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Word length perturbations in certain symmetric presentations of dihedral groups. (English) Zbl 1403.20034
The authors classify all small symmetric presentations of the dihedral group $$D_n$$ (here, “small” means a generating set with at most three elements and “symmetric” means that the generating set $$S$$ contains only nontrivial elements and if $$s\in S$$, then $$s^{-1}\in S$$).
The length of an element $$g\in D_n$$ with respect to such a symmetric generating set $$S$$ is simply the minimal number of letters in $$S$$ needed to write the word $$g$$. Two special integers are defined to measure somehow the effect on the length of the deletion of a letter or the replacement of a letter with another. Exact values for these two special integers are obtained for various symmetric generating sets and general bounds for other symmetric generating sets.
MSC:
 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F05 Generators, relations, and presentations of groups
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