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 \(G\).

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