A finite group is said to be a Frobenius group if there is a non-trivial subgroup
of
(known as the Frobenius complement of
) such that the conjugates
of
are “disjoint as possible” in the sense that
whenever
. This gives a decomposition
where the Frobenius kernel of
is defined as the identity element
together with all the non-identity elements that are not conjugate to any element of
. Taking cardinalities, we conclude that
A remarkable theorem of Frobenius gives an unexpected amount of structure on and hence on
:
Theorem 1 (Frobenius’ theorem) Let
be a Frobenius group with Frobenius complement
and Frobenius kernel
. Then
is a normal subgroup of
, and hence (by (2) and the disjointness of
and
outside the identity)
is the semidirect product
of
and
.
I discussed Frobenius’ theorem and its proof in this recent blog post. This proof uses the theory of characters on a finite group , in particular relying on the fact that a character on a subgroup
can induce a character on
, which can then be decomposed into irreducible characters with natural number coefficients. Remarkably, even though a century has passed since Frobenius’ original argument, there is no proof known of this theorem which avoids character theory entirely; there are elementary proofs known when the complement
has even order or when
is solvable (we review both of these cases below the fold), which by the Feit-Thompson theorem does cover all the cases, but the proof of the Feit-Thompson theorem involves plenty of character theory (and also relies on Theorem 1). (The answers to this MathOverflow question give a good overview of the current state of affairs.)
I have been playing around recently with the problem of finding a character-free proof of Frobenius’ theorem. I didn’t succeed in obtaining a completely elementary proof, but I did find an argument which replaces character theory (which can be viewed as coming from the representation theory of the non-commutative group algebra ) with the Fourier analysis of class functions (i.e. the representation theory of the centre
of the group algebra), thus replacing non-commutative representation theory by commutative representation theory. This is not a particularly radical depature from the existing proofs of Frobenius’ theorem, but it did seem to be a new proof which was technically “character-free” (even if it was not all that far from character-based in spirit), so I thought I would record it here.
The main ideas are as follows. The space of class functions can be viewed as a commutative algebra with respect to the convolution operation
; as the regular representation is unitary and faithful, this algebra contains no nilpotent elements. As such, (Gelfand-style) Fourier analysis suggests that one can analyse this algebra through the idempotents: class functions
such that
. In terms of characters, idempotents are nothing more than sums of the form
for various collections
of characters, but we can perform a fair amount of analysis on idempotents directly without recourse to characters. In particular, it turns out that idempotents enjoy some important integrality properties that can be established without invoking characters: for instance, by taking traces one can check that
is a natural number, and more generally we will show that
is a natural number whenever
is a subgroup of
(see Corollary 4 below). For instance, the quantity
is a natural number which we will call the rank of (as it is also the linear rank of the transformation
on
).
In the case that is a Frobenius group with kernel
, the above integrality properties can be used after some elementary manipulations to establish that for any idempotent
, the quantity
is an integer. On the other hand, one can also show by elementary means that this quantity lies between and
. These two facts are not strong enough on their own to impose much further structure on
, unless one restricts attention to minimal idempotents
. In this case spectral theory (or Gelfand theory, or the fundamental theorem of algebra) tells us that
has rank one, and then the integrality gap comes into play and forces the quantity (3) to always be either zero or one. This can be used to imply that the convolution action of every minimal idempotent
either preserves
or annihilates it, which makes
itself an idempotent, which makes
normal.
— 1. Idempotent theory —
Let be a finite group. Then we can form the Hilbert space
of complex functions
with inner product
where we use the usual averaging notation . This Hilbert space is also a complex *-algebra using the adjoint map
and the convolution product
For future reference, we record the adjoint relations
and
for any .
As an algebra, is isomorphic to the group algebra
of
, if one identifies each group element
with the “Dirac delta function”
at
, defined by
Note that
and
for any and
.
The convolution algebra will in general be non-commutative. However, it contains a commutative (in fact central) *-subalgebra
, namely the space of class functions, that is to say the functions
which are conjugation invariant in the sense that
for all . It is easy to see that
is a central *-subalgebra of
, thus if
are class functions then so is
and
, and one has
for any
. Indeed it is not hard to show that
is the centre of
, although we will not need this fact here.
