APPLICATIONS OF REPRESENTATION THEORY TO FINITE GROUP STUDY

CHRISTOPHER A. MCMILLEN

Abstract. Particular branches of representation theory, most notably the concept of exceptional characters and their importance in the study of trivial intersection subsets, leads to a succinct classification of finite groups of odd order satisfying the relation

1 ≠ x ∈ G ⟹ C_{G}(x) is abelian. Here we use C_{G}(x) to denote the centralizer of x in G. We shall explore this classification in this exposition.

1. Frobenius Groups

Definition. Let H be a proper, nontrivial subgroup of a group G. Then H is called a Frobenius subgroup of G iff:

x ∉ H ⟹ (H ∩ H^{x}) = 1,

where H^{x} is the conjugate subgroup x^{−1}Hx. A group G which contains a Frobenius subgroup is called a Frobenius group.

Lemma 1.1. Let H be a Frobenius subgroup of a Frobenius group G. ∀ nonidentity x ∈ H, C_{G}(x) ⊂ H.

Proof. y ∈ C_{G}(x) ⟹ x^{y} = x, and so 1 ≠ x ∈ (H ∩ H^{y}) ⟹ y ∈ H by the definition of a Frobenius subgroup.

We bring up the concept of Frobenius groups not only for the integral role they play in analyzing groups of odd order which satisfy the above centralizer condition, but also for their value in demonstrating the importance of representation theory to the study of finite group structure, as the following theorem demonstrates.

Theorem 1.2. Let H be a Frobenius subgroup of a Frobenius group G. We define N to be the set of elements in G not conjugate to any nonidentity element of H. Then |N| = |G : H|. Furthermore, N ◁ G and G = NH.

Proof. Let us denote by C_{1}, …, C_{k} the conjugacy classes of H, so that C_{1} = {1}. Define for i > 1

C_{i}^{∗} = {x ∈ G | x^{y} ∈ C_{i} for some y ∈ G} .

(Here we use x^{y} to denote the conjugate element y^{−1}xy.) By Lemma 1.1, C_{G}(x) ⊂ H for nonidentity x ∈ H, and so C_{H}(x) = C_{G}(x). x ∈ C_{i} for some i > 1. But since |G|/|C_{G}(x)| = |C_{i}^{∗}| and |H|/|C_{H}(x)| = |C_{i}|, if |G : H| = n then |C_{i}^{∗}| = n|C_{i}|. Since

Received by the editors December 16, 1998.

N = G − ⋃_{i>1} C_{i}^{∗},

|N| = n|H| − n (|H| − 1) ⟹ |N| = |G : H|.

To see that N is indeed a normal subgroup of G, set C_{1}^{∗} = N, and let us first show that the class function

ϕ^{∗}(C_{i}^{∗}) = ϕ(C_{i})

is an irreducible character on G, where ϕ an arbitrary irreducible character of H over C. Recall that for x ∈ G, the induced character ϕ^{G} is defined as

ϕ^{G}(x) = \frac{1}{|H|} ∑_{y∈G} ϕ^{o}(y^{−1}xy),

where

ϕ^{o}(z) = ϕ(z) for z ∈ H, and ϕ^{o}(z) = 0 for z ∉ H.

We see, therefore, that

ϕ^{∗} − ϕ(1)1_{G} = (ϕ − ϕ(1)1_{H})^{G} ,

where 1_{G} and 1_{H} represent the trivial characters on G and H, respectively. This follows since both sides of the above equation produce 0 when applied to x ∈ C_{1}^{∗}, while for x ∈ C_{i}^{∗}, i > 1, ϕ^{∗}(x) = ϕ^{G}(x) and 1_{G}(x) = (1_{H})^{G}(x) since |C_{i}^{∗}| = n|C_{i}|. Thus, ϕ^{∗} is a generalized character of G, by which we mean an integral linear combination ∑ a_{i}χ_{i} for χ_{i} irreducible characters of G. Since (ϕ^{∗}, ϕ^{∗})_{G} = ∑ a_{i}^{2}, ϕ^{∗} is itself irreducible on G iff (ϕ^{∗}, ϕ^{∗})_{G} = 1 and ϕ^{∗}(1) > 0. But ϕ^{∗}(1) = ϕ(1) > 0 since ϕ is irreducible on H, and (ϕ^{∗}, ϕ^{∗})_{G} = (1/|G|) ∑_{i} |ϕ^{∗}(C_{i}^{∗})|^{2}|C_{i}^{∗}| = (1/|H|) ∑_{i} |ϕ(C_{i})|^{2}|C_{i}| = (ϕ, ϕ)_{H} = 1, again by the irreducibility of ϕ.

