Cyclic group C_n - Knowlpedia

For an integer $n\geq1$ , the cyclic group $C_{n}$ is the group $\mathbb Z/n\mathbb Z$ of residue classes, with addition modulo $n$ .

Order and simplicity

It has $n$ elements, generated by the residue class of $1$ . Every subgroup is cyclic, and every subgroup is normal because the group is abelian. It is simple exactly when $n$ is prime. If $n>1$ is composite, a proper divisor gives a proper nontrivial subgroup. The case $n=1$ is the trivial group and is excluded from simplicity.

Catalogue convention

This is an additive group object. The ring with the same residue classes has additional multiplication, and belongs to different categories. The cyclic-group classification identifies every cyclic group of this order with this model.

References

J. S. Milne, Group Theory, v4.01, §1.16, p. 13: finite cyclic groups; §1, Lagrange theorem.