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I was asked the following: Let $D_{16}$ be the dihedral group of order $16$. What is $D_{16} / Z(D_{16})$?

I know that the center of $D_{16}$ har order $2$. So therefore, the quotient has order $16/2 = 8$. I know that there are $5$ groups of order $8$, but I am not sure which of these $D_{16} / Z(D_{16})$ is isomorphic to.

I am pretty sure that it is not $\mathbb{Z}_{8}$ since that would imply that $D_{16}$ is abelian (which it is not). So what is it?

EDIT: I see Find $G/Z(G)$ given the following information about the group? but I am not sure that being generated by two elements mean.

John Doe
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2 Answers2

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It cannot be the cyclic group of order $8$ because the factor with respect to the center cannot be a nontrivial cyclic group. (I mean in general. This is a well-known exercise problem.)

Hint: try to compute the order of elements in the factor group. (The center is the $2$-element group generated by the rotation with $\pi$.) Also note that it cannot be commmutative: pick two reflections such that the angle between the two axes is $\pi/8$. These elements do not commute in the factor group. So you just need to choose between the quaternion group and $D_8$: the order of elements will seal the deal.

A. Pongrácz
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$D_{16} / Z(D_{16})\cong D_8$.

yavar
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