A remark on the Brauer–Fowler theorems

Abstract

Let G be a finite group of even order that has no 2-rank 1. We will prove, using only elementary methods, that there is an involution $t \in G$ such that $|G| < |C_{G}(t)|^{6}$.

1 Introduction

In the process of the classification of finite simple groups, the following scheme (“local analysis”) has been used repeatedly. Let S be a known finite simple group, let $s \in S$ be an involution, and let $H=C_{S}(s)$. Assume that G is a finite simple group with an involution t such that $C_{G}(t) \simeq H$, so what can we say about G? The celebrated Brauer–Fowler paper [1] provides the upper bound to the order of G

$$\begin{aligned} \big |G \big | \le \left( \frac{1}{2}|H| \cdot \big (|H|+1 \big ) \right) \mathbf {!} \end{aligned}$$

and therefore it assures us that there are a finite number of cases to consider (the first Janko group $J_{1}$ was discovered by considering the case $H \simeq C_{2} \times A_{5}$).

Brauer–Fowler’s results, together with Feit–Thompson’s odd order theorem [4], are fundamental in the study of finite simple groups and are at the origin of the project (now theorem) of the classification of finite simple groups (CFSG).

We remember that a group G has p-rank r if a Sylow p-subgroup of G contains an elementary abelian subgroup of order $p^{r}$ but it does not contain elementary abelian subgroups of order $p^{r+1}$.

Using CFSG in all its strength, it was possible to establish that if a group of even order G does not have 2-rank 1, then there is an involution $t \in G$ such that

$$\begin{aligned} \big | G \big | < \big | C_{G}(t) \big |^{3} \end{aligned}$$

(1)

(see [11] and the discussion in [6]). If G has 2-rank 1, then, by the Brauer–Suzuki theorem [2], G admits a normal subgroup N of odd order such that $\overline{G}=G/N$ has a unique subgroup of order 2 (necessarily contained in the center of $\overline{G}$).

The purpose of this article is to prove, using only elementary methods (basically revisiting [1]), the following

Theorem 1

Let G be a group of even order and assume that G does not have 2-rank 1. Then

$$\begin{aligned} \big | G \big | < \big | C_{G}(t) \big |^{6} \end{aligned}$$

(2)

for some involution $t \in G$.

We will also show that, with some more work, the bound of Theorem 1 can be improved to

$$\begin{aligned} \big | G \big | < \left( \frac{ | C_{G}(t) |}{2} \right) ^{6}, \end{aligned}$$

(3)

which is a good approximation just in the case where $G=A_{5}$.

Theorem 1 can be refined in some special cases. In particular, we prove the following result.

Theorem 2

Let G be a group of even order and assume that G does not have 2-rank 1. If the centralizer of every involution of G is a 2-subgroup, then

$$\begin{aligned} \big | G \big | < 2 \cdot \big | C_{G}(t) \big |^{3} \end{aligned}$$

(4)

for some involution $t \in G$.

We remark that Suzuki has completely classified finite groups in which the centralizer of every involution is a 2-subgroup [9, part II].

2 Notation and basic results

In what follows by G we will denote a finite group of even order and by involution we will mean an element of order 2 of G.

If $X \subseteq G$, then ${\mathsf {Inv}}(X)=\{x \in X \mid x \not =1=x^{2} \}$ is the set of involutions in X; we define $\nu (X)=| {\mathsf {Inv}}(X)|$ and, if $x \in G$, $\nu (x)=|{\mathsf {Inv}}(C_{G}(x))|$. We also define

$$\begin{aligned} {\widetilde{\gamma }}=\max \big \{ | C_{G}(x) | \; \big | \; x \in {\mathsf {Inv}}(G) \big \} \;\;\; \text{ and } \;\;\; {\widetilde{\nu }}=\max \big \{ \nu (x) \; \big | \; x \in {\mathsf {Inv}}(G) \big \}. \end{aligned}$$

For an element $g \in G$, we will denote the order of g by |g|. For the rest, we will use the notation of Gorenstein’s book [5].

Lemma 3

([1, Theorem 3B]). Let G be a group having at least two conjugacy classes of involutions and let $t \in {\mathsf {Inv}}(G)$. Then

$$\begin{aligned} | G | \le | C_{G}(t) | \cdot {\widetilde{\nu }}^{2}, \end{aligned}$$

in particular, $|G| < {\widetilde{\gamma }}^{3}$.

