# Identifying A \(2p\) Group

**Algebra**Level 5

Take a non-abelian group \(G\) with order \(2p\) for an odd prime \(p\). Which of the following is true?

**Details and assumptions**

\(P_y\) is the rotational symmetry group of a right pyramid whose base is a regular \(2p\)-gon.

\(P_r\) is the rotational symmetry group of a right prism whose base is a regular \(p\)-gon.

The group operations of \(P_y\) and \(P_r\) are composition of rotations, and of \(\mathbb{Z}_p \times \mathbb{Z}_2\) and \(\mathbb{Z}_2^p\) are addition (component-wise).

\(e_G\) is the identity element of \(G\).