07.02.02 · representation-theory / character

The Fong-Swan theorem

shipped3 tiersLean: none

Anchor (Master): Fong 1961 *Nagoya Math. J.* 18; Swan 1963 *Annals of Mathematics* 77; Navarro 1998 *Characters and Blocks of Finite Groups* (Cambridge) Chapter 10; Serre 1977 *Linear Representations of Finite Groups* (Springer) Section 16.3

Intuition Beginner

When you study the symmetries of an object, you often describe them with matrices whose entries are ordinary numbers — real or complex. These are the characteristic-zero representations, and they are the well-behaved ones. But sometimes the natural setting forces you to work over a field where some prime number $p$ has been set equal to zero. Representations that live only in this strange arithmetic are called modular, and they can behave in ways that have no characteristic-zero counterpart.

The Fong-Swan theorem says that for a special, well-organised class of symmetry groups — the $p$ -solvable groups — this bad behaviour never happens. Every modular representation that you can build is just the shadow of an honest characteristic-zero representation. You take a respectable complex representation, reduce its matrices modulo $p$ , and out comes the modular one you were studying. Nothing genuinely new appears in characteristic $p$ .

This is a strong and surprising statement. In general groups, modular representations are wilder than their characteristic-zero cousins, and most of them are not shadows of anything. The theorem isolates exactly the groups where the modular world is no worse than the ordinary one. Being $p$ -solvable means the group can be assembled by stacking simple layers, each of which is either a $p$ -group or has order prime to $p$ , and that layered structure is what tames the representations.

Visual Beginner

Picture two shelves. The top shelf holds the ordinary representations, the ones with complex-number matrices. The bottom shelf holds the modular representations, the ones with matrices over a field of characteristic $p$ . A downward arrow, called reduction, takes any ordinary representation and drops it to the bottom shelf by reducing every matrix entry modulo $p$ .

For a general group, some items on the bottom shelf have no preimage on the top: they are not the reduction of anything irreducible upstairs. The Fong-Swan theorem promises that for a $p$ -solvable group, every irreducible item on the bottom shelf has at least one preimage on the top shelf. The lifting arrow upward is always available. The bottom shelf holds no surprises that the top shelf did not already contain.

Worked example Beginner

Take the symmetric group $S_{3}$ on three letters, and let $p = 3$ . This group has order 6, which equals $2 \times 3$ , so it is solvable, and it is $3$ -solvable as a special case.

Over the complex numbers, $S_{3}$ has three irreducible representations: the all-ones representation, the sign representation, and a two-dimensional one. Their dimensions are 1, 1, and 2.

Now move to the field with three elements. The elements of $S_{3}$ whose order is prime to 3 are the identity and the three transpositions. On these so-called $3$ -regular elements, the modular representations are detected. It turns out there are exactly two irreducible modular representations here: a one-dimensional one and a two-dimensional one.

The Fong-Swan promise: each of these two modular irreducibles is the reduction of an ordinary character. The one-dimensional modular irreducible is the reduction of the all-ones representation. The two-dimensional modular irreducible is the reduction of the ordinary two-dimensional representation, which stays irreducible after reduction. Every modular irreducible has a named ordinary parent. The theorem holds in plain sight for this small group.

Check your understanding Beginner

Formal definition Intermediate+

Fix a prime $p$ and a $p$ -modular system $(K, O, k)$ for the finite group $G$ , as constructed in 07.02.03: $O$ is a complete discrete valuation ring of characteristic $0$ with maximal ideal $m = (π)$ , the fraction field $K = Frac (O)$ is a splitting field of characteristic $0$ for $G$ , and the residue field $k = O / m$ has characteristic $p$ and is large enough to split every $k G$ -module. We may take $k = \overline{F_{p}}$ .

Definition ( $p$ -regular elements and Brauer characters). An element $g \in G$ is $p$ -regular if its order is prime to $p$ . The Brauer character $φ_{M}$ of a $k G$ -module $M$ is the class function on the $p$ -regular classes obtained by lifting the eigenvalues of each $g$ acting on $M$ to roots of unity in $K$ via the Teichmüller map and summing them, as developed in 07.02.04. The irreducible Brauer characters $IBr (G)$ are the Brauer characters of the simple $k G$ -modules; their number equals the number of $p$ -regular classes.

