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# Struct/The cyclic group with six elements

Title: The group with three elements.

There is a unique cyclic group, up to isomorphism, with six elements. One example is the group $(\ZZ / 6 \ZZ, +)$ of residues, mod $6$, displayed below.

Addition table for $\ZZ / 6 \ZZ$.
$+$ $\bar 0$ $\bar 1$ $\bar 2$ $\bar 3$ $\bar 4$ $\bar 5$
$\bar 0$ $\bar 0$ $\bar 1$ $\bar 2$ $\bar 3$ $\bar 4$ $\bar 5$
$\bar 1$ $\bar 1$ $\bar 2$ $\bar 3$ $\bar 4$ $\bar 5$ $\bar 0$
$\bar 2$ $\bar 2$ $\bar 3$ $\bar 4$ $\bar 5$ $\bar 0$ $\bar 1$
$\bar 3$ $\bar 3$ $\bar 4$ $\bar 5$ $\bar 0$ $\bar 1$ $\bar 2$
$\bar 4$ $\bar 4$ $\bar 5$ $\bar 0$ $\bar 1$ $\bar 2$ $\bar 3$
$\bar 5$ $\bar 5$ $\bar 0$ $\bar 1$ $\bar 2$ $\bar 3$ $\bar 4$