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class equation

Contents

Contents

Idea

The term “class equation” (or class formula, or orbit decomposition formula) refers to a basic type of counting argument that comes about by decomposing a finite G-set as a sum of its orbits. It has a number of fundamental applications in group theory.

Statement

Let GG be a group and let AA be a G-set (given by a homomorphism Ghom Set(A,A)G \to \hom_{Set}(A, A) of monoids, with which is associated an action α:G×AA\alpha: G \times A \to A). Recall that AA is connected in the category of G-sets if AA is inhabited and the action is transitive; in this case, choosing an element aAa \in A, there is a surjection of GG-sets GAG \to A sending 1a1 \mapsto a, and this induces an isomorphism G/Stab(a)AG/Stab(a) \cong A where Stab(a)Stab(a) is the stabilizer of aa and G/Stab(a)G/Stab(a) is the GG-set consisting of left cosets of Stab(a)Stab(a).

More generally, each GG-set AA admits a canonical decomposition as a coproduct of its connected components; the components are usually called the orbits of the action. Choosing a representative element a xa_x in each orbit xx, this means we have an isomorphism of GG-sets

A orbitsxG/Stab(a x).A \cong \sum_{orbits x} G/Stab(a_x).

By taking GG and AA to be finite and counting elements, we get an equation of the form

|A|= orbitsx|G||Stab(a x)|.{|A|} = \sum_{orbits x} \frac{{|G|}}{{|Stab(a_x)|}}.

We call an instance of this equation a class equation. By judicious choice of groups GG and GG-sets AA, often in combination with number-theoretic arguments, one can derive many useful consequences; some sample applications are given below.

Notice that reading the class equation equivalently as

orbitsx1|Stab(a x)|=|A||G|\sum_{orbits x} \frac{{1}}{{|Stab(a_x)|}} = \frac{|A|}{|G|}

it expresses the groupoid cardinality of the action groupoid of GG acting on AA.

Applications

Centers of pp-groups

Let pp be a prime; recall that a pp-group is a finite group whose order is a power of pp. A basic structural result is the following.

Proposition

A non-trivial pp-group GG has a nontrivial center Z(G)Z(G).

Proof

Let GG act on itself by the conjugation action G×GGG \times G \to G, (g,h)ghg 1(g, h) \mapsto g h g^{-1}. In this case an orbit Orb(h)Orb(h) is usually called the conjugacy class of hh, and Orb(h)Orb(h) is trivial (consists of exactly one element hh) iff hh belongs to Z(G)Z(G). In any case |Orb(h)|=|G||Stab(h)|{|Orb(h)|} = \frac{{|G|}}{{|Stab(h)|}} divides |G|=p n{|G|} = p^n, and therefore pp divides |Orb(h)|{|Orb(h)|} if hh is noncentral. In this case the class equation takes the form

|G|=|Z(G)|+ nontrivialorbitsx|G||Stab(a x)|{|G|} = {|Z(G)|} \; + \sum_{nontrivial\; orbits x} \frac{{|G|}}{{|Stab(a_x)|}}

and now since pp divides |G|{|G|} as well as each term in the sum over nontrivial orbits, it must also divide |Z(G)|{|Z(G)|}. In particular, Z(G)Z(G) has more than one element.

It follows by induction that pp-groups are solvable, since the center is a normal subgroup and the quotient G/Z(G)G/Z(G) is also a pp-group. Since a group obtained from an abelian group by repeated central extensions is nilpotent, pp-groups are in fact nilpotent.

Number of fixed points

An elementary observation that is frequently useful is that the number of fixed points of an involution on a finite set SS has the same parity as SS. This is a statement about /(2)\mathbb{Z}/(2)-sets; we generalize this to a statement about GG-sets for general pp-groups GG. (Again, a fixed point of a GG-set is an element whose orbit is a singleton.)

