are listed in this table. The total number of symmetry elements in a group is termed the order of the group (h). In each row and column of the group multiplication table, every symmetry elements occur 'once'. Obviously, we can apply the principle of Sudoku to load the grids without operating and verify thereafter. 

Illustration: Let us construct the group multiplication table for the water molecule and authenticate it is an abelian group.

Step1 Identify the order of the group (h) - Go after this sequence || E, i, C_n (C_n¹, C_n², C_n³, …, C_nⁿ = E) 𝜎h > 𝜎v > 𝜎d, S_n ||: 

Water There are 4 symmetry elements for a water molecule. They are,

1. Identity Element (E): Doing nothing or leaving the molecule unchanged or a rotation through 360° 
2. Axis of symmetry (C2): Rotation through 180° producing an indistinguishable configuration.
3. Two mirror planes of symmetry (2𝜎v): Plane containing the principal axis of rotation.

Step2 Pencil out a 4 grid and subdivide it into a further 4 small boxes (since there are 4 symmetry elements) i.e., 4x4.

Step3 To add on the 4 symmetry elements (E, C₂, 𝜎_v, 𝜎_v^') insert a row and a column i.e., 4x4 transfigure as 5x5. Since the element E is analogous to unity in the multiplication operation, any symmetry element operated with it gives that element. (Rule No.4)

	E	C2	𝜎v	𝜎v'
E	E	C2	𝜎v	𝜎v'
C2	C2
𝜎v	𝜎v
𝜎v'	𝜎v'

Step4 For a water molecule, the square of symmetry element yields us an identity element or the product of two similar symmetry elements nullifies the operation (Rule No.1 & 5). Thus, every element has its own inverse. This reveals that the inverse of C₂ is C₂^-1 similarly the inverse of  

𝜎_v is 𝜎_v^-1and  𝜎_v^' is 𝜎_v^'-1(Rule No.6).  Hence, the identity element defines the term 'Inverse'. 

	E	C2	𝜎v	𝜎v'
E	E	C2	𝜎v	𝜎v'
C2	C2	E
𝜎v	𝜎v		E
𝜎v'	𝜎v'			E

Furthermore, 

Results	Infers	Focus !
E= C22  = C2C2  = C2C2-1	The inverse of C2 is C2-1	C2-1= C2
E= C33 = C3C32 = C3C3-1	The inverse of C3 is C32	C3-1= C32
E= C44 = C4C43 = C4C4-1	The inverse of C4 is C43	C4-1= C43
E= C66 = C6C65 = C6C6-1	The inverse of C6 is C65	C6-1= C65

Step5 Fill out the remaining boxes as per the Sudoku puzzle. 

🤔	E	C2	𝜎v	𝜎v'
E	E	C2	𝜎v	𝜎v'
C2	C2	E	𝜎v'	𝜎v
𝜎v	𝜎v	𝜎v'	E	C2
𝜎v'	𝜎v'	𝜎v	C2	E

Step5 Property No. 2& 3 are verified. Commutative law of combination [ AB= BA, BC= CB...] holds good (Property No.2). Thus, the water molecule and other related ones belong to the abelian group. Associative law of combination [A(BC)= (AB)C, ) (AB)(BC)= (ABC)D...] also holds true (Property No.3)

Task: Complete the group multiplication table for Ammonia (6 symmetry elements)

🤔	E	C3	C32	𝜎v'	𝜎v''	𝜎v'''
E	E	C3	C32	𝜎v'	𝜎v''	𝜎v'''
C3	C3	C32	E	𝜎v'''	𝜎v'	𝜎v''
C32	C32	E	C32	𝜎v''	𝜎v'''	𝜎v'
𝜎v'	𝜎v'	𝜎v''	𝜎v'''	E	C3	C32
𝜎v''	𝜎v''	𝜎v'''	𝜎v'	C32	E	C3
𝜎v'''	𝜎v'''	𝜎v'	𝜎v''	C3	C32	E

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