Cyclic Group C_8

The cyclic group C_8 is one of the three Abelian groups of the five groups total of group order 8. Examples include the integers modulo 8 under addition (Z_8) and the residue classes modulo 17 which have quadratic residues, i.e., {1,2,4,8,9,13,15,16} under multiplication modulo 17. No modulo multiplication group is isomorphic to C_8.


The cycle graph of C_8 is shown above. The cycle index is


Its multiplication table is illustrated above.

The elements A_i satisfy A_i^8=1, four of them satisfy A_i^4=1, and two satisfy A_i^2=1.

Because the group is Abelian, each element is in its own conjugacy class. There are four subgroups: {1}, {1,D}, {1,B,D,F}, and {1,A,B,C,D,E,F,G} which, because the group is Abelian, are all normal. Since C_8 has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

See also

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.

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