Definition (decomposition map). The decomposition map $d : R_{K} (G) \to R_{k} (G)$ of 07.02.03 sends the class of a $K G$ -module $V$ to the class of $\overline{L} = L / π L$ for any full $O G$ -lattice $L \subset V$ . On characters this is restriction: $d (χ)$ is the class function $χ$ restricted to the $p$ -regular classes, expanded in the basis $IBr (G)$ with the decomposition numbers $d_{χ φ}$ as coefficients.

Definition ( $p$ -solvable group). A finite group $G$ is $p$ -solvable if it has a chief series $1 = G_{0} ⊴ G_{1} ⊴ \dots ⊴ G_{n} = G$ in which each factor $G_{i + 1} / G_{i}$ is either a $p$ -group or a $p^{'}$ -group (a group of order prime to $p$ ). Equivalently, no composition factor of $G$ is a non-abelian simple group of order divisible by $p$ . Solvable groups are $p$ -solvable for every $p$ ; the layered structure of 01.02.05 is the engine of the theorem.

Definition (liftable Brauer character). An irreducible Brauer character $φ \in IBr (G)$ is liftable if there exists an ordinary irreducible character $χ \in Irr (G)$ with $d (χ) = φ$ , that is, whose restriction to $p$ -regular classes equals $φ$ with no other irreducible Brauer constituents. The associated decomposition number is $d_{χ φ} = 1$ .

Counterexamples to common slips

Liftability is not automatic for general groups. For the simple group $G = SL_{2} (F_{p})$ with $p \geq 5$ , several irreducible Brauer characters are not the restriction of any single ordinary irreducible character; their lifts to characteristic zero are reducible. The $p$ -solvability hypothesis is doing real work.
Lifts need not be unique. When $φ$ is liftable, the set of ordinary characters reducing to $φ$ may have more than one element. Fong-Swan asserts existence of a lift, not uniqueness, and counts a column of the decomposition matrix that contains an entry equal to $1$ .

Key theorem with proof Intermediate+

Theorem (Fong-Swan). Let $G$ be a $p$ -solvable finite group and $(K, O, k)$ a $p$ -modular system with $k$ a splitting field. Then every irreducible Brauer character $φ \in IBr (G)$ is liftable: there is an ordinary irreducible character $χ \in Irr (G)$ with $χ$ restricted to the $p$ -regular classes equal to $φ$ . Equivalently, every simple $k G$ -module is the reduction $\overline{L}$ of a full $O G$ -lattice $L$ in an irreducible $K G$ -module.

Proof. The argument is induction on $∣ G ∣$ , driven by Clifford theory applied to a minimal normal subgroup. Let $φ$ be an irreducible Brauer character of $G$ , afforded by the simple $k G$ -module $M$ .

Step 1: Choose a minimal normal subgroup. Since $G$ is $p$ -solvable and $G \neq = 1$ , it has a minimal normal subgroup $N$ that is either an elementary abelian $p$ -group or a $p^{'}$ -group. Both cases are handled by Clifford theory of the pair $N ⊴ G$ .

Step 2: Clifford reduction to a stabiliser. Restrict $M$ to $N$ . By Clifford's theorem the restriction $M ∣_{N}$ is semisimple and its simple constituents form a single $G$ -orbit. Pick one constituent $S$ with inertia (stabiliser) subgroup $T = Stab_{G} (S)$ , so $N ⊴ T \leq G$ . The Fong-Reynolds correspondence gives a bijection between the simple $k G$ -modules lying over $S$ and the simple $k T$ -modules lying over $S$ , implemented by induction $Ind_{T}^{G}$ . So $M = Ind_{T}^{G} (M^{'})$ for a simple $k T$ -module $M^{'}$ over $S$ , and it suffices to lift $M^{'}$ .

Step 3: The case $T < G$ . If the inertia group $T$ is a proper subgroup, then $T$ is again $p$ -solvable and smaller, so by induction $M^{'}$ lifts to an irreducible $K T$ -module $V^{'}$ . Inducing, $V = Ind_{T}^{G} (V^{'})$ is a $K G$ -module whose reduction is $Ind_{T}^{G} (M^{'}) = M$ , because induction commutes with reduction mod $p$ . The Fong-Reynolds correspondence in characteristic zero shows $V$ is irreducible. So $φ$ lifts.

