Formation and cleavage of three-membered rings

One of the commonest three-membered ring systems is the three-membered cyclic ether, otherwise known as an epoxide, or oxirane. The first thing to 

Write down the three possible hydrolysis mechanisms that may occur depending upon the pH of the solution. Pay particular attention to whether or not the substrate or intermediate is protonated, and also the nature of the attacking nucleophile. 

The formation of the vicinal diol, i.e. the 1,2-diol, may thus be achieved under a wide range of conditions. 

One of the bonds in the three-membered ring may break and so form an allylic carbonium ion that is stabilised due to delocalisation. This ion may then react further to form the final substituted product. 

This reaction may proceed in the reverse direction. For example, suggest the pathway whereby the allylic compound, 3-bromo-2-methylpropene, is converted to the 1-methylcyclopropyl cation. 

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