- #1

CAF123

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## Homework Statement

We were given the following argument showing that ##D_6 \cong D_3 \times \mathbb{Z}_2##.

What we did was take two subgroups H and K of G such that their intersection was trivial and that ##hk = kh## for all h and k. H = {symmetries of 6-gon that preserved a triangle inside} and K = <g

^{3}>. By some theorem, we proved this shows that HK is isomorphic to H x K which in turn is isomorphic to ##D_3 \times \mathbb{Z}_2##

The question is: Does a similar argument apply to ##D_n## where n is even ##\geq 6##?

## The Attempt at a Solution

I presume the question is asking if we can write ##D_n \cong D_x \times \mathbb{Z}_y## for some x and y?

What I have done so far is the following:

Let G = D

_{n}. Choose H and K such that |H| . |K| = 2n. For the above argument to work, necessarily H and K are subgroups of G. So by Lagrange, the orders of H and K have to divide the order of D

_{n}. This means: $$|D_n| = a|H|, |D_n| = b|K|\,\Rightarrow\,a|H| = b|K|.$$Since H and K are subgroups, they are subsets so H,K contained in G. But since we want the intersection to be trivial (i.e ##H \cap K = \left\{e\right\}##, we have that ##|H| \neq |K|##. I am not sure if this helps me at all and I am unsure of how to proceed.

Many thanks.