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AbstractThe twin group Tn{T_{n}}is a right-angled Coxeter group generated by n-1{n-1}involutions, and the pure twin group PTn{\mathrm{PT}_{n}}is the kernel of the natural surjection from Tn{T_{n}}onto the symmetric group on n symbols.In this paper, we investigate some structural aspects of these groups.We derive a formula for the number of conjugacy classes of involutions in Tn{T_{n}}, which, quite interestingly, is related to the well-known Fibonacci sequence.We also derive a recursive formula for the number of z-classes of involutions in Tn{T_{n}}.We give a new proof of the structure of Aut(Tn){\operatorname{Aut}(T_{n})}for n≥3{n\geq 3}, and show that Tn{T_{n}}is isomorphic to a subgroup of Aut(PTn){\operatorname{Aut}(\mathrm{PT}_{n})}for n≥4{n\geq 4}.Finally, we construct a representation of Tn{T_{n}}to Aut(Fn){\operatorname{Aut}(F_{n})}for n≥2{n\geq 2}.
Forum Mathematicum – de Gruyter
Published: Sep 1, 2020
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