Let $G$ be a finite group that acts on a set $X$. For each $g$ in $G$ let $X^g$ denote the set of elements in $X$ that are fixed by $g$. Then the number of orbits is

Problems

• Necklace of Beads
• TheBeautifulBoard
• Magic Bracelet
• Lucy and the Flowers
• Sorting Machine
• Pizza Toppings
• Alphabet soup²
• Drum Decorator¹
• Count the Necklaces
• Cube Coloring

See also

• Pólya enumeration theorem

External links

• Burnsides Lemma
• Burnside's lemma: orbit-counting theorem (cube and necklace colorings)
• Burnside's lemma
• Burnside's Lemma