Homework SourseLet H and K be two subgroups of a group G We define a HK as
Let H and K be two subgroups of a group G. We define a HK as a subset of G as HK = {hk|h H, k K}. Show that if HK = G, H K = {e} and hk = kh, h H, k K, then H × K is isomorphic to G.
Let H and K be two subgroups of a group G. We define a HK as a subset of G as HK = {hk|h H, k K}. Show that if HK = G, H K = {e} and hk = kh, h H, k K, then H × K is isomorphic to G.
Let H and K be two subgroups of a group G. We define a HK as a subset of G as HK = {hk|h H, k K}. Show that if HK = G, H K = {e} and hk = kh, h H, k K, then H × K is isomorphic to G.
Solution
First show that hkh-1k-1=e for all hH, kK. Then show that the function
f:H×KG given by f(h, k) =hk is a group isomorphism.
For all hH, kK, we have
hkh-1k-1= (hkh-1)k-1=h(kh-1k-1)HK
so hkh-1k-1=e
. This means that hk=kh
Now prove that f is an isomorphism,
For h, h\'H and k, k\'K, we compute:
f(h, k)·f(h\', k\') = (hk)(h\'k\')
= h(kh\')k\'
= h(h\'k)k\'
= (hh\')(kk\')
= f(hh\', kk\')
= f((h, k)·(h\', k\'))
Hence f is a homomorphism. Since HK=G, any element gG
may be written as g=hk for some hH, kK, and
f(h, k) =hk = g, so f is surjective. Finally, if (h, k)ker f, then hk=e, so
h=k-1HK={e}, so (h, k) = (e, e). Hence f is injective.
