Let \(p>0\) be a fixed number. The authors are interested in subharmonic solutions of the system \[ \ddot{u}(t) = f(t,u(t)),\ u(t)\in V \] where \(f(t,u)\) is a continuous map, \(p\)-periodic with respect to the temporal variable. More precisely, let \(V := \mathbb{R}^k\) and let \(p = 2 \pi\) without loss of generality. Assume that \(f: \mathbb{R}\times V \rightarrow V\) is a continuous function satisfying the following symmetry conditions:

(\(S_1\))

For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(t+2\pi,x) = f(t,x)\) (dihedral symmetry);

(\(S_2\))

For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(-t,x) = f(t,x)\) (time-reversibility);

(\(S_3\))

For all \(t \in \mathbb{R}\) and \(x \in V\) we have \(f(t,-x) = -f(t,x)\) (antipodal \(\mathbb{Z}_2\)-symmetry).

The symmetric properties of the system of study allow reformulation of the problem of existence of the subharmonic \(2\pi m\)-periodic solutions as a question about the operator equation \(\mathcal{F}(u)=0\) with \(D_m\times \mathbb{Z}_2\)-symmetries in the functional space \(\mathcal{E} := C_{2\pi m}^2(\mathbb{R};V)\). The authors introduce an additional symmetry to the system of study before proving several results on the existence and multiplicity of subharmonic solutions. Namely, let \(\Gamma\) be a finite group then

(\(S_4\))

For all \(t \in \mathbb{R}\), \(x \in V\), and \(\sigma \in \Gamma\), we have \(f(t,\sigma x) = \sigma f(t,x)\) (\(\Gamma\)-equivariant).

The last condition allows for the restatement of the original problem as the \(G\)-equivariant operator equation with respect to the full group \[ G := \Gamma \times D_m \times \mathbb{Z}_2. \] If the isotropy group \(G_u\) of a solution \(u\) satisfies \(\{ e \} \times \mathbb{Z}_m \times\{ 1 \} \nleq G_u\), then \(u\) is a subharmonic solution.
The authors prove several novel results in the paper. Most notably Theorems 2.6 and 2.10. The main technical tool is Brower \(\textbf{G}\)-equivariant degree theory. Given a group \(G\) corresponding \(\textbf{G}\)-equivariant Brower degree is computed using the computer algebra system GAP. In addition, the authors discuss the bifurcation problem of subharmonic solutions in the case of a system depending on an extra parameter \(\alpha\). The paper is clear and easy to follow.