Now, since ϕ^{∗}(n) = ϕ^{∗}(1) for n ∈ N by definition, N is contained in the kernel of the representation with character ϕ^{∗} for any given irreducible character ϕ of H over C. But since C is a field of characteristic 0, given any nonidentity x ∈ H ∃ an irreducible representation ρ of H s.t. x ∉ ker ρ. (This is a highly nontrivial result, stated in [2, Proposition 1.4 of Chapter 6], which requires Clifford’s Theorem, among other things, for its proof.) Thus, for any x ∈ C_{i}^{∗}, i > 1, ∃ an irreducible character ϕ of H s.t. x is not contained in the kernel of the representation with character ϕ^{∗}. Hence, N is precisely the intersection of the kernels of all representations with character ϕ^{∗} corresponding to some irreducible character ϕ of H. Thus, N ◁ G as claimed.

Since N ◁ G, N ∩ H = {1}, and |N| = |G : H|, it follows by elementary group theory that G = NH. (See [1, Section 4 of Chapter 1], if necessary, for details.)

Definition. The normal subgroup N of the Frobenius group G in Theorem 1.2 is called the Frobenius kernel of G.

It turns out that the Frobenius kernel N of a Frobenius group G necessarily satisfies a particular centralizer condition, and that in fact this condition is sufficient to classify any proper, nontrivial normal subgroup N of an arbitrary group G as a Frobenius kernel, and G as a Frobenius group. This fact is subsumed in the following theorem, which will prove vital in later applications.

Theorem 1.3. Let N be a Frobenius kernel of the Frobenius group G. N is a nontrivial, proper, normal subgroup of G satisfying

1 ≠ x ∈ N ⟹ C_{G}(x) ⊂ N. (1)

Conversely, a nontrivial, proper, normal subgroup N of an arbitrary group G which satisfies (1) is a Frobenius kernel, and G is a Frobenius group.

Proof. We begin by proving the necessity of (1). Let G be a Frobenius group with Frobenius kernel N. Since a Frobenius subgroup H of G is proper and nontrivial by definition, it follows that since G = NH, N must be proper and nontrivial as well. For i > 1, take x ∈ C_{i}^{∗}, where the C_{i}^{∗} are defined as before with respect to the conjugacy classes C_{i} of a Frobenius subgroup H of G. x^{y} ∈ C_{i} for some y ∈ G, and hence x ∈ H^{y^{−1}}. But it can be trivially verified that H^{y^{−1}} satisfies the properties specified in (1), and so H^{y^{−1}} is itself a Frobenius subgroup of G. By Lemma 1.1, C_{G}(x) ⊂ H^{y^{−1}}. But by definition, (N − {1}) contains no elements conjugate to any element of H, and so (N − {1}) ∩ H^{y^{−1}} = ∅. Thus, ∀x ∈ C_{i}^{∗}, i > 1, (N − {1}) ∩ C_{G}(x) = ∅, i.e. no nonidentity element of N commutes with any element of C_{i}^{∗} for i > 1. We conclude that given nonidentity x ∈ N, C_{G}(x) ⊂ N.

Now we show that (1) is sufficient to characterize a proper, nontrivial subgroup N ◁ G as a Frobenius kernel of the Frobenius group G. Various group theoretic arguments show that since C_{N}(x) = 1 for x ∉ N, |N| is relatively prime to |G : N|, and hence ∃ a subgroup H ⊂ G s.t. NH = G and N ∩ H = 1. (The means to prove this assertion, though based on fundamental tenets of group theory, lies far astray of the scope of this exposition. See [2, page 280] for details.) In fact, H is a Frobenius subgroup of G. To see this, we will show that, for x ∈ G,

