Applications of Representation Theory to Finite Group Study

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 =6 x ∈ G =⇒ CG(x) is abelian.

Here we use CG(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.ThenH is called a Frobenius subgroup of G iff: x/∈ H =⇒ (H ∩ Hx)=1, where Hx is the conjugate subgroup x−1Hx. 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, CG(x) ⊂ H. y y Proof. y ∈ CG(x)=⇒ x = x,andso1=6 x ∈ (H ∩ H )=⇒ 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 C G and G = NH.

Proof. Let us denote by C1,...,Ck the conjugacy classes of H,sothatC1 = {1}. Deﬁne for i>1 ∗ y Ci = {x ∈ G | x ∈ Ci for some y ∈ G} . y −1 (Here we use x to denote the conjugate element y xy.) By Lemma 1.1, CG(x) ⊂ H for nonidentity x ∈ H,andsoCH (x)=CG(x). x ∈ Ci for some i>1. But

Received by the editors December 16, 1998. 1 APPLICATIONS OF REPRESENTATION THEORY TO FINITE GROUP STUDY 2

∗ ∗ since |G|/|CG(x)| = |Ci | and |H|/|CH (x)| = |Ci|,if|G : H| = n then |Ci | = n|Ci|. Since [ ∗ N = G − Ci , i>1 |N| = n|H|−n (|H|−1) =⇒|N| = |G : H|. ∗ To see that N is indeed a normal subgroup of G,setC1 = N, and let us first show that the class function ∗ ∗ φ (Ci )=φ(Ci) is an irreducible character on G,whereφ an arbitrary irreducible character of H over C. Recall that for x ∈ G, the induced character φG is defined as X G 1 o −1 φ (x)=| | φ (y xy), H y∈G where φo(z)=φ(z)forz ∈ H, and φo(z)=0forz/∈ H. We see, therefore, that ∗ G φ − φ(1)1G =(φ − φ(1)1H ) , where 1G and 1H represent the trivial characters on G and H, respectively. This ∗ follows since both sides of the above equation produce 0 when applied to x ∈ C1 , ∗ ∗ G G ∗ while for x ∈ Ci , i>1, φ (x)=φ (x)and1G(x)=(1H ) (x) since |Ci | = ∗ n|Ci|.Thus,φ isP a generalized character of G, by which we mean an integral ∗ ∗ Plinear combination aiχi for χi irreducible characters of G. Since (φ ,φ )G = 2 ∗ ∗ ∗ ∗ ∗ ai , φ is itself irreducible on G iff (φ ,φ )G =1andφ (1)P> 0. But φ (1) = ∗ ∗ | | | ∗ ∗ |2| ∗| φ(1) > 0P since φ is irreducible on H, and (φ ,φ )G =(1/ G ) i φ (Ci ) Ci = 2 (1/|H|) i |φ(Ci)| |Ci| =(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 ∈ Ci , 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 C G as claimed. Since N C 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. APPLICATIONS OF REPRESENTATION THEORY TO FINITE GROUP STUDY 3

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) 1 =6 x ∈ N =⇒ CG(x) ⊂ N.

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 ∈ Ci ,wheretheCi are defined as before with respect y to the conjugacy classes Ci of a Frobenius subgroup H of G. x ∈ Ci for some −1 −1 y ∈ G, and hence x ∈ Hy . But it can be trivially verified that Hy satisfies −1 the properties specified in (1), and so Hy is itself a Frobenius subgroup of G. y−1 By Lemma 1.1, CG(x) ⊂ H . But by definition, (N −{1}) contains no elements y−1 ∗ conjugate to any element of H,andso(N −{1}) ∩ H = ∅.Thus,∀x ∈ Ci , i>1, (N −{1}) ∩ CG(x)=∅, i.e. no nonidentity element of N commutes with any ∗ element of Ci for i>1. We conclude that given nonidentity x ∈ N, CG(x) ⊂ N. Now we show that (1) is sufficient to characterize a proper, nontrivial subgroup N C G as a Frobenius kernel of the Frobenius group G. Various group theoretic arguments show that since CN (x)=1forx/∈ N, |N| is relatively prime to |G : N|, and hence ∃ a subgroup H ⊂ G s.t. NH = G and N ∩ H =1.(Themeansto 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 ∩ Hx =16 =⇒ x ∈ H, from which the assertion follows. H must be nontrivial and proper, since N is non- ∩ x 6 ∃ ∈ x ∈ trivial and proper and G = HN.IfH H =1, some nonidentity h H s.t. h 0 −1 0 H as well. Since G = HN, x = h0n for h0 ∈ H, n ∈ N.Now hh n−1hh n ∈ N 0 −1 0 0 −1 0 since N is a normal subgroup of G. However, hh n−1hh n = hh hh n = 0 −1 0 −1 0 hh hx ∈ H as well, and since N ∩ H =1,weseethat hh n−1hh n =1, 0 0 i.e. hh and n commute. But since h =1,6 hh ∈/ N, and by (1), therefore, n =1. So x = h0 ∈ 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) NG(K)=H,whereNG(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, CG(x) ⊂ H. APPLICATIONS OF REPRESENTATION THEORY TO FINITE GROUP STUDY 4

