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Question

Are Propyl group, Butyl group also included in Alkyl group

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Solution

Yes they are included in alkyl group.
Propane has three carbons and eight hydrogens. If one of the six end hydrogens is replaced by a bond to something else, you end up with a normal propyl group. If one of the two middle hydrogens is replaced by a bond to something else, you end up with an isopropyl group.

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