H ∩ H^{x} ≠ 1 ⟹ x ∈ H,

from which the assertion follows. H must be nontrivial and proper, since N is nontrivial and proper and G = HN. If H ∩H^{x} ≠ 1, ∃ some nonidentity h ∈ H s.t. h^{x} ∈

H as well. Since G = HN, x = h′n for h′ ∈ H, n ∈ N. Now (h^{h′})^{−1} n^{−1}h^{h′}n ∈ N

since N is a normal subgroup of G. However, (h^{h′})^{−1} n^{−1}h^{h′}n = (h^{h′})^{−1} h^{h′n} =

(h^{h′})^{−1} h^{x} ∈ H as well, and since N ∩ H = 1, we see that (h^{h′})^{−1} n^{−1}h^{h′}n = 1,

i.e. h^{h′} and n commute. But since h ≠ 1, h^{h′} ∉ N, and by (1), therefore, n = 1. So x = h′ ∈ H, as we desired. Since H is a Frobenius subgroup and hence G a Frobenius group, we can then easily verify that N satisfies the conditions outlined in Theorem 1.2 and conclude that N is the Frobenius kernel of G.

2. Trivial Intersection Subsets and Exceptional Characters

Definition. Let K be a subset of a subgroup H of a group G. K is called a trivial intersection subset with respect to G and H (or, in shorthand, a TI -subset in G) iff:

(i) N_{G}(K) = H, where N_{G}(K) represents the normalizer of K in G; (ii) Elements of K which are conjugate in G are likewise conjugate in H; (iii) ∀ nonidentity x ∈ K, C_{G}(x) ⊂ H.

Trivial intersection subsets are in fact characterized succinctly by a condition similar to that which defines Frobenius subgroups. This characterization will oftentimes prove more useful that the rather cumbersome definition above, and so we state it below for convenience.

Lemma 2.1. Let K be a nonempty, nontrivial subset of a group G, and set H = N_{G}(K). K is a TI-subset in G iff:

(K ∩ K^{x}) ⊂ {1} ∀x ∈ (G − H). (2)

Proof. Assume K is a TI -subset in G, and suppose ∃ x ∈ G s.t. nonidentity k ∈ (K ∩ K^{x}). k ∈ K and k = (k′)^{x} for some k′ ∈ K. So k and k′ are conjugate in G; by property (ii) of the above definition, they are conjugate in H, i.e. ∃ h ∈ H s.t. k = (k′)^{h}. So (k′)^{x} = (k′)^{h}, implying that xh^{−1} ∈ C_{G}(k′). But since k is nonidentity, so too is its conjugate k′, and so by property (iii), C_{G}(k′) ⊂ H. Thus, xh^{−1} ∈ H ⟹ x = (xh^{−1})h ∈ H, proving that (2) must hold if K is a TI -subset in G.

Conversely, assume K satisfies (2). For nonidentity x ∈ K, y ∈ C_{G}(x) ⟹ x^{y} = x, and so nonidentity x ∈ (K ∩ K^{y}) ⟹ y ∈ H by (2). Thus, property (iii) of the definition holds. If k′ = k^{x} for nonidentity k, k′ ∈ K and x ∈ G, (2) implies that x ∈ H, and so property (ii) of the definition holds. Property (i) is satisfied by definition.

For any subset K contained in a subgroup H, let us denote by gc(H) the set of all generalized characters of H, and define

gc(H; K) = {θ ∈ gc(H) | θ(x) = 0 ∀x ∈ (H − K)} .

The following lemma displays a couple of important properties possessed by the elements of gc(H; K) when K is a TI -subset.

Lemma 2.2. Let K be a TI-subset with respect to G and H, and let θ ∈ gc(H; K). Then

θ^{G}(x) = θ(x) ∀x ∈ (K − 1).

Furthermore, if θ(1) = 0, then for ϕ ∈ gc(H; K) we must have

(θ^{G}, ϕ^{G})_{G} = (θ, ϕ)_{H} .

Proof. Let nonidentity x ∈ K. We saw in our proof of Lemma 2.1 that if x^{y} ∈ K for some y ∈ G, y ∈ H by necessity. So, coupling the fact that θ ∈ gc(H; K) vanishes on (H − K) with the formula for computing the induced character, we see that θ^{G}(x) = (1/|H|) ∑_{y∈G} θ^{o}(y^{−1}xy) = (1/|H|) ∑_{y∈H} θ(x) = θ(x), proving the first equation of Lemma 2.2.