Proof

Let $C=C_{G}(t)$ and $m=|G: C|$, we must show that $m \le {\widetilde{\nu }}^{2}$. By hypothesis, there is an involution $u \in G$ which is not conjugate to t and we set $D=C_{G}(u)$. Let ${\mathsf {Inv}}(D)=\{ u_{1}, u_{2},\ldots , u_{\ell } \}$ and set $D_{i}=C_{G}(u_{i})$, $1\le i \le \ell$, then $\ell \le {\widetilde{\nu }}$ and

$$\begin{aligned} \left| \bigcup _{i=1}^{\ell } {\mathsf {Inv}}(D_{i}) \right| < \sum _{i=1}^{\ell } \big | {\mathsf {Inv}}(D_{i}) \big | \le \ell \cdot {\widetilde{\nu }} \le {\widetilde{\nu }}^{2}. \end{aligned}$$

Now t has exactly m conjugates $t_{1}, t_{2}, \ldots , t_{m}$ in G. It will suffice to show that $t_{j} \in \bigcup _{i=1}^{\ell } D_{i}$ for every $1\le j \le m$. But $t_{j}$ is not conjugate to u and hence there is $v_{j} \in {\mathsf {Inv}}(G)$ such that $[u,v_{j}]=1=[t_{j},v_{j}]$, $1 \le j \le m$. Then $v_{j} \in D_{i}$ for some $1\le i \le \ell$, so $m \le {\widetilde{\nu }}^{2}$. Since $|G|=|C_{G}(t)| \cdot m$, the lemma follows. $\square$

Let $x \in G$. The set

$$\begin{aligned} C^{*}_{G}(x)=\{g \in G \mid x^{g}=x \; \text{ or } \; x^{g}=x^{-1} \} \end{aligned}$$

is the extended centralizer of x in G. It is immediate that $C_{G}^{*}(x)$ is a subgroup of G and $C_{G}(x) \le C_{G}^{*}(x) \le N_{G}(\langle x \rangle )$. An element $x \in G$ is strongly real if there is $t \in {\mathsf {Inv}}(G)$ such that $x^{t}=x^{-1}$.

For $x \in G$, let $\beta (x)$ be the number of $(t,u) \in {\mathsf {Inv}}(G) \times {\mathsf {Inv}}(G)$ such that $x=tu$.

Lemma 4

Let $x \in G$. Then we have:

• (a) If x is not strongly real, $\beta (x)=0$.

• (b) If $x=1$, $\beta (x)=\nu (G)$.

• (c) If $x \in {\mathsf {Inv}}(G)$, $\beta (x)=\nu \big ( C_{G}(x) \big ) -1 \le |C_{G}(x)|-2$.

• (d) If $x \not \in {\mathsf {Inv}}(G) \cup \{1\}$, then $\beta (x)=\nu \big (C_{G}^{*}(x) {\setminus } C_{G}(x) \big )$.

• (e) For any $x \in G$, $\beta (x) \le |C_{G}(x)|$.

Proof

See [5, Lemma 9.1.5]. $\square$

We define

$$\begin{aligned} {\widetilde{\beta }}=\max \big \{ \beta (x) \; \big | \; x \in G, x \not \in \{1 \} \cup {\mathsf {Inv}}(G) \big \} \end{aligned}$$

and ${\widetilde{\beta }}=0$ if the set of strongly real elements of G is $\{1 \} \cup {\mathsf {Inv}}(G)$.

Lemma 5

Let G be a group having a unique conjugacy class of involution. Then

$$\begin{aligned} \big | G \big | \le ({\widetilde{\beta }}+1) \cdot {\widetilde{\gamma }} \cdot ({\widetilde{\gamma }}-1). \end{aligned}$$

(5)

In particular, if $\, {\widetilde{\beta }} \ge {\widetilde{\gamma }}$, then $|G| \le {\widetilde{\beta }} \cdot {\widetilde{\gamma }}^{2}$.

Proof

Let $t, x_{0}=1, x_{1}, \ldots , x_{k}$ be representatives of the conjugacy classes of strongly real elements of G with $|t|=2$ and $|x_{j}|>2$ for $1 \le j \le k$. Let $\nu =\nu \big (C_{G}(t) \big )$. Since ${\widetilde{\gamma }}=| C_{G}(t) |$, we have $\nu (G)=|G| \cdot {\widetilde{\gamma }}^{-1}$. By the definition of ${\widetilde{\beta }}$, Lemma 4, and by taking into account that $\nu \le {\widetilde{\gamma }}-1$, we have

