The table that decides every action

Counting what a group cannot tell apart settles a real question with an average: the number of orbits of a group acting on a set is the mean number of points its elements hold still. Sixty-five thousand arrangements become eight hundred and five structures, and the arithmetic is a single sum.

It answers how many and nothing else. Two actions with the same orbit count are the same to it, and they need not resemble each other at all: four points in one orbit of four, and four points in four orbits of one, are both “an action on four points” and one of them has a symmetry the other does not. The object that tells them apart is a table.

The example is worth having before the general statement. Take 4mm, the class of a square with its four mirrors, acting on the plane. A point in general position has four… eight images, and its stabiliser is trivial. A point on a mirror has four images and a stabiliser of order two. A point on the axis has one image and a stabiliser of everything. Those are three different actions, and every action of 4mm on a finite set is built from them and from the few others in between. Listing the pieces is what the table does.

The table #

List the conjugacy classes of subgroups of the group, smallest first, and let the entry in row i and column j be

the number of cosets of Hⱼ that Hᵢ holds still.

Column j is the coset space G/Hⱼ — the orbit of a point whose stabiliser is Hⱼ — and the column records how much of that orbit each subgroup fixes.

Two properties do all the work and both are immediate.

The entries in the first row deserve one sentence, since they are the sizes of the coset spaces and every other row is bounded by them. A subgroup of order h has |G|/h cosets, so the first row runs from |G| down to one — eight, four, four, four, two, two, two, one for 4mm. Every other entry is at most the one above it in the same column, because a subgroup fixes a subset of what the identity fixes, and that bound is what keeps the table small.

Two remarks about the ordering before going on, because both matter and neither is obvious.

The rows and columns are indexed by conjugacy classes of subgroups rather than by subgroups. That is forced: two conjugate subgroups have coset spaces that are isomorphic as actions, so keeping them apart would make the table singular by having identical columns. Conjugacy is exactly the equivalence that makes the columns distinct, and it is the same equivalence an orbit is what the invariants cannot tell apart uses one level down, on points instead of on subgroups.

And the ordering is by size, not by containment. Containment is a partial order and the table needs a total one; sorting by order refines containment, which is all triangularity requires. Two subgroups of the same order that are not contained in one another give a zero in both of the off-diagonal positions they share, so the table is triangular in any refinement of containment and the particular choice does not matter.

It is worth saying what the entry means in words once more, because “the number of cosets of Hⱼ that Hᵢ fixes” is compact and easy to misread. A coset gHⱼ is a point of the orbit; Hᵢ fixes it when every element of Hᵢ, applied to that point, gives the point back. Since Hᵢ acts by left multiplication, that condition is Hᵢ gHⱼ = gHⱼ, which rearranges to g⁻¹Hᵢg ⊆ Hⱼ. So the entry counts the ways Hᵢ can be conjugated into Hⱼ, divided by how many of those give the same coset — and every mark in this collection was computed by testing that containment directly rather than by a formula.

Invertible means complete #

An action of a finite group decomposes into orbits, and each orbit is a coset space of the stabiliser of any of its points. So an action is a list of multiplicities — how many orbits of each type — and the fixed-point count of a subgroup is the table’s row applied to that list.

Because the table is invertible, the map runs backwards. Given the number of points each subgroup class holds still, the multiplicities come out, one at a time, from the bottom.

That is a stronger statement than it looks. It says the fixed-point vector is a complete invariant for finite actions: two actions with the same fixed-point counts are isomorphic, not merely equinumerous. Burnside’s lemma extracts one number from that vector and throws the rest away.

There is a subtlety in “complete” worth naming, because the word is used in two senses in the literature. The table determines the decomposition of an action into orbits — which orbit types and how many of each — and that is what makes two actions isomorphic. It does not determine the group: two non-isomorphic groups can have tables of the same shape, and distinguishing them needs something else. So the table is a complete invariant for actions of a fixed group and not for groups themselves, which is the right way round for the use below.

One consequence deserves its own line because it is a genuine strengthening of the lemma rather than a restatement. Two actions with the same number of orbits and the same number of points can still differ: eight points in two orbits of four, or eight points in one orbit of six and two of one. The fixed-point vectors of those two differ in the entries for the subgroups that stabilise the singletons, and the table separates them at once. Burnside’s average sees the same pair of numbers in both cases.

The columns are Wyckoff positions #

The crystallographic reading is not an analogy; it is the same object under a different name.

A point of a crystal has a site symmetry — the subgroup of operations fixing it — and its orbit has as many points as the group has cosets of that subgroup. That is a column of the table. The set of columns is the set of position types, which is what a Wyckoff table lists, and Wyckoff positions is where this collection computes them from a space group.

The last column of that figure is worth dwelling on because it is a fact about crystals stated in group-theoretic terms. A normal site symmetry fixes every point of its own orbit — the whole orbit shares it — while a non-normal one fixes only the point it belongs to and whichever others happen to lie on the same element. In a crystal that is the difference between a mirror plane every equivalent atom lies on and a mirror plane each atom has its own copy of.

And the triangularity says something a reader may have taken for granted: a more symmetric site cannot be a special case of a less symmetric one. The zeros above the diagonal are exactly the statement that a point with a large site symmetry never appears in the orbit of a point with a small one.

The reading also runs backwards, and that is where it earns its place. Given a crystal structure and a count of how many atoms lie on each kind of symmetry element, the table says which combinations of Wyckoff positions are possible — and refuses the combinations that are not. A structure report claiming a certain number of atoms with a certain set of site symmetries is making a claim about a vector, and the vector either decomposes into non-negative integers or it does not. That is a check on a published structure available from the space group alone.

