THE ORBIT–STABILIZER orbit size times stabilizer size equals the group — symmetry, counted
When a symmetry groupG acts on a set, each element sits in an orbit (everywhere the symmetries can send it) and has a stabilizer (the symmetries that fix it in place). The orbit–stabilizer theorem says these multiply to the whole group: |orbit| × |stabilizer| = |G|. It is the counting backbone of symmetry. Take a square with its 8 symmetries (the dihedral group D₄): a corner can go to any of 4 corners (orbit 4), and 2 symmetries fix that corner (identity and a reflection), and indeed 4 × 2 = 8. The invariant here is |G| itself — however you split a symmetric object into orbit and stabilizer, the product returns the group's order.
THE TECHNIQUE |orbit| × |stabilizer| = |G|
The 8 symmetries of a square acting on its corners. The demo multiplies a corner's orbit by its stabilizer and recovers |G|=8: live demo
HISTORY & CREDIT Lagrange lineage (cosets)
“Counting symmetric arrangements needs case-by-case bookkeeping.” — orbit times stabilizer is the group order, always. The split changes; the product is nailed to |G|. Symmetry does the counting for you. cited
1770s · Lagrange — the coset theorem (subgroup order divides group order), the parent result. orbit–stabilizer · the stabilizer is a subgroup; the orbit is its coset space, so |orbit|=|G|/|stab|. now · the foundation of counting under symmetry — Burnside (dart 290), Sylow theory, Pólya enumeration.
The stabilizer is a subgroup, the orbit is its cosets, and Lagrange makes them multiply to |G|. Symmetry always balances its own books. Lagrange
RECOMMEND FOR I-13 orbit × stabilizer = |G|, computed
On the canonical compiler, a square corner has orbit 4 and stabilizer 2, and 4 × 2 = 8 = |D₄|:
$ i13 run inv_orbitstab.i13 # |orbit| * |stabilizer|
orbit = 4
stabilizer = 2
group_order = 8 -- = |D4|, however you split it
Recommend: orbit–stabilizer is the invariant |G| recovered by a multiplication, and it is where the batch turns algebraic. The “transformation” is the choice of which element you look at: different corners can have different-looking orbits and stabilizers, but the product is always |G|. i13 multiplies 4 × 2 = 8 and recovers the group order — a foreign shape (group action) enacted as one arithmetic step. It is the counting law under symmetry, and the direct setup for Burnside (dart 290), where the same group-action bookkeeping counts distinct objects.