# 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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