Group-theoretic unification of Kaprekar’s constant 6174 and Trigg’s constant 2538

We can use group-theoretic inversion as a mnemonic, or sleight-of-hand, to give Kaprekar’s constant 6174 and Trigg’s constant 2538 a common source. The technique is analogous to what happens in Umbral Calculus. In Umbral Calculus, indices are treated as exponents. Our approach will the the reverse: we will treat exponents as indices. The one and only exponent in question is -1, that is, the multiplicative-inverse operator, but instantiated several times. The key formula that we will use is the well-known fact that the inverse of a product is equal to the reverse product of the inverses, that is, that (ab)^(-1) = (b^(-1))(a^(-1)). So, dropping the exponent, we can say that we have transformed ‘ab’ into ‘ba’. With this mnemonic, or sleight-of-hand, in mind, let us now consider the following two expressions: ((ab)^(-1))(cd)^(-1)) and ((ab)(cd))^(-1). The first expression can be verbalized as ‘the product of the inverses of two products’, and the second one can be verbalized as ‘the inverse of the product of two products’.
To get the iterate for the Kaprekar process, we drop the exponent in the first expression and apply the exponent twice in the second expression, and then drop the exponent. The two resulting expressions are, respectively, abcd and dcba, with the expression
abcd – dcba being the iterate for the Kaprekar process, with the understanding that ‘abcd’ represents the largest number that can be made from the digits of the original number in question.
To get the iterate for the Trigg process, we apply the exponent one time to each side, getting (b^(-1))(a^(-1))(d^(-1))(c^(-1)) and ((cd)^(-1))(ab^(-1)), and then drop the exponent, getting the two expressions badc and cdab, badc – cdab being the iterate for the Trigg process.
So, the Kaprekar process and the Trigg process can be commonly-sourced from Group Theory.
Here is a place in the literature where the Trigg process is described.
keywords: Recreational Mathematics, Number Theory, unification