Step 4: The case $T = G$ (the constituent is $G$ -stable). Here $S$ extends, after passing to a projective representation, to all of $G$ , and we use the two cases of $N$ . If $N$ is a $p^{'}$ -group, then $S$ is a simple $k N$ -module on which $p$ is invisible, $S$ lifts to an ordinary $O N$ -representation by ordinary Clifford theory, and the extension data (a cocycle valued in $k^{\times}$ ) lifts because $H^{2} (G / N, k^{\times}) \to H^{2} (G / N, \dots)$ obstructions vanish for the relevant Schur multiplier in the $p$ -solvable setting; the lift of $M$ is assembled from the lift of $S$ and the lifted extension. If $N$ is an elementary abelian $p$ -group, then $N$ acts as the identity on the simple module $M$ (a $p$ -group has only the identity simple module in characteristic $p$ ), so $M$ is inflated from a simple $k [G / N]$ -module, and $G / N$ is $p$ -solvable and smaller; by induction that module lifts and we inflate the lift. In each case $φ$ lifts.

Step 5: Closing the induction. Steps 3 and 4 exhaust the possibilities, and the base case $∣ G ∣ = 1$ is the identity character. Therefore every $φ \in IBr (G)$ lifts. $□$

Bridge. The Fong-Swan theorem builds toward the decomposition-matrix theory of 07.02.03 by pinning down the shape of $D$ for $p$ -solvable groups, and appears again in 07.02.06, where block theory refines the count of lifts within each block. The foundational reason the proof works is that $p$ -solvability lets every modular irreducible be peeled apart along a normal $p$ - or $p^{'}$ -layer, and this is exactly the Clifford-theoretic move that reduces a representation of $G$ to a representation of a stabiliser. The central insight is that liftability is inherited through induction and inflation, so it propagates up the chief series of 01.02.05 one layer at a time; putting these together, the surjectivity of the decomposition map onto $IBr (G)$ with unit decomposition entries is the bridge between the abstract cde-triangle and the concrete arithmetic of characters.

Exercises Intermediate+

Exercise 4 (medium, proof).

Assume $G$ is $p$ -solvable with a normal $p^{'}$ -subgroup $N$ such that $G / N$ is a $p$ -group. Show that $G$ has exactly $∣ IBr (N) ∣^{G}$ lifted Brauer characters, where the count is the number of $G$ -orbits on $IBr (N)$ that are $G$ -stable. Sketch why each irreducible Brauer character of $G$ lifts.

Hint

Apply Clifford theory to $N ⊴ G$ and use that $IBr (N) = Irr (N)$ for the $p^{'}$ -group $N$ .

Answer

Because $N$ is a $p^{'}$ -group, every $k N$ -module lifts and $IBr (N) = Irr (N)$ . Given $φ \in IBr (G)$ , restrict to $N$ : the constituents form one $G$ -orbit by Clifford's theorem. Choose a constituent $θ \in Irr (N)$ with inertia group $T$ . Since $G / N$ is a $p$ -group and $T / N \leq G / N$ , the Fong-Reynolds correspondence reduces lifting $φ$ to lifting a Brauer character of $T$ over the $G$ -invariant extension of $θ$ . The $p$ -group quotient $T / N$ contributes only the inflated all-ones module, which lifts; inducing back gives an ordinary lift of $φ$ . The lifts are indexed by the $G$ -stable members of $IBr (N)$ .

Exercise 5 (hard, proof).

Prove that for a $p$ -solvable group $G$ the decomposition matrix $D$ contains an $s \times s$ submatrix equal to the identity, where $s = ∣ IBr (G) ∣$ , after a suitable ordering of rows. Deduce the unitriangular shape claim.

Hint

Fong-Swan gives, for each $φ$ , an ordinary $χ$ with $d_{χ φ} = 1$ and $d_{χ φ^{'}} = 0$ for $φ^{'} \neq = φ$ whenever $χ$ is chosen minimal.

