Is $D_{2n}$ isomorphic to $D_n \times \Bbb{Z}_2$ for all $n$? For all odd $n$?

Asked Nov 15 '15 at 10:47

Active Nov 15 '15 at 12:09

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Is $D_{2n}$ isomorphic to $D_n \times \Bbb{Z}_2$ for all $n$? For all odd $n$?

I just want to see if my thinking is sound here. My thought process is this.

$\mathbb{Z}_2 \cong \{e,j\} \subset D_n$

where $e$ is the identity and $j$ is a reflection about some axis.

let $\phi \colon D_n \times \{e,j\} \to D_{2n}$ defined by the map

$(a,b) \mapsto ab$

This is a surjective homomorphism and because the sizes are the same, it is an isomorphism.

Is this right? Have I done enough?

edited Nov 15 '15 at 11:18

A.P.

asked Nov 15 '15 at 10:47

user139985

I take it you are using $D_n$ for the symmetries of a regular $n$-gon. $D_4$ is certainly not isomorphic to $D_2\times{\bf Z}_2$, as one is abelian, not the other. – Gerry Myerson Nov 15 '15 at 11:21
It is only for when n is odd. – user139985 Nov 15 '15 at 11:31
Could you say more about why your mapping is a homomorphism? (Keep in mind that $a \in D_{n}$ and $b \in {e, j}$, and that $a$ and $b$ probably have customary (and different) interpretations as elements of $D_{2n}$.) – Andrew D. Hwang Nov 15 '15 at 11:32
1

Related. See also this. – A.P. Nov 15 '15 at 12:15

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