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Suppose I'm dealing with strings of symbols $S = AAa$ where $A = aaa$ and that in order to describe the permutations $AaA, aAA$, I'm indexing the symbols as they occur (left-to-right works): $S_1 = A_2 A_3 a_4$. If I now consider all permutations of indexed symbols on the right of "$=$" (essentially all permutations of $\{2,3,4\}$, then there are many uninteresting ones such as $A_3 A_2a_4$.

Can these uninteresting permutations be filtered out while still retaining a group structure? Or is there another way to view the symmtries: $AAa$, $AaA$, $aAA$? The method must apply to longer strings as well.

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    Could you give a precise mathematical definition of "uninteresting ones"? – 2017-02-22
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    @J.-E.Pin: If the group of permutations $S_n$ acts on the strings of length $n$, the uninteresting permutations $\sigma$ with respect to the string $s$ seem to be those $\sigma$ such that $\sigma s = s$. – 2017-02-22
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    @AlexM. yes those that, when the index is dropped, result in a non-change of the underlying string. – 2017-02-22

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