Now let θ(1) = 0. We write (θ^{G})_{H} = θ + θ_{0}, where θ_{0} is a generalized character on H. Since θ(1) = 0, θ^{G}(1) = 0 by definition, and so θ = θ^{G} on all of K, i.e. θ_{0} vanishes on K. Since ϕ ∈ gc(H; K) vanishes on (H − K), (θ_{0}, ϕ)_{H} = 0. Coupling this fact with Frobenius reciprocity yields (θ^{G}, ϕ^{G})_{G} = ((θ^{G})_{H} , ϕ)_{H} = (θ + θ_{0}, ϕ)_{H} = (θ, ϕ)_{H}, and so the second equation of Lemma 2.2 is proved.

Using the lemmas we have developed about a TI -subset K contained in a subgroup H of a group G, we now establish a correspondence between certain irreducible characters of H and a set of irreducible characters of G.

Theorem 2.3. Let K be a TI-subset with respect to G and H. Suppose ∃ a set of irreducible characters of H, {ϕ_{1}, …, ϕ_{w}}, satisfying w ≥ 2 and

ϕ_{i}(x) = ϕ_{j}(x) ∀i, j and ∀x ∈ {H − K, 1} .

Then ∃ a set of irreducible characters of G, {χ_{1}, …, χ_{w}}, and a sign ϵ = ±1 satisfying

ϕ_{i}^{G} = ϵχ_{i} + Δ,

where Δ is independent of i and either Δ ≡ 0 on G or Δ is a character of G.

Proof. Suppose {ϕ_{1}, …, ϕ_{w}} exists as above. Fix any i and j, and set θ = (ϕ_{i} −ϕ_{j}). Note that θ ∈ gc(H; K) s.t. θ(1) = 0; thus, by Lemma 2.2, (θ^{G}, θ^{G})_{G} = (θ, θ)_{H} = (ϕ_{i} − ϕ_{j}, ϕ_{i} − ϕ_{j})_{H} = 2 due to the irreducibility of ϕ_{i} and ϕ_{j}. Also, θ(1) = 0 ⟹ θ^{G}(1) = 0, so we see that θ^{G} is the difference of two irreducible characters on G.

If w = 2, then it follows automatically that θ^{G} = (ϕ_{1} − ϕ_{2})^{G} = χ_{1} − χ_{2}, where χ_{1} and χ_{2} are suitably chosen irreducible characters on G.

For w = 3, note that ((ϕ_{1} − ϕ_{2})^{G}, (ϕ_{1} − ϕ_{3})^{G})_{G} = 1. So denote the irreducible character of G contained by both (ϕ_{1} − ϕ_{2})^{G} and (ϕ_{1} − ϕ_{3})^{G} by χ_{1}, and denote its multiplicity in each by ϵ = ±1. We see therefore that, following the proof for the case w = 2, we are left with (ϕ_{1} − ϕ_{2})^{G} = ϵ(χ_{1} − χ_{2}) and (ϕ_{1} − ϕ_{3})^{G} = ϵ(χ_{1} − χ_{3}) for suitably chosen irreducible characters χ_{2} and χ_{3} on G. It also follows that (ϕ_{2} − ϕ_{3})^{G} = (ϕ_{1} − ϕ_{3})^{G} − (ϕ_{1} − ϕ_{2})^{G} = ϵ(χ_{2} − χ_{3}).

