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Cyclic Group C_7


C_7 is the cyclic group that is the unique group of group order 7. Examples include the point group C_7 and the integers modulo 7 under addition (Z_7). No modulo multiplication group is isomorphic to C_7. Like all cyclic groups, C_7 is Abelian.

CyclicGroupC7CycleGraph

The cycle graph is shown above, and the group has cycle index is

 Z(C_7)=1/7x_1^7+6/7x_7.

The elements A_i of the group satisfy A_i^7=1, where 1 is the identity element.

CyclicGroupC7Table

Its multiplication table is illustrated above and enumerated below.

C_71ABCDEF
11ABCDEF
AABCDEF1
BBCDEF1A
CCDEF1AB
DDEF1ABC
EEF1ABCD
FF1ABCDE

Because it is Abelian, the group conjugacy classes are {1}, {A}, {B}, {C}, {D}, {E}, and {F}. Because 7 is prime, the only subgroups are the trivial group and the entire group. C_7 is therefore a simple group, as are all cyclic graphs of prime order.


See also

Cyclic Group, Cyclic Group C2, Cyclic Group C3, Cyclic Group C4, Cyclic Group C5, Cyclic Group C6, Cyclic Group C8, Cyclic Group C9, Cyclic Group C10, Cyclic Group C11, Cyclic Group C12

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Cyclic Group C_7." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclicGroupC7.html

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