 TOPICS  # Cyclic Group C_9

The cyclic group is one of the two Abelian groups of group order 9 (the other order-9 Abelian group being ; there are no non-Abelian groups of order 9). An example is the integers modulo 9 under addition ( ). No modulo multiplication group is isomorphic to . Like all cyclic groups, is Abelian. The cycle graph of is shown above. The cycle index is  Its multiplication table is illustrated above.

The numbers of elements satisfying for , 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: , and . Because the group is Abelian, these 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 C8, Cyclic Group C10, Cyclic Group C11, Cyclic Group C12

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## Cite this as:

Weisstein, Eric W. "Cyclic Group C_9." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclicGroupC9.html