Proof isn't always possible

Retraction: It is with regret that I do not delete the previous post. I am an individual, not an organization, and I failed to adequately double-check my results, but I am leaving the post to maintain some accountability. I am able to definitively say that the script for 2chars_from_file.py DOES NOT yield an exhaustive list. It develops the ciphertext for all 2 Char combinations of the FIRST 10, three character ciphertexts. This is not due to a logic error, but rather there is a virus that corrupts the indentation of the two write statements.  This may be verified by downloading the relevant code, and examining it, and running it if needed, against the relevant datafile. It is difficult to be exhaustively correct about ALL of the scripts. This calls ALL of my results in question, with regard to the qualification of a mathematical group. If Vigenere IS a group, the resulting substitution is still valid, but there exist 6 Char passwords that will accomplish the same encryption in ONE STE

I have for some time speculated that the 1500's Cipher, " Vigenere ," is not a mathematical  group . Not to bury the lead, IT IS NOT!  However, this is not completely satisfying for the question I was asking.   Vigenere could be represented as a Matrix function such that f(P) = C and f(C) = P. (P for plaintext, C for ciphertext.) Early, I observed that when the operation is repeated with passwords of differing lengths, the periodicity is equal to the product of the unique prime factors of the password lengths. In encryption, there is also a concept of the amount of ciphertext necessary to calculate or infer the password. This is called the " Unicity distance ." In Vigenere, the unicity distance is the same as the period. Since the idea in encryption is to extend the period, I amused myself with double and triple encrypting. Around 1995, I worried that a double encrypted message of a specific period might be comparable to a message encrypted ONCE, with a single p