# Cyclic group

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.

In organic chemistry, the phenyl group or phenyl ring is a cyclic group of atoms with the formula C6H5.

Cyclic Groups Proof: If x is a generator so is its inverse

Examples of Cyclic Groups

Direct Products of Finite Cyclic Groups Video 1

Proof that Z x Z is not a cyclic group

398 is the number of integers with complexity 22.

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

Proof that (R, +) is not a Cyclic Group

Every Cyclic Group is Abelian Proof

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

The cyclic group Z26 underlies Caesar's cipher.

Cyclic Group

Cyclic Groups

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

Proof: Let G = (a) be a cyclic group generated by a. Its trivial subgroup {e} containing the identity in of G, is cyclic. Let H be any nontrivial subgroup G.