1. Let H and K be subgroups of a group G. If G=HK and g=hk, where h is in H and k is in K, is
there any relationship among |g|,|h|,and |k|? What if G=HxK?
Solution
If hk = h'k' then (h')^-1h = k'(k)^-1 = x. Since x is in (H intersect K) = 1, then h = h', k = k'.
This means that any element g in G can be expressed uniquely as g = hk with h in H an k in K.
Since both H and K are normal then (hk)(h'k') = h(kh'k^-1)kk' where kh'k^-1 is in H. Using
the same reasoning (hk)(h'k') = hh'(h'^-1kh')k' where h'^-1kh' is in K.
Since h(kh'k^-1)kk' = hh'(h'^-1kh')k', and the expression for (hk)(h'k') as h"k" is unique,
then looking at the factors in each expression it follows that kh'k^-1 = h' and h'^-1kh'= k.
So G = H?K.