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Math Help - central product of two groups

  1. #1
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    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\}. )
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  2. #2
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    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:

    central product of two groups-mhfgroups11a.png

    central product of two groups-mhfgroups11b.png
    Thanks from xixi
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  3. #3
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    Re: central product of two groups

    Thanks, nice and precise proof!
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