Given a finite sequence formed by the numbers 1, 2, 3, and 4, we are allowed to replace some of the numbers with others according to the following rules
- If we have two distinct and adjacent numbers among 1, 2, and 3, we can replace the two numbers with the third number. For example, we can replace 13 or 31 with 2. We can also apply this rule in reverse, so 2 can be replaced with 13 or 31.
- The sequence 123 (in that order) can be replaced by 4. We can also replace 4 with 123.
Show that given any finite sequence of the numbers 1, 2, 3, and 4, we can reach exactly one number after a series of moves using the 2 above rules.
I've managed to prove that it is possible to reach one number using the two above rules, but I'm not able to prove that the one number is unique. I was thinking we could exploit some sort of invariant, but I can't find one. The fact that the order matters in condition 2 is a bit troublesome. Any help on the problem is appreciated.
We map the symbols $1,2,3,4$ to the elements $a,b,c,e$, respectively, of the Klein 4-group, $V$, $e$ being the identity. In $V$, we have $aa=bb=cc=e$, $ab=ba=c$, $ac=ca=b$, $bc=cb=a$. A string of symbols taken from $1,2,3,4$ then corresponds to an element of $V$. The transformation $12\to3$ corresponds to the equality $ab=c$; similarly for all the other symbol transformations, including $123\to4$, which is $abc=e$. So, the transformations don't change the value of the element in $V$.
Now using transformations to reduce a string to a single symbol corresponds to evaluating a product of group elements as a single element of $V$, and of course no matter what sequence of steps you take to evaluate the group element, a given string of symbols will always yield the same group element, since the transformations don't change the value of the group element.