Proposition

If GG is a pp-group acting on a set AA, then

  • |A||Fix(A)|modp{|A|} \equiv {|Fix(A)|} \; mod p.

As special cases, if there is just one fixed point, then |A|1modp{|A|} \equiv 1 \; mod p, and if pp divides |A|{|A|}, then pp divides |Fix(A)|{|Fix(A)|}.

Proof

The class equation takes the form

|A|=|Fix(A)|+ nontrivialorbitsx|G||Stab(a x)|{|A|} = {|Fix(A)|} \; + \sum_{nontrivial\; orbits x} \frac{{|G|}}{{|Stab(a_x)|}}

where pp divides each summand over nontrivial orbits on the right, since GG is a pp-group. Now reduce mod pp.

Wedderburn’s theorem

In the theory of finite projective planes, an important result is that a projective plane is Pappian if it is Desarguesian. The purely algebraic version of this is Wedderburn’s theorem:

Theorem

A finite division ring DD is commutative.

Proof

(Witt) The center of DD is a field; if DD is of characteristic p>0p \gt 0, then the center FF has q=p fq = p^f elements for some ff. We may regard DD as an FF-vector space of dimension nn, whence the number of elements of its multiplicative group D ×D^\times is q n1q^n-1.

The conjugation action of D ×D^\times on itself yields a decomposition

D ×F ×+ nontrivialorbitsx|D ×||Stab(a x)|D^\times \cong F^\times \; + \sum_{nontrivial\; orbits x} \frac{{|D^\times|}}{{|Stab(a_x)|}}

where again the elements of the center F ×=Z(D ×)F^\times = Z(D^\times) correspond to the trivial orbits. The stabilizer of any a xa_x, together with 00, forms a division ring (strictly) intermediate between FF and DD; usually this is called the centralizer C(a x)C(a_x) of a xa_x. Putting d x=dim F(C(a x))d_x = dim_F(C(a_x)), the division ring C(a x)C(a_x) has q d xq^{d_x} elements, and notice d xd_x divides nn because n/d xn/d_x is just the dimension of DD seen as a vector space (module) over C(a x)C(a_x). Thus |Stab(a x)|=q d x1{|Stab(a_x)|} = q^{d_x} - 1, and we have a class equation

q n1=q1+ xq n1q d x1.q^n - 1 = q - 1 + \sum_x \frac{q^n - 1}{q^{d_x} - 1}.

Now let ζ\zeta \in \mathbb{C} be any primitive n thn^{th} root of unity. Since zζz - \zeta divides each polynomial z n1z^n-1 and z n1z d x1\frac{z^n - 1}{z^{d_x} - 1}, so does prim.ζ(zζ)\prod_{prim.\; \zeta} (z-\zeta). It follows that the algebraic integer prim.ζ(qζ)\prod_{prim.\; \zeta} (q - \zeta) divides each of the integers q n1q^n-1 and q n1q d x1\frac{q^n - 1}{q^{d_x} - 1}, and hence divides q1q-1 according to the class equation. But also |qζ||q1|{|q - \zeta|} \geq {|q-1|} for any root of unity ζ\zeta. Thus |qζ|=|q1|{|q - \zeta|} = {|q-1|} and n=1n = 1, i.e., F=Z(D)F = Z(D) is all of DD as was to be shown.

Sylow theorems

Let GG be a finite group of order nn, and suppose that pp is a prime that divides nn; say n=p fun = p^f u where pp does not divide uu. Recall that a Sylow p-subgroup is a pp-subgroup of maximal order p fp^f.

A fundamental fact of group theory is that Sylow pp-subgroups exist and they are all conjugate to one another; also the number of Sylow pp-subgroups is 1modp\equiv 1 \; mod p.

Existence of Sylow pp-subgroups can be proven by exploiting the same type of argument as in the proof of Proposition :

Theorem

If GG has order nn and p kp^k is a prime power dividing nn, then there is a subgroup of GG of order p kp^k.

