In combinatorics, a necklace of length $n$ is an equivalence class of strings of length $n$, under rotation, so $abcde = bcdea$. A bracelet is an equivalence class of strings under rotation and reflection (so in addition $abcde = edcba$).
Given a set $A$ which we call the alphabet, the set of length $n$ strings are the ordered tuples $A^n$ which we can interpret as the set of all mappings $\{0,1,2,\ldots,n-1\} \to A$.
Let $S_n$ be the symmetric group of order $n$, it acts naturally on $\{0,1,2,\ldots,n-1\}$, which induces an action on $A^n$. Two particular subgroups can be distinguished: the cyclic group $C_n$, which is generated by $i \mapsto i +1 \pmod n$, and the dihedral group $D_n$, which has the additional generator $i \mapsto n -1 - i$.
The orbits of $C_n$ in $A^n$ are called necklaces. They are the equivalence class of strings where two strings are equivalent if one can cyclically rotate one to the other. For example:
$$ abcde \mapsto bcdea \mapsto cdeab \mapsto deabc \mapsto eabcd\mapsto abcde $$
shows the orbit of the string $abcde$ under the action $C_n$. These strings are all regarded as the same necklace.
The orbits of $D_n$ in $A^n$ are called bracelets. In addition to the cyclic rotations, we also identify two strings if they are mirror images
$$ abcde \leftrightarrow edcba.$$
