Question

Prove that if H, K are subgroups of a group G and H Ų K = G. Then either H=G or K=G.

Answer 1

Here is your answer
...
Suppose that HK is a subgroup. Then for every x∈HK there exists h∈H,k∈Ksuch as x=hk. But x−1 is in HK too, and x−1=(hk)−1=k−1h−1∈KH, therefore HK⊆KH. Conversely, for k∈K and h∈H we have k∈HK and h∈HK. Since HK is a subgroup we get kh∈HK, so KH⊆HK.

Now suppose HK=KH.

1° e∈H,e∈K so e∈HK.

2° For x,y∈HK there exists hx,hy∈Hand kx,ky∈K such that x=hxkx and y=hyky, so xy=hxkxhyky, but KH=HKso you can find h∈H,k∈K such as kxhy=hk and therefore xy=hxhkky∈HK.

3° Let x=hk. Then x−1=k−1h−1∈KH, but since KH=HK, x−1∈HK.

So HK is a subgroup.