Cyclic Group C_9
The cyclic group C_9 is one of the two Abelian groups of group order 9 (the other order-9 Abelian group being C_3×C_3; there are no non-Abelian groups of order 9). An example is the integers modulo 9 under addition (Z_9). No modulo multiplication group is isomorphic to C_9. Like all cyclic groups, C_9 is Abelian.
The cycle graph of C_9 is shown above. The cycle index is
| Z(C_9)=1/9x_1^9+2/9x_3^3+2/3x_9. |
Its multiplication table is illustrated above.
The numbers of elements satisfying A^i=1 for i=1, 2, ..., 9 are 1, 1, 3, 1, 1, 3, 1, 1, 9.
Because the group is Abelian, each element is in its own conjugacy class. There are three subgroups: {1}, {1,C,F} and {1,A,B,C,D,E,F,G,H}. Because the group is Abelian, these are all normal. Since C_9 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 C8, Cyclic Group C10, Cyclic Group C11, Cyclic Group C12Explore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Cyclic Group C_9." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CyclicGroupC9.html