We extend two standard theorems on groups to gyrogroups: the direct product theorem and the cancellation theorem for direct products. Firstly, we prove that under a certain condition a gyrogroup $G$ can be decomposed as the direct product of two subgyrogroups. Secondly, we prove that finite gyrogroups can be cancelled in direct products: if $A\cong B$, then $A\times H\cong B\times K$ or $H\times A\cong K\times B$ implies $H\cong K$, where $A,B,H,$ and $K$ are finite gyrogroups.

This article studies connections between group actions and their corresponding vector spaces. Given an action of a group $G$ on a non-empty set $X$, we examine the space $L(X)$ of scalar-valued functions on $X$ and its fixed subspace: $L^G(X)=\{f\in L(X):f(a\cdot x)=f(x)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}a\in G,x\in X\}$. In particular, we show that $L^G(X)$ is an invariant of the action of $G$ on $X$. In the case when the action is finite, we compute the dimension of $L^G(X)$ in terms of fixed points of $X$ and prove several prominent results for $L^G(X)$, including Bessel's inequality and Frobenius reciprocity.

We investigate the metric geometry of gyrogroups, a class of group-like structures whose binary operation is generally nonassociative. In particular, we extend the notion of the word metric from finitely generated groups to gyrogroups. This extension enables any gyrogroup to be viewed as a metric space, providing a suitable framework for proving a Mazur–Ulam-type theorem and for analyzing its algebraic and combinatorial structure via the associated right Cayley graph in a manner analogous to the classical setting of groups.