Let G be a group. Show that Gneq HcupKG =HcupK for any two subgroups Hleq GHleq G and Kleq GKleq G.

necessaryh 2020-11-06 Answered
Let G be a group. Show that GqHKG=HK for any two subgroups HGHGandKGKG.
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delilnaT
Answered 2020-11-07 Author has 94 answers
Let G is a group and H,K are two non trivial subgroup of G. If possible, let G=HK. Now Since, H not equal to G and K not equal to G, so we must have HH not equal to K and H and K can not be subset of each other. Therefore there exists hHKandkKH. Now since G is a group and h,kHK=G, then hkG=HK. So, hk is either in H or in K. Suppose hkH, since H is a subgroup and h∈H, so h1H.h1H. Therefore,
h1(hk)=(h1×h)k=kH, which is against our choice, contradiction. Similarly, we can show that hk can not be in K. Henc, $G $ not equal to HK, for non trivial subgroups H and K.
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