 TOPICS  # Cyclic Group C_8

The cyclic group is one of the three Abelian groups of the five groups total of group order 8. Examples include the integers modulo 8 under addition ( ) and the residue classes modulo 17 which have quadratic residues, i.e., under multiplication modulo 17. No modulo multiplication group is isomorphic to . The cycle graph of is shown above. The cycle index is  Its multiplication table is illustrated above.

The elements satisfy , four of them satisfy , and two satisfy .

Because the group is Abelian, each element is in its own conjugacy class. There are four subgroups: , , , and which, because the group is Abelian, are all normal. Since has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

Cyclic Group, Cyclic Group C2, Cyclic Group C3, Cyclic Group C4, Cyclic Group C5, Cyclic Group C6, Cyclic Group C7, Cyclic Group C9, Cyclic Group C10, Cyclic Group C11, Cyclic Group C12, Dihedral Group D4, Finite Group C2×C4, Finite Group C2×C2×C2, Quaternion Group

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## Cite this as:

Weisstein, Eric W. "Cyclic Group C_8." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclicGroupC8.html