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- If H and K are arbitrary subgroups of G, prove that HK=KH if and only if HK is a subgroup of G.Let be a group of order 24. If is a subgroup of , what are all the possible orders of ?Let H and K be subgroups of a group G and K a subgroup of H. If the order of G is 24 and the order of K is 3, what are all the possible orders of H?
- 19. With and as in Exercise 18, prove that is a subgroup of . Exercise18: 18. If is a subgroup of , and is a normal subgroup of , prove that .With H and K as in Exercise 18, prove that K is a normal subgroup of HK. Exercise18: If H is a subgroup of G, and K is a normal subgroup of G, prove that HK=KH.Let be a subgroup of a group with . Prove that if and only if
- 27. Suppose is a normal subgroup of order of a group . Prove that is contained in , the center of .15. Prove that on a given collection of groups, the relation of being a homomorphic image has the reflexive property.Find a subset of Z that is closed under addition but is not subgroup of the additive group Z.