A group structure for Tamil

We can form a group structure for Tamil alphabets in many ways; simply we may apply residue classes modulo N or symmetric group of permutations modulo N for any cardinality. However, one interesting group structure with applications is the abstraction of 247 Tamil letters written on a torus; in this essay I will attempt to describe it and show that it forms a group.

We consider the 247 Tamil letters formed by 1 ayudha letter and 12 uyir letters for 13 vowels, and 18 mei letters for 18 consonants and 216 uyirmei or conjugate letters [247 = 13 + 18 + 216]. By consider a mapping of 13 vowels to Z13[residue classes modulo 13] and 18 uyirmei letters + ayutha letter to Z19 [residue classes modulo 19].

Fig. 1: The Cayley table for Z13 can represent Uyir letters.
Fig. 2: The Cayley table for Z19 can represent Mei letters (with modification)

Representation

Further we may represent each uyirmei letter as a index into a 2D table formed by rows of mei letters, and columns of uyir letters. So, for example letter ‘கு = க் + ஊ’ can be written as 6 + 1*13 = 19. Uyir letters are all represented from [0-12], Mei letters are represented as multiples of 13, [13, 26, 39, .. 234] for [க், ச், … ல், வ், ழ், ள்]. Uyirmei letters form everything in between.

The general representation of a letter can be: t = a + 13*b, where a goes from [0-12] and b goes from [0-18]. This representation pegs ‘ஃ’ at the origin. In the direct product of Z13 and Z19 this will be represented as (a,b)

Letter representation in the product group: Z13 x Z19

Result

Further since we showed uyir and mei letters can be embedded into the Z13, and Z19 residue classes and we know 247 factors neatly into 2 primes 13 and 19, we may use the Chinese remainder theorem (which guarantees that given two sets of residue classes which are co-prime, we can form a residue class with a unique representation for the direct-sum [direct-product] of the underlying sets). In our case we are guaranteed that Z13 x Z19 direct sum structure forms an isomorphic group in Z247. This is the key result in this easy:

Tamil letters [247] have a direct product representation in group Z247 which is isomorphic to the direct product of Z13, Z19 as mapping the uyir and mei group representations.

Key result – Group representation for Tamil alphabets

While Chinese remainder theorem guarantees a ring structure, I don’t know the second operator which can take role of product to make the ring structure possible at this writing.