Cyclic Group C_10
The cyclic group C_(10) is the unique Abelian group of group order 10 (the other order-10 group being the non-Abelian D_5). Examples include the integers modulo 10 under addition (Z_(10)) and the modulo multiplication groups M_(11) and M_(22) (with no others). Like all cyclic groups, C_(10) is Abelian.
The cycle graph of C_(10) is shown above. The cycle index is
| Z(C_(10))=1/(10)x_1^(10)+1/(10)x_2^5+2/5x_5^2+2/5x_(10). |
Its multiplication table is illustrated above.
The numbers of elements satisfying A^i=1 for i=1, 2, ..., 10 are 1, 2, 1, 2, 5, 2, 1, 2, 1, 10.
Because the group is Abelian, each element is in its own conjugacy class. There are four subgroups: {1}, {1,E}, {1,B,D,F,H}, and {1,A,B,C,D,E,F,G,H,I}. Because the group is Abelian, these are all normal. Since C_(10) 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 C9, Cyclic Group C11, Cyclic Group C12Explore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Cyclic Group C_10." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CyclicGroupC10.html