# Math Help - central product of two groups

1. ## central product of two groups

How can I prove that central product of $D_{8}$ by $D_{8}$ is isomorphic to central product of $Q_{8}$ by $Q_{8}$, i.e. $D_{8}\circ D_{8} \cong Q_{8}\circ Q_{8}$.
(If G is a group and H,k be its subgroups, then G is central product of H by K if we have: 1)G=HK. 2)hk=kh for all h in H and all k in K, then we write G= $H \circ K$.
Hint: We have $H \circ K \cong (H \times K)/D$ where $D=kerf$ and $f:H \times K \rightarrow H \circ K$ is a homomorphism for which we have $f(h,k)=hk$. Then $D=kerf=\{(h,h^{-1})|h \in H \cap K\}$. )

2. ## Re: central product of two groups

Hi,
My first thought was that this is easy. Just compute in the dihedral group and the quaternion group. However, the details turn out to be somewhat lengthy. Here they are:

3. ## Re: central product of two groups

Thanks, nice and precise proof!