Part 1 of 13. Emmy Noether's work in physics and algebra keeps asking what stays fixed when a group acts. This part sets up the vocabulary: group actions, orbits, stabilizers and invariant functions.
This series follows one instinct of Emmy Noether's: to understand a system, find what stays the same when you act on it, and let that do the work.
Rotating the plane by sends to . A quantity computed from the point is invariant if no rotation changes it. Figure 1 carries a point through a full turn and traces five quantities along the way.
Figure 1. Each row plots one quantity of the rotated point P for every angle from 0° to 360°, with a dot at the current angle, and gives its spread (largest minus smallest value over the turn). Drag the hollow start point P₀ to try another orbit, or use the arrow keys on it. Rows with zero spread are drawn in green.
Only stays flat, and with it anything computed from it, such as the distance to the origin. The rotations form a group: two rotations compose to a rotation, each can be undone, and rotating by zero does nothing. An invariant of the group is a quantity that every member leaves alone.
Most groups in Noether's work are finite. The first we look at closely is , the rigid motions that carry a square onto itself: the identity , rotations by 90°, 180°, 270°, and four reflections. Figure 2 lists all 64 ways of composing two of them.
Figure 2. The Cayley table of , computed by multiplying the 2×2 matrices. Row , column holds : do , then . Click a cell to watch the square perform both moves, or a row label to apply one element. Red cells are pairs where . Corners start as A, B, C, D counter-clockwise from the upper right.
Invariance is a property of a function and a group together. Figure 3 tests seven polynomials against six groups acting on the plane, from the trivial group up to , all rotations and reflections. Here , is the four rotations of the square, and is the flip across the x-axis.
Figure 3. Each cell is the largest change over every element of the group and ten seeded sample points ; is sampled at 360 rotations and 360 reflections. Zero cells are green. Click a cell: the strip plots at every point of one orbit, and an invariant collapses to a single dot.
Along each group contains the last, so it imposes more conditions and fewer functions pass. keeps and , which a general rotation breaks.
A group action of on a set assigns to each and a point , so that
The orbit is every point reachable from ; the stabilizer is the subgroup that leaves where it is. For a finite group,
Figure 4. acting on the plane. Drag the red point (it snaps onto the dashed mirror lines and the origin), or use the arrow keys. Amber dots are its orbit, each labelled with the elements that send the red point there. Mirror lines of reflections in the stabilizer turn green.
Every orbit point carries the same number of labels, : if sends somewhere, so does for each in the stabilizer. The eight elements split evenly over the orbit, which is the whole proof of the theorem.
Colour the four corners of a square with colours. Colourings that differ by a symmetry are the same square seen from another side, so the honest count is the number of orbits. Summing orbit-stabilizer over the colourings counts orbits without listing them:
Figure 5. Left: every colouring of the corners, grouped into -orbits by applying all eight elements. Right: how many colourings each element fixes, with their average drawn as a line. Click an orbit to light up its stabilizer, or a bar to outline the colourings that element fixes.
This is usually called Burnside's lemma, though Burnside's 1897 book credits Frobenius (1887), and Cauchy knew it in 1845.
The symmetric group shuffles three numbers . A polynomial fixed by all six shuffles is symmetric, and the symmetric polynomials form a ring. Waring stated in 1770, and Gauss proved in 1816, that the whole ring is generated by three of them:
The proof is an algorithm. Take the largest term in dictionary order. Symmetry forces , so has the same leading term; subtract a multiple of it and repeat. The leading term drops each time, so it stops.
Figure 6. The reduction run in exact integer arithmetic. Each row removes one leading term. The polynomial is fixed only by the three cyclic shuffles, and the algorithm stops on a leading term whose exponents are out of order. The last line evaluates both sides at .
The discriminant preset turns into the discriminant of the cubic , whose roots are . The next part starts from that invariant. Whether a ring of invariants is generated by a finite list was the question of Noether's 1907 dissertation. Her advisor Paul Gordan had proved it for binary forms in 1868 by explicit computation, and Hilbert proved it for forms in any number of variables in 1890 by an existence argument Gordan distrusted.
One word does two jobs. A function is invariant when for every ; an object is fixed when . The circle in Figure 1 is fixed as a set, and is an invariant function.