Let us call a class function an idempotent if
. Thus for instance
,
, and
are idempotents, and the convolution of any two idempotents is again an idempotent. Note that if
is idempotent, then the linear transformation
on
is also idempotent and also self-adjoint (since
and
are class functions and thus commute with each other), and thus
is an orthogonal projection onto the space
. The projection
similarly projects
to the subspace
. We define the rank
of the idempotent
to be the rank of
on
, or equivalently the dimension of the space
. Thus for instance
has rank
,
has rank equal to the number of conjugacy classes of
, and
has rank one.
We place an ordering on idempotents by declaring if
, thus for instance
, and
for any idempotents
. The relation
is equivalent to the range of the projection
being a subspace of the range of the projection
. In particular this shows that
is a partial order on idempotents, and that
whenever
. If
, we see that
is also an idempotent with
. From this we see that every non-zero idempotent
is either minimal (in the sense that there are no other non-zero idempotents
such that
, or can be split up as the sum of two strictly smaller idempotents. From this we see that we can decompose
into the sum
of minimal idempotents
, whose ranges
then form an orthogonal decomposition of
(and similarly the
form an orthogonal decomposition of
); and by convolving this fact with any other idempotent
we see that all other idempotents
take the form
for some subset
of
. Conversely, one can show that
is an idempotent for every
. Thus, the structure of idempotents is completely determined by the set of minimal idempotents.
The theory thus far relied only on elementary algebra; in particular, one could replace the complex numbers here by any other field of characteristic not a factor of and still get essentially an identical theory (replacing the complex conjugate by some other automorphism, such as the trivial automorphism). But the next fact is less elementary (being analogous to Plancherel’s theorem for finite groups) and relies crucially on the algebraically closed nature of
(or closely related facts, such as Liouville’s theorem).
Proof: Let be an idempotent of rank greater than one, then
contains a function
which is not a scalar multiple of
. The linear transformation
is a normal transformation on
. Since
is not a multiple of
,
is not a multiple of the identity, thus by the spectral theorem there is an eigenspace
which is a proper non-trivial subspace of
. This space is a convolution ideal of
; if we let
be the orthogonal projection from
to
, then
commutes with convolutions and in particular
for all
. Applying this to
we conclude that
is an idempotent, with
, thus
is not minimal. The claim follows (note that minimal idempotents by definition cannot be of rank zero).
Now we record some integrality properties of idempotents. We have the following trivial but crucial fact:
Lemma 3 Let
be a finite-dimensional vector space, and let
be an idempotent linear transformation. Then the trace
of
is equal to the dimension
of the range of
. In particular, the trace is a natural number.
Proof: This follows by computing the trace in a basis of together with a basis of some complementary subspace of
.
Corollary 4 Let
be an idempotent in
, and let
be a subgroup of
. Then the quantity
is a natural number.
Proof: We introduce the left and right translation operators
and
on for
, together with the projection operator
introduced previously. Observe that the three operators commute with each other. As such, the operator
is idempotent. A short computation shows that the trace of is equal to
and the claim now follows from Lemma 3.
— 2. Proof of Frobenius’ theorem —
We now begin the proof of Frobenius’ theorem. We begin with an elementary consequence of Corollary 4:
Lemma 5 Let
be a finite group with Frobenius complement
and Frobenius kernel
. Then for any idempotent
in
, the quantity
is an integer.
Proof: When the quantity (4) vanishes, so we may assume (by the decomposition
) that the idempotent
has mean zero.
Applying Corollary 4 to the subgroups ,
,
and
of
, we conclude that the expressions
are natural numbers (and hence integers). We now consider various linear combinations of these quantities. We write (6) as
Observe that as is a class function the quantity
is invariant with respect to conjugation of either
or
(or both). From this and (1), we see that (6) is equal to
Subtracting this from (5), we conclude that
is an integer; comparing this with (4), we see that it suffices to show that
is an integer.
If we multiply (8) by and isolate the terms where
or
is equal to
, we obtain
Subtracting two copies of (7), we conclude that
is an integer. Using the conjugation invariance of in
and
, together with (1), we can rewrite this expression as
On the other hand, as has mean zero, we have
for any , and similarly
for any . Thus we can rewrite the previous expression as
but from (1) is closed under conjugation, and hence (9) is an integer as required.
On the other hand, we have an alternate representation for the quantity (4):
Lemma 6 Let
be a finite group with Frobenius complement
and Frobenius kernel
. Then for any idempotent
in
, the quantity (4) is equal to
, where
is the orthogonal projection to the space
of class functions of mean zero supported on
. In particular, since
is a subspace of
, the quantity (4) lies between
and
.
