(a) Let p be an odd prime and G a group of order 2p. Show that...

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(a) Let p be an odd prime and G a group of order 2p. Show that G is either cyclic i. e. G=Z₂, or G = Dp. What can you deduce from this result? (b) Let G be a group with subgroups H and K such that HK = {1}, the elements of H commute with those of K and HK = G. Show that G = H× K. (c) List the element of (i) Z, OZA (ii) Z, Z. Find the order of each element in (i). are the groups cyclic ? (a) Let p be an odd prime and G a group of order 2p. Show that G is either cyclic i. e. G=Z₂, or G = Dp. What can you deduce from this result? (b) Let G be a group with subgroups H and K such that HK = {1}, the elements of H commute with those of K and HK = G. Show that G = H× K. (c) List the element of (i) Z, OZA (ii) Z, Z. Find the order of each element in (i). are the groups cyclic ?

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a To prove that a group G of order 2p where p is an odd prime is either cyclic ie GZ 2p or isomorphic to the dihedral group of order 2p ie GD p well use Sylows theorems First lets denote n p as the nu... View the full answer