4. Suppose (G, ), (H, ), and (K, %) are groups with H,K C G. Show that (HNK, ) is a group. Solution: Solution goes here. Solution : - Let's understand this in a simple approach. here given that ( (, *) in a group to that is satisfied the following properties, closure, asociativity, identity and invoke. Ako given Hikk can and (h, *), ( 12, k) both are groups, 10 they allo satisfied above, four conditions. Now we have to prove that (rink, x) in a group, that is we have to satisfy the above four propertice. 1. closure: - Since know that H and K are groups to they satisfy the closure property that is Va, bEH implies that akbEH similarly for Va,b ER =] axbEk There fore axb E tink, ( since this quantity percent in both to it also be in intouection). 2. Auociativity:- Given that Hand K both are groups to clearly they maintain asociativity property tan hence Hink allo satisfy the anociativity. ies Vaibick tinks a* ( b* c ) = (axb) * C3. Identity:- Here Git and K are groups Clearly they exist identity, say e is such CEG, CEHand eEK Property of group days that, if identity exists then it is unique. Therefore eth , CEK = ) eEtink 4 . Inverve :- Since H and R have the invoice such that V C EH , K , VCEH,K , = ) CHET = CIC=e Since in above get that identity exist in ANK AO ACEHInk, JJ excel = ctxe = e Hence inverne exlet in tink. There fore Hink show all four properties under operation * hence ( Hink , * ) is a group. Nate:- I have provided all the method to colve the