Proof: The quantity can be expanded as
which can be expanded further as
This simplifies to
and a routine computation (using the conjugation invariance of ) shows that this is the same quantity as (4).
Let be a finite group with Frobenius complement
and Frobenius kernel
, and let
be a minimal idempotent of
. By Lemma 3, Lemma 5, and Lemma 6, we see that the quantity (4) is an integer between
and
, and is thus exactly
or exactly
. From Lemma 6, we thus conclude that the space
either contains the range
of
, or is orthogonal to this range. Summing over minimal idempotents, we conclude that
is the direct sum of the ranges
of some minimal idempotents
; the same is thus true of the orthogonal complement
of
in
, that is to say the space of class functions that are constant on
. In particular, this space
is closed under convolution, which implies that
is constant on
. However, this convolution attains a maximum value of
at the set
, and hence
, thus
is a subgroup. As
was already known to be closed under conjugation, Theorem 1 follows.
Remark 1 As mentioned previously, one can use character theory to describe the minimal idempotents as the functions of the form
, where
are the irreducible characters of
. The integrality property in Corollary 4 can then be obtained by taking the representation of
associated to
and decomposing it into irreducible representations of
; we omit the details. Thus one can view the above argument as being a disguised version of the usual character-theoretic proof. However, the point is that one can prove Corollary 4 without knowing the relationship between minimal idempotents, irreducible characters, and irreducible representations. (Indeed, Corollary 4 seems to be weaker than the fact from character theory that the restriction or induction of a character is again a character (and in particular decomposes into a natural number combination of irreducible characters); see this MathOverflow post for some further discussion.)
— 3. The solvable case —
In the case when the Frobenius complement is solvable, an elementary proof of Theorem 1 was discovered by Grun and by Shaw, based on the transfer homomorphism. To describe this argument, we first consider the model case when
is abelian. We then enumerate the right cosets of
as
for some
. This enumeration is not unique; in addition to permutations of the
, we have the freedom to arbitrarily multiply each
on the right by an element
of
to obtain
without affecting the coset:
.
Regardless of this freedom, we can define the transfer homomorphism for
by noting that for each coset
, one has
for some unique
depending on
and
, thus
for some
depending on
, and
. We then set
One observes that this map is in fact independent of the choice of coset enumeration ; permutations of
are irrelevant as
is assumed to be abelian, and shifting one of the
by an element
of
causes one of the
to be multiplied by
, byt another of the
to be divided by
, leaving
unchanged. Once one knows that
is independent of the choice of coset enumeration, it is not difficult to verify the homomorphism property
, for instance by using one enumeration
to compute
and
, and another enumeration
to compute
.
We now make the crucial observation that if is an abelian Frobenius complement, then the transfer map
is the identity on
, thus
for all
. To see this, let
be the order of
. We claim that the action
of
on the quotient space
consists of a single fixed point at
, together with orbits of size exactly
. Indeed,
is clearly fixed by
since
, and if we have
for some coset
and some
, then
lies in
and so
is a multiple of
, whence the claim. If we then use a coset enumeration that consists of
together with
-tuples of the form
for each orbit of
, we see that
as claimed.
As is a surjective homomorphism from
to
, the kernel is a normal subgroup of order
; also, as
is the identity on
, this kernel is disjoint from
, and hence also from every conjugate of
. From (1) we conclude that the kernel is
, and hence
is normal as required.
Now we adapt the above argument to the case when is solvable instead of abelian. Here, one can still define a transfer map
, but it takes values in the abelianisation
of
, defined by
One can verify as before that is independent of the choice of enumeration, and is a homomorphism. The previous argument also shows that the restriction of
to
is the quotient map from
to
. Thus, the kernel
is a normal subgroup of
whose order is
and which intersects
in precisely
. From this we see that
is the union of
and the conjugates of
, and so
is also a Frobenius group with Frobenius complement
and Frobenius kernel
. The claim then follows by an induction of the derived length of
.
Remark 2 This argument in fact shows that to prove Frobenius’s theorem, it suffices to do so in the case that
is a perfect group.