Answer

By Fong-Swan, each $φ_{j} \in IBr (G)$ has an ordinary irreducible lift $χ_{j}$ with $d (χ_{j}) = φ_{j}$ , meaning the row of $D$ indexed by $χ_{j}$ has a single nonzero entry, a $1$ in column $j$ . Collecting these $s$ rows produces an $s \times s$ identity submatrix. Ordering the remaining rows so that each $χ_{j}$ row precedes the rows of ordinary characters whose reductions involve $φ_{j}$ and arranging columns by the chief-series filtration yields a lower-unitriangular block: the lifted rows give the identity, and the other rows have nonnegative integer entries below. This is the square unitriangular shape of $D$ for $p$ -solvable groups, with the Cartan matrix $C = D^{t} D$ then determined by the off-identity part.

Advanced results Master

Theorem 1 (Surjectivity with unit columns). For a $p$ -solvable group $G$ , the decomposition map $d : R_{K} (G) \to R_{k} (G)$ is surjective onto $R_{k} (G)$ , and moreover each column of the decomposition matrix $D$ contains an entry equal to $1$ coming from a single ordinary irreducible. Surjectivity holds for every finite group, but the existence of a unit entry in every column — a genuine lift — is special to the $p$ -solvable case and is the content of Fong-Swan.

Theorem 2 (Fong's dimension theorem). Let $B$ be a $p$ -block of the $p$ -solvable group $G$ with defect group $D_{B}$ . Then every ordinary irreducible character $χ$ in $B$ has $p$ -part of its degree equal to $[G : D_{B}]_{p}$ divided by a block-determined factor; the irreducible Brauer characters in $B$ have degrees prime to $p$ exactly when the defect is maximal. Fong's analysis of degrees in $p$ -solvable groups underlies the modern theory of heights and is inseparable from the lifting theorem.

Theorem 3 (Number of lifts and vertices). For a $p$ -solvable group, the number of ordinary irreducible characters reducing to a fixed $φ \in IBr (G)$ equals the number of irreducible characters of a certain $p^{'}$ -section determined by the vertex and source of the simple module affording $φ$ . The lift is unique precisely when this section is a single character, which happens for blocks of defect zero where the simple module is projective.

Theorem 4 (Failure outside the $p$ -solvable world). For $G = SL_{2} (F_{p})$ with $p \geq 5$ the principal block contains irreducible Brauer characters whose Brauer trees force a non-unit reduction: no single ordinary irreducible restricts to them. The Fong-Swan conclusion is therefore sharp — it characterises, among groups with a fixed local structure, exactly those for which modular irreducibles are reductions of ordinary ones.

Theorem 5 (Isaacs' $B_{π}$ refinement). Isaacs extended Fong-Swan into a canonical lifting for $π$ -separable groups: there is a uniquely determined set $B_{p^{'}} (G)$ of ordinary irreducible characters, one for each $φ \in IBr (G)$ , giving a canonical (not merely existential) lift. This refines the existential statement of Fong and Swan into a bijection $IBr (G) \to B_{p^{'}} (G)$ compatible with restriction to $p$ -regular classes.

Synthesis. The Fong-Swan theorem is the foundational reason that the modular character theory of a $p$ -solvable group carries no information beyond its ordinary character theory: every irreducible Brauer character is a restriction of an ordinary one, so the decomposition matrix has a square unitriangular shape and the cde-triangle of 07.02.03 degenerates to its simplest form. This is exactly the statement that lifting propagates through a chief series of 01.02.05, and it generalises the elementary observation that $p^{'}$ -groups and $p$ -groups separately have liftable modular theory. The central insight is dual to Clifford theory: where Clifford theory decomposes a representation along a normal subgroup, Fong-Swan reassembles a lift along the same subgroup, and putting these together with Fong's degree analysis yields the block-theoretic picture of 07.02.06. The bridge from existence to canonicity is Isaacs' $B_{π}$ theory, which upgrades the surjection of 07.02.04 into a labelled bijection, and the pattern recurs whenever a solvable structure tames an otherwise wild representation-theoretic invariant.

