In organic chemistry, the phenyl group or phenyl ring is a cyclic group of atoms with the formula C6H5. Phenyl groups are closely related to benzene. Phenyl groups have six carbon atoms bonded together in a hexagonal planar ring, five of which are bonded to individual hydrogen atoms, with the remaining carbon bonded to a substituent.

The circle of fifths may be endowed with a cyclic group structure

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Cyclic Group

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A Cayley graph for the free product of the cyclic groups C3 and C5 (with the natural generating set). Drawn in TikZ :)

Direct Products of Finite Cyclic Groups Video 1

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Proof that Z x Z is not a cyclic group

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