— 4. The even order case —
When is even, Cauchy’s theorem shows that
contains at least one non-trivial involution – an element of order two, since otherwise
would split up into the identity and pairs
, contradicting the even nature of
. One can then analyse the action of these involutions combinatorially; this was first done by Bender (and we use an argument from this text of Passman). Indeed, if
is an involution in
, then as discussed in the previous section, the action
of
on
fixes
and breaks the rest of
into orbits of size two; in particular,
is odd, and there are
pairs of cosets in
that are swapped by
. More generally, any involution in a conjugate
of
will fix
and then swap
pairs among the other cosets of
.
Suppose that contains at least two involutions; then by (1)
contains at least
involutions, each of which swaps
pairs in
. But there are only
such pairs; thus by the pigeonhole principle there are two cosets, say
and
, which are swapped by two different involutions
in
, thus
and
. But this implies that
fixes both
and
, and thus lies in
, and is thus trivial by (1), contradicting the disjoint nature of
and
. Thus
contains exactly one involution. The same argument then shows that
cannot contain any involutions. Thus the set
of involutions has order exactly
, consisting of the involution of
and all of its conjugates.
If are involutions, then by the above discussion
and
lie in different conjugates of
, and so the actions
and
fix different cosets of
. Also, we have seen that they swap different pairs of cosets. As such, the combined action
cannot fix any elements of
, and thus by (1) lies in
. We conclude that
; since
and
have the same order, this implies that
for all
, and hence
. This makes
a group (and thus a normal group, since it is conjugation invariant), as desired.
Remark 3 In this setting we can even establish the additional fact that
is abelian. Indeed, if
is an involution, then
, so for every
,
is an involution, which implies that
. Thus the inversion map
is a homomorphism on
, which forces
to be abelian.
8 comments
Comments feed for this article
24 May, 2013 at 6:15 pm
David Roberts
Theorem 1:
“with Frobenius complement H and Frobenius complement K.” <– Frob. kernel K.
[Corrected, thanks – T.]
26 May, 2013 at 3:35 am
Mikhail Katz
The statement of Frobenius theorem at the beginning of the post still says “frobenius complement K”.
[Fixed properly this time, thanks – T.]
26 May, 2013 at 7:47 am
John Mangual
Wikipedia expresses Frobenius groups in terms of permutation actions. Is it the same?
“A Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point.”
And then it says the Dihedral group is an example of a Frobenius group. And the groups of transformations
—
Lemma 5 is very interesting. I thought when
is a character it was always zero.
26 May, 2013 at 8:37 am
Terence Tao
Yes, they are equivalent; this is discussed in my previous post at https://terrytao.wordpress.com/2013/04/12/the-theorems-of-frobenius-and-suzuki-on-finite-groups/ .
A minimal idempotent
takes the form
for some irreducible character
. One can then simplify
as
for any
from the Fourier inversion formula for characters, so the first sum in Lemma 5 simplifies to
. It turns out in fact that in the irreducible case, the quantity in Lemma 5 vanishes when
is an exceptional character associated with a character in
, and is equal to 1 otherwise (this is basically the content of Lemma 6).
14 June, 2013 at 9:44 am
Andrew Drayton
Great article, thanks! I believe that the semidirect product is backwards in the statement of Frobenius thm. In other words, I seen
, since
is the normal subgroup (see e.g. Rotman).
[Corrected, thanks – T.]
29 May, 2015 at 1:05 pm
adriana82100
Could you advise please: in the Proof of Proposition 2 you wrote “if we let {P_V} be the orthogonal projection from {L^2(G)^G} to {V}, then {P_V} commutes with convolutions”.Why does {P_V} commute with convolutions?
[
is translation-invariant, hence
commutes with translations and hence with convolutions. -T.]
6 November, 2015 at 3:17 pm
Juri
I try to use your argument about idempotent functions to proof a more general theorem about Wielandt-Groups (it’s a group G with proper subgroup H and H0 is the smallest normal subgroup of H such that H intersected with gH(g^-1) is contained in H0 for each g from G\H). The problem is to show that the quantity (4) in the Lemma 5 is an integer. How did you find the right linear combinations of (5), (6), (7) and (8) in the proof of this Lemma? I will be grateful if you can give me some hint.
6 November, 2015 at 9:27 pm
Terence Tao
I no longer remember my exact thought processes here, but I think I was basically running Remark 1 in reverse, namely starting with the usual character-based proof, trying to write everything in terms of the idempotents
instead of the characters
, and then rewriting various integrality properties of
in terms of traces.