For w ≥ 4, the preceding paragraph allows us to again derive (ϕ_{1} − ϕ_{2})^{G} = ϵ(χ_{1} − χ_{2}), (ϕ_{1} − ϕ_{3})^{G} = ϵ(χ_{1} − χ_{3}), and (ϕ_{2} − ϕ_{3})^{G} = ϵ(χ_{2} − χ_{3}) for suitably chosen irreducible characters χ_{1}, χ_{2}, and χ_{3} on G. So take (ϕ_{1} − ϕ_{i})^{G} for i ≥ 4. Since ((ϕ_{1} − ϕ_{i})^{G}, (ϕ_{1} − ϕ_{j})^{G})_{G} = 1 for both j = 2 and j = 3, we see that (ϕ_{1}−ϕ_{i})^{G} must contain χ_{1} with multiplicity ϵ. For if not, (ϕ_{1} − ϕ_{i})^{G} = −ϵ(χ_{2} + χ_{3}) ⟹ χ_{2}(1) + χ_{3}(1) = 0, a contradiction since χ_{2} and χ_{3} are both irreducible on G. So we may write (ϕ_{1} − ϕ_{i})^{G} = ϵ(χ_{1} − χ_{i}) for i ≥ 4, where the χ_{i} are suitably chosen irreducible characters over G, and by induction we can obtain (ϕ_{i}−ϕ_{j})^{G} = ϵ(χ_{i}−χ_{j}) ∀i, j.

Since we have shown that (ϕ_{i}−ϕ_{j})^{G} = ϵ(χ_{i}−χ_{j}) ∀i, j for some set {χ_{1}, …, χ_{w}} of irreducible characters on G, we are left only to re-express this relation as ϕ_{i}^{G}−ϵχ_{i} = ϕ_{j}^{G} − ϵχ_{j} = Δ, where we note that Δ is a character on G (or Δ ≡ 0), and that Δ is independent of i. Thus, ϕ_{i}^{G} = ϵχ_{i} + Δ, as desired.

Definition. The set {χ_{1}, …, χ_{w}} of irreducible characters on G derived in Theorem 2.3 are termed the exceptional characters associated with the set {ϕ_{1}, …, ϕ_{w}} of irreducible characters on H.

Exceptional characters prove vital in proving our main theorem due to the following lemma, whose proof, though perhaps within our grasp, contains too much gory algebraic detail to be very insightful to prove. We refer the interested reader to Suzuki’s development of this proposition.

Lemma 2.4. Let K_{1}, …, K_{s} be the representatives of the conjugacy classes of maximal abelian subgroups in a group G. If every nontrivial irreducible character of G is an exceptional character for some K_{i}, then N_{G}(K_{i}) = K_{i} for some i.

Proof. This is equivalent to what follows part (d) of Example 1 in [2, Section 2 of Chapter 6], and relies heavily upon size arguments developed in Theorems 2.12 and 2.13 of the same.

We state one final lemma that will aid us greatly in the proof of our main theorem.

Lemma 2.5. Let K be a nontrivial abelian subgroup of a group G which satisfies

1 ≠ x ∈ K ⟹ C_{G}(x) = K. (3)

Notationally, we set H = N_{G}(K), l = |H : K|, and w = (|K| − 1)/l. Then K is a TI-subset with respect to G and H. Furthermore, if K ≠ H, then H is a Frobenius group with Frobenius kernel K. The nontrivial linear characters on K induce precisely w irreducible characters ϕ_{1}, …, ϕ_{w} on H of like degree, all of which vanish outside of K.

Proof. To verify that K is a TI -subset with respect to G and H, we show that 1 ≠ x ∈ (K ∩ K^{y}) ⟹ y ∈ H, from which (2) of Lemma 2.1 follows. So if 1 ≠ x ∈ (K ∩ K^{y}) for some y ∈ G, then 1 ≠ x^{y^{−1}} ∈ K as well, and so C_{G}(x) = C_{G}(x^{y^{−1}}) = K. But then K^{y} = (C_{G}(x^{y^{−1}}))^{y} = C_{G}((x^{y^{−1}})^{y}) = C_{G}(x) = K, and so by definition y ∈ N_{G}(K) = H.

