Asked by Jackie
If H is a subgroup of G and K is a normal subgroup of G,prove that HK = KH
Answers
Answered by
MathMate
By definition, HK contains all products of the form hk, where h∈H, and k∈K.
Similarly, by definition, KH contains all products of the form kh, where k∈K, and h∈H.
Next we need to prove that every product hk∈HK is also a member of KH.
By definition of normalcy of K in G, for k∈K,
xkx<sup>-1</sup> = k ∀x∈G.
Let h∈H, and k∈K.
hk = (hkh<sup>-1</sup>)h
=kh ∀h∈H and k∈K
so HK ⊆KH.
Similarly, prove that KH⊆HK.
Hence conclude that HK=KH.
Similarly, by definition, KH contains all products of the form kh, where k∈K, and h∈H.
Next we need to prove that every product hk∈HK is also a member of KH.
By definition of normalcy of K in G, for k∈K,
xkx<sup>-1</sup> = k ∀x∈G.
Let h∈H, and k∈K.
hk = (hkh<sup>-1</sup>)h
=kh ∀h∈H and k∈K
so HK ⊆KH.
Similarly, prove that KH⊆HK.
Hence conclude that HK=KH.