Trivial intersection subsets are in fact characterized succinctly by a condition similar to that which defines Frobenius subgroups. This characterization will of- tentimes 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,andsetH = NG(K). K is a TI-subset in G iff: (2) (K ∩ Kx) ⊂{1}∀x ∈ (G − H). Proof. Assume K is a TI -subset in G, and suppose ∃ x ∈ G s.t. nonidentity k ∈ (K ∩ Kx). k ∈ K and k =(k0)x for some k0 ∈ K.Sok and k0 are conjugate in G; by property (ii) of the above definition, they are conjugate in H, i.e. ∃ h ∈ H 0 h 0 x 0 h −1 0 s.t. k =(k ) .So(k ) =(k ) , implying that xh ∈ CG(k ). But since k is 0 0 nonidentity, so too is its conjugate k , and so by property (iii), CG(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 ∈ CG(x)=⇒ xy = x, and so nonidentity x ∈ (K ∩ Ky)=⇒ y ∈ H by (2). Thus, property (iii) of the definition holds. If k0 = kx for nonidentity k, k0 ∈ 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)thesetof 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)whenK is a TI -subset. Lemma 2.2. Let K be a TI-subset with respect to G and H,andletθ ∈ gc(H; K). Then θG(x)=θ(x) ∀x ∈ (K − 1). Furthermore, if θ(1) = 0, then for φ ∈ gc(H; K) we must have G G θ ,φ G =(θ, φ)H . Proof. Let nonidentity x ∈ K. We saw in our proof of Lemma 2.1 that if xy ∈ K for some y ∈ G, y ∈ H by necessity. So, coupling the fact that θ ∈ gc(H; K) vanishes on (H − K) withP the formula for computingP the induced character, we see G o −1 that θ (x)=(1/|H|) y∈G θ (y xy)=(1/|H|) y∈H θ(x)=θ(x), proving the first equation of Lemma 2.2. G Now let θ(1) = 0. We write θ H = θ + θ0,whereθ0 is a generalized character on H. Since θ(1) = 0, θG(1) = 0 by definition, and so θ = θG on all of K, ∈ − i.e. θ0 vanishes on K. Since φ gc(H; K) vanishes on (H K), (θ0,φ)H =0. G G G Coupling this fact with Frobenius reciprocity yields θ ,φ G = θ H ,φ H = (θ + θ0,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. APPLICATIONS OF REPRESENTATION THEORY TO FINITE GROUP STUDY 5

Theorem 2.3. Let K be a TI-subset with respect to G and H.Suppose∃ asetof irreducible characters of H, {φ1,...,φw}, satisfying w ≥ 2 and

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

Then ∃ a set of irreducible characters of G, {χ1,...,χw},andasign = 1 satisfying G φi = χi +∆, where ∆ is independent of i and either ∆ ≡ 0 on G or ∆ is a character of G. Proof. Suppose {φ1,...,φw} exists as above. Fix any i and j,andset θ =(φi −φj). ∈ G G Note that θ gc(H; K)s.t.θ(1) = 0; thus, by Lemma 2.2, θ ,θ G =(θ, θ)H = (φi − φj ,φi − φj )H = 2 due to the irreducibility of φi and φj .Also,θ(1) = 0 =⇒ θG(1) = 0, so we see that θG is the diﬀerence of two irreducible characters on G. G G If w = 2, then it follows automatically that θ =(φ1 − φ2) = χ1 − χ2,where χ1 and χ2 are suitably chosen irreducible characters on G. − G − G For w =3,notethat (φ1 φ2) , (φ1 φ3) G = 1. So denote the irreducible G G character of G contained by both (φ1 − φ2) and (φ1 − φ3) by χ1, and denote its multiplicity in each by = 1. We see therefore that, following the proof for the G G case w = 2, we are left with (φ1 − φ2) = (χ1 − χ2)and(φ1 − φ3) = (χ1 − χ3) for suitably chosen irreducible characters χ2 and χ3 on G. It also follows that G G G (φ2 − φ3) =(φ1 − φ3) − (φ1 − φ2) = (χ2 − χ3). G For w ≥ 4, the preceding paragraph allows us to again derive (φ1 − φ2) = G G (χ1 − χ2), (φ1 − φ3) = (χ1 − χ3), and (φ2 − φ3) = (χ2 − χ3) for suitably G chosen irreducible charactersχ1, χ2,andχ3 on G.Sotake(φ1 − φi) for i ≥ 4. − G − G − G Since (φ1 φi) , (φ1 φj) G =1forbothj =2andj =3,weseethat(φ1 φi) G must contain χ1 with multiplicity . For if not, (φ1 − φi) = −(χ2 + χ3)=⇒ χ2(1) + χ3(1) = 0, a contradiction since χ2 and χ3 are both irreducible on G.So G we may write (φ1 − φi) = (χ1 − χi)fori ≥ 4, where the χi are suitably chosen G irreducible characters over G, and by induction we can obtain (φi−φj) = (χi−χj) ∀i, j. G Since we have shown that (φi−φj) = (χi−χj) ∀i, j for some set {χ1,...,χw} of G irreducible characters on G, we are left only to re-express this relation as φi −χi = G φj − χj = ∆, where we note that ∆ is a character on G (or ∆ ≡ 0), and that ∆ G is independent of i.Thus,φi = χi + ∆, as desired.

