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An element $g$ in a group $G$ is called reversible if $g$ is conjugate to $g^{-1}$ in $ G $. An element $g$ in $G$ is strongly reversible if $ g $ is conjugate to $g^{-1}$ by an involution in $G$. The group of affine transformations of…

An element $g$ of a group is called {\em reversible} if it is conjugate in the group to its inverse. This paper is about reversibles in the group $G$ of formally-invertible pairs of formal power series in two variables, with complex…

An element $g$ in a group $G$ is called reversible (or real) if it is conjugate to $g^{-1}$ in $G$, i.e., there exists $h$ in $G$ such that $g^{-1}=hgh^{-1}$. The element $g$ is called strongly reversible if the conjugating element $h$ is…

An element of a group is said to be reversible if it is conjugate to its inverse. We characterise the reversible elements in the group of diffeomorphisms of the real line, and in the subgroup of order preserving diffeomorphisms.

A group element is called a generalized torsion if a finite product of its conjugates is equal to the identity. We prove that in a nilpotent or FC-group, the generalized torsion elements are all torsion elements. Moreover, we compute the…

Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the identity, then $g$ is called a generalized torsion element. The minimum number of conjugates in such a product is…

An element $g$ in a group $G$ is called \emph{reciprocal} if there exists $h \in G$ such that $g^{-1}=hgh^{-1}$. The reciprocal elements are also known as `real elements' or `reversible elements' in the literature. We classify the…

A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a…

An element of a group is called \emph{reversible} if it is conjugate to its inverse, and \emph{strongly reversible} if it can be expressed as a product of two involutions. We study strongly reversible elements in the Riordan group and in…

We show that a group whose generalized torsion elements are torsion elements (which we call a $TR^{*}$-group) is torsion-by-$R^{*}$ group, an extension of torsion group by a group without generalized torsion elements. We also discuss a…

A group element is called generalized torsion if a finite product of its conjugates is equal to the identity. We show that in a finitely generated abelian-by-finite group, an element is generalized torsion if and only if its image in the…

An element of a group is called \emph{reversible} if it is conjugate to its inverse. While reversibility in the quaternionic M\"{o}bius group $\mathrm{PSL}(2,\mathbb{H})$ has traditionally been studied using geometric and dynamical methods,…

An element $g$ of a group is called reversible if it is conjugate in the group to its inverse. An element is an involution if it is equal to its inverse. This paper is about factoring elements as products of reversibles in the group…

Let $G$ be a group. We say that an element $f\in G$ is {\em reversible in} $G$ if it is conjugate to its inverse, i.e. there exists $g\in G$ such that $g^{-1}fg=f^{-1}$. We denote the set of reversible elements by $R(G)$. For $f\in G$, we…

Let $G$ be a group. An element $g$ in $G$ is called reversible if it is conjugate to $g^{-1}$ within $G$, and called strongly reversible if it is conjugate to its inverse by an order two element of $G$. Let $\textbf{H}_{\mathbb H}^n$ be the…

It is known that a bi-orderable group has no generalized torsion element, but the converse does not hold in general. We conjecture that the converse holds for the fundamental groups of 3-manifolds, and verify the conjecture for…

In a group, a generalized torsion element is a non-identity element whose some non-empty finite product of its conjugates yields the identity. Such an element is an obstruction for a group to be bi-orderable. We show that the Weeks…

A non-trivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for…

An element $f$ of a group $G$ is reversible if it is conjugated in $G$ to its own inverse; when the conjugating map is an involution, $f$ is called strongly reversible. We describe reversible maps in certain groups of interval exchange…

An element of a group is called reversible if it is conjugate to its own inverse. Reversible elements are closely related to strongly reversible elements, which can be expressed as a product of two involutions. In this paper, we classify…