The rows have a reading too, and it is the one an experiment supplies. Row i says how many points of each orbit type the subgroup Hᵢ holds still — so if a measurement can count the atoms lying on a mirror, it has measured one row’s worth of one action, and the table converts that into a constraint on the whole structure. A vector of such counts, one per symmetry element, is exactly the input the inversion wants.

How large the tables are #

Thirty-three is a small number for an object that classifies every possible action of a group of order forty-eight, and the smallness is the point. There are infinitely many sets a group can act on and only thirty-three ways it can act indecomposably, so every action of m3̅m on any finite set whatever is a list of thirty-three non-negative integers. Nothing else about the set survives.

The third of those is the one that makes the inversion trustworthy. Back-substitution through a triangular system always produces something; what it need not produce is a list of non-negative integers, and a vector that is not the fixed-point count of any action fails on exactly that. So the arithmetic carries its own validity check, and a fixed-point vector that comes out of a measurement rather than a construction can be tested rather than assumed.

A note on the tables’ sizes, since the numbers look arbitrary. The classes whose subgroup count equals their class count are the ones with no conjugacy to do — every subgroup normal — and those are exactly the abelian classes together with a few others. The largest gaps are in the cubic classes, where a single kind of subgroup can occur in three, four or six orientations, and the gap is a count of orientations rather than of anything new. How many axes there are is a Sylow count measures the same orientation multiplicities at the prime level; here they appear as the difference between two columns of a census.

There is one crystallographic reading of the completeness that is worth spelling out because it sounds stronger than it is. Two arrangements of atoms on the sites of a crystal that fix the same number of points under every subgroup are the same arrangement up to relabelling — as sets with an action. They are not necessarily the same structure: which atom is which is extra data, and two arrangements can have identical orbit structure with different species on corresponding orbits. So the table settles the combinatorics of the sites and leaves the chemistry entirely open, which is the same division what counts as a bond draws elsewhere.

There is a limit on the reverse reading worth stating in the same breath, because it is easy to over-promise. Counting the atoms on each kind of element gives a fixed-point vector only if the count is of all of them, orbit by orbit, and a diffraction experiment does not deliver that directly — it delivers intensities. What is available in practice is a partial vector with some entries known and others not, and a triangular system with missing entries constrains rather than determines. That is still useful and it is a weaker statement than the inversion above.

One more property of the census is worth a sentence, because it says how much of the difficulty is conjugacy. Summed over the thirty-two, there are a great many more subgroups than classes of them, and the excess is entirely orientation: the same abstract subgroup sitting on different axes. So the table’s size is a measure of how much of a class’s structure survives the question “which way is it facing”, and the cubic classes lose the most.

What this is for #

Two uses beyond the tidiness.

The first is that it turns a question about a structure into a question about a vector. Asking what the possible arrangements of some species on the sites of a crystal are is asking which actions of the class occur, and the answer is a list of multiplicities constrained by the total number of sites. That is a linear problem over the non-negative integers rather than an enumeration, and the difference in cost is the difference between a table lookup and a search.

The second is that it is the finite shadow of a much larger apparatus. The table of marks is the multiplication table of the Burnside ring — the ring whose elements are formal differences of finite G-sets — and the invertibility above is the statement that the ring embeds in a product of copies of the integers, one per subgroup class. That embedding is what makes computations in it finite. None of that machinery is built here, and the table is the part of it a crystallographer would recognise.

One more consequence of the triangularity, and it is the practical one. Solving a triangular system is n² operations rather than n³, and it never needs a pivot: the diagonal entries are positive by construction, so no row exchange can be required and no numerical judgement is made anywhere. A decomposition computed this way is exact in integers from start to finish, and the assertion that the multiplicities come out integral is checking the input rather than the arithmetic.

A last note on the arithmetic’s honesty. Every entry of every table here was computed by enumerating cosets and testing the containment g⁻¹Kg ⊆ H directly, rather than by any formula — and the properties that make the table useful were then asserted rather than assumed: the triangularity is checked entry by entry, the diagonal is checked positive, and the first row is checked against the coset space sizes it must equal. Those three checks would each fail loudly on a subgroup enumeration that had missed a subgroup or merged two conjugacy classes, which is the failure mode a construction like this actually has.

One more thing the table does not do, and it is the one a reader coming from representation theory will ask about. The table of marks classifies actions on sets; the character table classifies actions on vector spaces. The two are related — a set gives a permutation representation, so there is a map from one to the other — and the map is neither injective nor surjective: non-isomorphic actions can give isomorphic permutation representations, and most representations do not come from a set at all. A character does not know its basis is the other table, and the two answer different questions about the same group.

A last observation about the shape of the argument, since it recurs in this collection and is worth naming. The table is useful because it is triangular, and it is triangular because of an inequality — a subgroup does not fit inside a smaller one. A great many complete invariants in mathematics are built that way: order the pieces by some size, arrange that the interaction between two pieces vanishes when the sizes go the wrong way, and the resulting system inverts. Selling’s reduction has the same shape, with a quantity that decreases at every step, and so does the back-substitution in the extension arithmetic. It is the cheapest structure a classification can have.

Where this stops #

The table is computed here for point groups and not for space groups. A space group is infinite, its subgroups of finite index are the ones a Wyckoff table is about, and the same construction applies with “conjugacy class of subgroup” replaced by “class of subgroup of finite index” — but the list is infinite, and the triangularity that makes the finite case invertible has to be replaced by a filtration. That is real machinery and this collection does not carry it.

And the completeness is for actions on sets. Two crystal structures with the same fixed-point vector have the same orbit structure and can still be different structures, because a structure is a set of points with positions rather than an abstract set. The table says which orbit types occur and how many; where the orbits sit is the moduli of the free parameters and is not a group-theoretic question at all.