Now assume K ≠ H. K is a nontrivial, proper, normal subgroup of H; furthermore, K satisfies the conditions of (1) in Theorem 1.3, so it follows immediately that K is a Frobenius kernel of the Frobenius group H. Let L be a Frobenius subgroup of H s.t. H = KL and K ∩ L = {1}. We see that |L| = |H : K| = l. Some group theoretic arguments can be employed to show that, since an element x ∈ L has order prime to |K|, the conjugate character ρ^{x} of a nontrivial linear character ρ on K is distinct ∀ x ∈ L, and that in fact this set of l characters comprises all conjugate characters of ρ. Furthermore, Clifford’s Theorem tells us that, given an irreducible character ϕ on H s.t. ϕ_{K} contains ρ, ϕ_{K} in fact contains ρ^{x} ∀ x ∈ L, and hence ρ^{H} = ϕ. (These arguments are fleshed out in [2, p. 287]; Clifford’s Theorem can be found in [2, p.264].) More importantly from our perspective, since there are |K| − 1 nontrivial linear characters on the abelian group K, there must be (|K| − 1)/l = w irreducible characters on H induced by the nontrivial linear characters on K. Label these ϕ_{1}, …, ϕ_{w}. Since K ◁ H, no element of (H − K) is conjugate to any element of K, and so ϕ_{i} must vanish outside of K by the definition of the induced character ∀i. Also, ϕ_{i}(1) = |H : K| = l, so the induced characters are of like degree.

3. Finite Group Study Applications

Theorem 3.1. Let G be a group of odd order satisfying

(4) 1 ≠ x ∈ G ⟹ C_{G}(x) is abelian.

Then G is either an abelian group or a Frobenius group.

Proof. Let us take a set of representatives K_{1}, …, K_{s} of the conjugacy classes of maximal abelian subgroups of G. For nonidentity x ∈ K_{i}, we note that C_{G}(x) is abelian, and that since K_{i} is abelian, K_{i} ⊂ C_{G}(x). But K_{i} is maximal, so C_{G}(x) = K_{i}. We see that K_{i} satisfies the properties of Lemma 2.5, and so K_{i} is a TI -subset of G. Set H_{i} = N_{G}(K_{i}). If K_{i} = H_{i}, then (H_{i} ∩ H_{i}^{x}) = {1} for x ∈ (G − H_{i}) by Lemma 2.1, and so H_{i} is a Frobenius subgroup and G a Frobenius group. So we assume G is nonabelian and that K_{i} ≠ H_{i} ∀i and derive a contradiction.

Let l_{i} = |H_{i} : K_{i}| and w_{i} = (|K_{i}| − 1)/l_{i}. We know by Lemma 2.5 that there exists a set of w_{i} irreducible characters on H_{i} of like degree which vanish outside of K_{i}. Since |G| is odd, so too are l_{i} and |K_{i}|, and so w_{i} must be even and hence ≥ 2. So we employ Theorem 2.3 to construct a family E_{i} of w_{i} exceptional characters on G.

Note that for i ≠ j, nonidentity elements of K_{i} and K_{j} are not conjugate, since x ∈ (K_{i} − {1}) and x^{y} ∈ (K_{j} − {1}) ⟹ K_{i}^{y} = (C_{G}(x))^{y} = C_{G}(x^{y}) = K_{j}, contradicting the fact that K_{i} and K_{j} are not conjugate. Hence, each K_{i} contains w_{i} nontrivial conjugacy classes distinct from those of any K_{j}, j ≠ i. Since any element of G must be contained in some maximal abelian subgroup, we see that G has precisely ∑ w_{i} + 1 conjugacy classes as a result, and hence there are ∑ w_{i} + 1 irreducible representations on G.

But for each K_{i}, we have defined a family E_{i} of w_{i} exceptional characters on G. If χ_{1} and χ_{2} are distinct exceptional characters for K_{i}, we can easily verify from the definition of the induced character that χ_{1} = χ_{2} on K_{j}, j ≠ i, since the irreducible characters on H from which χ_{1} and χ_{2} were induced must vanish outside of K_{i}. So if χ_{1} were also an exceptional character for some K_{j}, χ_{2} would be an exceptional character for K_{j} likewise, since χ_{1} and χ_{2} are identically defined on K_{j}. But then χ_{1} = χ_{2} on K_{i}, contradicting the distinctness of χ_{1} and χ_{2}. Thus, the ∑ w_{i} exceptional characters so far defined are all unique. We see that the ∑ w_{i} nontrivial irreducible characters on G must all be exceptional characters for some K_{i}. But then, by Lemma 2.4, K_{i} = N_{G}(K_{i}) = H_{i} for some i. Contradiction.

References

1. Michio Suzuki, Group theory I, Springer-Verlag, New York, US, 1982.

2. , Group theory II, Springer-Verlag, New York, US, 1986.