central product of two groups

How can I prove that central product of by is isomorphic to central product of by , i.e. .

(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= .

Hint: We have where and is a homomorphism for which we have . Then . )

2 Attachment(s)

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:

Attachment 28044

Attachment 28045

Re: central product of two groups

Thanks, nice and precise proof!