Proof

First we show that Sylow subgroups exist. We start by observing that if a group HH has a pp-Sylow subgroup PP, then so does any subgroup GG. First note that if we let GG act on H/PH/P by left translation, then the stabilizer of any element hPh P is GhPh 1G \cap h P h^{-1}, a pp-group since hPh 1h P h^{-1} is. Then note that since H/PH/P has cardinality prime to pp, so must one of its connected components G/Stab(a x)G/Stab(a_x) in its GG-set decomposition

H/P orbitsxG/Stab(a x),H/P \cong \sum_{orbits\; x} G/Stab(a_x),

and this makes Stab(a x)Stab(a_x) a pp-Sylow subgroup of GG.

Then, if GG is any group, and n=ord(G)n = ord(G), apply this observation to the embedding

GCayleyPerm(|G|)S nGL n(/(p))=HG \stackrel{Cayley}{\hookrightarrow} Perm({|G|}) \cong S_n \hookrightarrow GL_n(\mathbb{Z}/(p)) = H

where we embed the symmetric group S nS_n via permutation matrices into the group HH of n×nn \times n invertible matrices over /(p)\mathbb{Z}/(p). The group HH has order (p n1)(p np)(p np n1)(p^n - 1)(p^n - p)\ldots (p^n - p^{n-1}), with maximal pp-factor p n(n1)/2p^{n(n-1)/2}. It thus has a pp-Sylow subgroup given by unitriangular matrices, i.e., upper-triangular matrices with all 11's on the diagonal. Therefore pp-Sylow subgroups PP exist for any GG.

Finally, note that by Proposition , PP is solvable and therefore has a composition series

{1}=P 0P 1P\{1\} = P_0 \subset P_1 \subset \ldots \subset P

where each P kP_k has order p kp^k.

Theorem

If HH is a pp-subgroup of GG and PP is a Sylow pp-subgroup, then g 1HgPg^{-1} H g \subseteq P for some gGg \in G. In particular, all Sylow pp-subgroups are conjugate to one another.

Proof

GG acts on the set of cosets G/PG/P as usual by left translation, and we may restrict the action to the pp-subgroup HH. By maximality of PP, we see |G/P|{|G/P|} is prime to pp, and so by Proposition , |Fix H(G/P)|{|Fix_H(G/P)|} is also prime to pp. In particular, Fix H(G/P)Fix_H(G/P) has at least one element, say gPg P. We infer that hgP=gPh g P = g P for all hHh \in H, or that g 1hgP=Pg^{-1} h g P = P for all hHh \in H, and this implies that g 1HgPg^{-1} H g \subseteq P.

Theorem

The number of Sylow pp-subgroups of GG is 1modp\equiv 1 \; mod p.

Proof

Let YY be the set of Sylow pp-subgroups; GG acts on YY by conjugation. As all Sylow pp-subgroups are conjugate, there is just one orbit of the action, and the stabilizer of an element PYP \in Y is just the normalizer N G(P)N_G(P) (by definition of normalizer). Thus YG/N G(P)Y \cong G/N_G(P) as GG-sets.

Restrict the action to the subgroup PP. Of course the element PYP \in Y is a fixed point of this restricted action, and if QQ is any other fixed point, it means xQx 1=Qx Q x^{-1} = Q for all xPx \in P, whence PN G(Q)P \subseteq N_G(Q). Now: P,QP, Q are both Sylow pp-subgroups of N G(Q)N_G(Q) and are therefore conjugate to each other (as seen within the group N G(Q)N_G(Q)). But QQ is already fixed by the conjugation action in its stabilizer N G(Q)N_G(Q), so we conclude P=QP = Q. We conclude Fix P(Y)Fix_P(Y) has exactly one element. From |Y||Fix P(Y)|modp{|Y|} \equiv {|Fix_P(Y)|} \; mod p, the theorem follows.

Last revised on October 1, 2018 at 13:59:54. See the history of this page for a list of all contributions to it.