Deﬁnition. The set {χ1,...,χw} of irreducible characters on G derivedinTheo- rem 2.3 are termed the exceptional characters associated with the set {φ1,...,φw} 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 K1,...,Ks 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 Ki,thenNG(Ki)=Ki 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. APPLICATIONS OF REPRESENTATION THEORY TO FINITE GROUP STUDY 6

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

(3) 1 =6 x ∈ K =⇒ CG(x)=K.

Notationally, we set H = NG(K), l = |H : K|,andw =(|K|−1)/l. Then K is a TI-subset with respect to G and H. Furthermore, if K =6 H,thenH is a Frobenius group with Frobenius kernel K. The nontrivial linear characters on K induce precisely w irreducible characters φ1,...,φw 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 =6 x ∈ (K ∩ Ky)=⇒ y ∈ H, from which (2) of Lemma 2.1 follows. So if y y−1 1 =6 x ∈ (K ∩ K )forsomey ∈ G,then1=6 x ∈ K as well, and so CG(x)= y y y−1 y y−1 y−1 CG(x )=K.ButthenK = CG(x ) = CG( x )=CG(x)=K,and so by definition y ∈ NG(K)=H. Now assume K =6 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}.Weseethat|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 ρx 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 x irreducible character φ on H s.t. φK contains ρ, φK in fact contains ρ ∀ x ∈ L, and hence ρH = 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,theremust be (|K|−1)/l = w irreducible characters on H induced by the nontrivial linear characters on K. Label these φ1,...,φw. Since K C H,noelementof(H − K)is conjugate to any element of K,andsoφi must vanish outside of K by the definition of the induced character ∀i.Also,φi(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 =6 x ∈ G =⇒ CG(x) is abelian. Then G is either an abelian group or a Frobenius group.

Proof. Let us take a set of representatives K1,...,Ks of the conjugacy classes of maximal abelian subgroups of G. For nonidentity x ∈ Ki,wenotethatCG(x) is abelian, and that since Ki is abelian, Ki ⊂ CG(x). But Ki is maximal, so CG(x)=Ki.WeseethatKi satisﬁes the properties of Lemma 2.5, and so Ki x is a TI -subset of G.SetHi = NG(Ki). If Ki = Hi,then(Hi ∩ Hi )={1} for x ∈ (G − Hi) by Lemma 2.1, and so Hi is a Frobenius subgroup and G a Frobenius group. So we assume G is nonabelian and that Ki =6 Hi ∀i and derive a contradiction. APPLICATIONS OF REPRESENTATION THEORY TO FINITE GROUP STUDY 7

Let li = |Hi : Ki| and wi =(|Ki|−1)/li. We know by Lemma 2.5 that there exists a set of wi irreducible characters on Hi of like degree which vanish outside of Ki. Since |G| is odd, so too are li and |Ki|,andsowi must be even and hence ≥ 2. So we employ Theorem 2.3 to construct a family Ei of wi exceptional characters on G. Note that for i =6 j, nonidentity elements of Ki and Kj are not conjugate, since y y y y x ∈ (Ki −{1})andx ∈ (Kj −{1})=⇒ Ki =(CG(x)) = CG(x )=Kj, contradicting the fact that Ki and Kj are not conjugate. Hence, each Ki contains wi nontrivial conjugacy classes distinct from those of any Kj, j =6 i. Since any element of G Pmust be contained in some maximal abelian subgroup, we seeP that G has precisely wi + 1 conjugacy classes as a result, and hence there are wi +1 irreducible representations on G. But for each Ki, we have defined a family Ei of wi exceptional characters on G.Ifχ1 and χ2 are distinct exceptional characters for Ki, we can easily verify from the definition of the induced character that χ1 = χ2 on Kj, j =6 i, since the irreducible characters on H from which χ1 and χ2 were induced must vanish outside of Ki.Soifχ1 were also an exceptional character for some Kj, χ2 would be an exceptional character for Kj likewise, since χ1 and χ2 are identically defined on Kj. ButP then χ1 = χ2 on Ki, contradicting the distinctness of χ1 and χ2.Thus,theP wi exceptional characters so far defined are all unique. We see that the wi nontrivial irreducible characters on G must all be exceptional characters for some Ki. But then, by Lemma 2.4, Ki = NG(Ki)=Hi 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.