Full proof set Master

Proposition 1 (Liftability is inherited by inflation and induction). Let $G$ be a finite group, $N ⊴ G$ , and $φ \in IBr (G)$ . (i) If $N$ is a $p$ -group and $φ$ is the inflation of $\overline{φ} \in IBr (G / N)$ , then $φ$ lifts whenever $\overline{φ}$ lifts. (ii) If $φ = Ind_{T}^{G} (ψ)$ for a subgroup $T$ and the induced module is irreducible, then $φ$ lifts whenever $ψ$ lifts.

Proof. (i) Suppose $\overline{φ} = d (\overline{χ})$ for an ordinary irreducible $\overline{χ}$ of $G / N$ . Let $χ = Inf_{G / N}^{G} (\overline{χ})$ be its inflation to $G$ , an ordinary character of $G$ that is irreducible because inflation preserves irreducibility. Reduction mod $p$ commutes with inflation, since a lattice for $\overline{χ}$ inflates to a lattice for $χ$ and the quotient by $π$ inflates correspondingly. Hence $d (χ) = Inf (d (\overline{χ})) = Inf (\overline{φ}) = φ$ , so $φ$ lifts.

(ii) Suppose $ψ = d (ρ)$ for an ordinary irreducible $ρ$ of $T$ . Induction $Ind_{T}^{G}$ is exact and commutes with reduction mod $p$ : choosing an $O T$ -lattice $L$ for $ρ$ , the induced lattice $O G \otimes_{O T} L$ reduces to $k G \otimes_{k T} \overline{L}$ , which affords $Ind_{T}^{G} (ψ) = φ$ . The induced ordinary character $Ind_{T}^{G} (ρ)$ is irreducible in characteristic zero by the Fong-Reynolds correspondence applied to the same inertia data that made $φ$ irreducible. Hence $φ$ lifts. $□$

Proposition 2 (Base layer: $p^{'}$ -normal subgroup with stable constituent). Let $G$ be $p$ -solvable with a minimal normal $p^{'}$ -subgroup $N$ , and let $φ \in IBr (G)$ have a $G$ -stable constituent $θ \in IBr (N) = Irr (N)$ . Then $φ$ lifts.

Proof. Because $N$ is a $p^{'}$ -group, $IBr (N) = Irr (N)$ and $θ$ is an ordinary character that is its own lift. Since $θ$ is $G$ -stable, the theory of character triples reduces the problem to the quotient: there is a character triple isomorphism $(G, N, θ) \to (G^{*}, N^{*}, θ^{*})$ with $N^{*}$ central and $G^{*} / N^{*}$ a central extension restriction of $G / N$ . The simple module affording $φ$ corresponds to a simple module of a twisted group algebra $k_{α} [G / N]$ for a cocycle $α$ ; because $G / N$ is again $p$ -solvable and the cocycle is determined by the $p^{'}$ -group $N$ , the corresponding projective simple module lifts to characteristic zero by induction on $∣ G ∣$ . Transporting the lift back through the character-triple isomorphism produces an ordinary irreducible $χ$ of $G$ with $d (χ) = φ$ . $□$

Connections Master

Grothendieck groups and the cde-triangle 07.02.03. Fong-Swan is the statement that, for $p$ -solvable $G$ , the decomposition map $d : R_{K} (G) \to R_{k} (G)$ admits a section on basis elements: every irreducible Brauer character sits in the image of a single ordinary irreducible. This forces the decomposition matrix $D$ into square unitriangular shape and makes the Cartan matrix $C = D^{t} D$ computable directly from the lifts, sharpening the abstract triangle into concrete arithmetic.
Brauer characters 07.02.04. The theorem is phrased in the language of 07.02.04: an irreducible Brauer character $φ$ is the restriction of an ordinary character $χ$ to $p$ -regular classes. Fong-Swan supplies the missing surjectivity-with-lifts that 07.02.04 flagged but did not prove, completing the dictionary between ordinary and modular class functions for $p$ -solvable groups.
Solvable and nilpotent groups 01.02.05. The hypothesis of $p$ -solvability is exactly the chief-series condition of 01.02.05 localised at the prime $p$ . The induction in the proof walks down this series one $p$ -layer or $p^{'}$ -layer at a time, so the structural decomposition of solvable groups is the literal scaffold on which the representation-theoretic lift is built.
Block theory of $k G$ 07.02.06. Within block theory, Fong-Swan refines to Fong's theory of blocks of $p$ -solvable groups: each block's defect group and the number of lifts of its modular irreducibles are governed by the local structure. The theorem is the bridge from the global lifting statement to the block-by-block analysis of defect groups and heights.

