Question

Find a sequence of letter swaps from the word Limped to the word Dimple knowing that eaxh step you can only swap two adjacent letters. This is in my abstract algebra class and i feel like im just not doing it correctly.

Answer 1

LIMPED LIMPDE LIMDPE LIDMPE LDIMPE DLIMPE DILMPE DIMLPE DIMPLE

In the context of Abstract Algebra, consider the set W = {L,I,M,P,E,D} of all letters of the word LIMPED.
Let S_W be the set of all bijections of W onto itself. Then, S_W forms a group under composition of functions.
This group is called the Symmetric Group on W. The elements of S_W are called the permutations of W.

Suppose that the initial arrangement of the letters is in the "configuration" LIMPED.
Each permutation in S_W "acts" on this configuration to produce another configuration. (The precise notion is that of a Group
Action of the group S_W on the set of all arrangements(configurations) of the letters L,I,M,P,E,D).

For example, the identity permutation operates on this configuration to produce the same configuration LIMPED.

The permutation which interchanges two adjacent letters is called a 2-cycle or a Transposition of adjacent letters.
For example, the Transposition (E D) operates on LIMPED to give rise to the configuration LIMPDE. Here, (E D) denotes the permutation which interchanges E and D and keeps all other letters fixed.

Applying two permutations successively on LIMPED is equivalent to composing the two permutations in S_W at first, and then
applying the product on LIMPED.

Thus, to reach the configuration DIMPLE from LIMPED (by only swapping two adjacent letters at each step), the permutation that is to be applied on LIMPED is:

(L P) o (L M) o (L I) o (D L) o (D I) o (D M) o (D P) o (D E)  (where the compositions occur from right to left).