The twin group {T_{n}} is a right-angled Coxeter group generated by {n-1} involutions, and the pure twin group {\mathrm{PT}_{n}} is the kernel of the natural surjection from {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 {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 {T_{n}}.
We give a new proof of the structure of {\operatorname{Aut}(T_{n})} for {n\geq 3}, and show that {T_{n}} is isomorphic to a subgroup of {\operatorname{Aut}(\mathrm{PT}_{n})} for {n\geq 4}.
Finally, we construct a representation of {T_{n}} to {\operatorname{Aut}(F_{n})} for {n\geq 2}.
For a natural number m, a Lie algebra L over a field k is said to be of breadth type (0, m) if the co-dimension of the centralizer of every non-central element is of dimension m. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type (0, 3) over Fq of odd characteristics up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, C and R. We also discuss 2-step nilpotent Camina Lie algebras.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.