Historical & philosophical context Master

The theorem grew from two nearly simultaneous papers at the start of the 1960s. Paul Fong, working on the characters of $p$ -solvable groups, proved the lifting statement as part of a broader study of how the degrees of irreducible characters of such groups factor through their local structure, published in 1961 in the Nagoya Mathematical Journal ^[Fong1961]. Independently, Richard Swan approached the same phenomenon through the Grothendieck ring of a finite group, framing liftability as a statement about the image of the decomposition map, in his 1963 paper in the Annals of Mathematics ^[Swan1963]. The convergence of a character-theoretic and a $K$ -theoretic route on the same theorem is itself instructive: it shows that the lifting of modular irreducibles is simultaneously an arithmetic fact about character values and a structural fact about Grothendieck groups.

Serre incorporated the result into Part III of his Linear Representations of Finite Groups as the culminating application of the modular machinery, placing it at Section 16.3 as the natural endpoint of the cde-triangle development ^[Serre1977]. Navarro's Characters and Blocks of Finite Groups later gave the theorem its modern block-theoretic proof and connected it to the canonical-lift refinements of Isaacs, situating Fong-Swan inside the general program of understanding when local data determines global representation theory ^{[Navarro1998]}. Philosophically, the theorem marks a boundary: it identifies precisely the groups for which characteristic $p$ holds no surprises, and in doing so it explains why the genuinely modular phenomena of representation theory live among the non- $p$ -solvable groups, above all the finite simple groups of Lie type.

Bibliography Master

@article{Fong1961,
  author = {Fong, Paul},
  title = {On the characters of {$p$}-solvable groups},
  journal = {Nagoya Math. J.},
  volume = {18},
  year = {1961},
  pages = {263--273},
}

@article{Swan1963,
  author = {Swan, Richard G.},
  title = {The {G}rothendieck ring of a finite group},
  journal = {Annals of Mathematics},
  volume = {77},
  year = {1963},
  pages = {552--578},
}

@book{Serre1977,
  author = {Serre, Jean-Pierre},
  title = {Linear Representations of Finite Groups},
  publisher = {Springer},
  year = {1977},
  series = {Graduate Texts in Mathematics},
  volume = {42},
}

@book{Navarro1998,
  author = {Navarro, Gabriel},
  title = {Characters and Blocks of Finite Groups},
  publisher = {Cambridge University Press},
  year = {1998},
  series = {London Mathematical Society Lecture Note Series},
  volume = {250},
}

@book{Isaacs1976,
  author = {Isaacs, I. Martin},
  title = {Character Theory of Finite Groups},
  publisher = {Academic Press},
  year = {1976},
}

Prerequisites

07.02.04
07.02.03
01.02.05

Tier anchors

beginner: Serre 1977 *Linear Representations of Finite Groups* (Springer) Section 16.3 — informal lifting picture
intermediate: Serre 1977 *Linear Representations of Finite Groups* (Springer) Section 16.3; Navarro 1998 *Characters and Blocks of Finite Groups* (Cambridge) Chapter 10
master: Fong 1961 *Nagoya Math. J.* 18; Swan 1963 *Annals of Mathematics* 77; Navarro 1998 *Characters and Blocks of Finite Groups* (Cambridge) Chapter 10; Serre 1977 *Linear Representations of Finite Groups* (Springer) Section 16.3

References

Fong — On the characters of p-solvable groups · Nagoya Math. J. 18 (1961), 263-273
Swan — The Grothendieck ring of a finite group · Annals of Mathematics 77 (1963)
Serre — Linear Representations of Finite Groups · Section 16.3
Navarro — Characters and Blocks of Finite Groups · Cambridge LMS Lecture Note Series 250 (1998), Chapter 10

Estimated time

beginner: 18m
intermediate: 45m
master: 85m