$$\begin{aligned} \nu (G)^{2}&= \sum _{g \in G} \beta (g) \le (\nu -1)\frac{|G|}{{\widetilde{\gamma }}} +\nu (G) + \sum _{j=1}^{k}\beta (x_{i}) \frac{|G|}{| C_{G}(x_{i})|}&\\&\le \nu \frac{|G|}{ {\widetilde{\gamma }}}+ {\widetilde{\beta }} \cdot \sum _{j=1}^{k} \frac{|G|}{|C_{G}(x_{j})|} \le \nu \cdot \frac{|G|}{{\widetilde{\gamma }}} +{\widetilde{\beta }} \cdot \big (1 - \frac{1}{{\widetilde{\gamma }}} \big ) \cdot |G|&\end{aligned}$$

and hence, since $\nu \le {\widetilde{\gamma }}-1$,

$$\begin{aligned} \frac{|G|}{{\widetilde{\gamma }}^{2}} = \frac{\nu (G)^{2}}{|G|} \le \frac{\nu }{{\widetilde{\gamma }}} +\big (1 - \frac{1}{{\widetilde{\gamma }}} \big ) \cdot {\widetilde{\beta }} \le \big (1-\frac{1}{{\widetilde{\gamma }}} \big ) \cdot \big (1+{\widetilde{\beta }} \big ). \end{aligned}$$

Thus the lemma is proved. $\square$

Lemma 6

Let H be a group of odd order admitting an automorphism $\phi$ of order 2. Let $F=C_{H}(\phi )$ and $I=\{ h \in H \mid h^{\phi }=h^{-1} \}$. Then the following conditions hold:

• (a) $H=FI=IF$, $F \cap I =\{1 \}$, and $| I |=|G:F|$.

• (b) Two elements of F conjugate in H are conjugate in F.

• (c) If $|I| > \frac{1}{3} | H |$, then $I=H$, H is abelian, and $\phi$ is the inversion on H.

• (d) We have

  $$\begin{aligned} \big | {\mathsf {Inv}}(H \langle \phi \rangle ) \big |= |I|=|H:F|. \end{aligned}$$

Proof

For (a) and (b), see [5, Lemma 10.4.1]. If $| H | < 3 \cdot |I|$, then $| F | <3$ and, since H has odd order, $F=\{1 \}$.

Let $\tau$ be an involution of $H \langle \phi \rangle$. Since H has odd order and $H\langle \phi \rangle =H \cup \phi H$, we must have $\tau =\phi h$ for some $h \in H$. Now $1=\tau ^{2}=hh^{\phi }$ implies $h^{\phi }=h^{-1}$ and $h \in I$. $\square$

Lemma 7

Let H be a group of odd order and let $\langle \phi \rangle \times \langle \tau \rangle \le \mathrm {Aut}(H)$ be an elementary abelian group of order 4. Then

$$\begin{aligned} \big | H \big | \cdot \big | C_{H}(\langle \phi , \tau \rangle ) \big |^{2} = \big | C_{H}(\phi ) \big | \cdot \big | C_{H}(\tau ) \big | \cdot \big | C_{H}(\phi \tau ) \big |. \end{aligned}$$

Proof

See [10] for a proof of this result of Brauer (and for other interesting generalizations). $\square$

Wall, in [12], gave a neat classification of finite groups with at least half of their elements being involutions (for an alternative proof that does not use character theory, see [8]). In the following lemma, we collect some results that we will use in our proofs.

Lemma 8

Let H be a group of even order, then we have:

• (a) If H is not an elementary abelian 2-group, then $\nu (H) < \frac{3}{4} \cdot | H|$.

• (b) If H is not a 2-group, then $\nu (H) < \frac{2}{3} \cdot | H |$.

Proof

Point (a) is well known: see [12, Lemma 7].

If H is not a 2-group, then H is of type I in the classification of [12, p. 261]. In particular, $H=A \rtimes \langle t \rangle$, where t is an involution that induces the inversion on the (abelian) subgroup A. Since |A| is not a 2-group, it contains an element of order at least 3 and $\nu (A) < \frac{1}{3} | A |$, so $\nu (H) < \frac{2}{3}$. $\square$

3 Proof of the theorems

In the next proofs, we will make use of the following basic result, the proof of which is elementary.

Let H be a group and $A,B \le H$, then

$$\begin{aligned} \big |G: A \cap B \big | \le \big | G:A \big | \cdot \big |G : B \big |. \end{aligned}$$

Proof of Theorem 1

Let G be a group of even order and of 2-rank greater than 1. If G has at least two conjugacy classes of involutions, we can conclude by Lemma 3.

Assume that G has a unique conjugacy class of involution and let C be the centralizer (in G) of an involution. From Lemma 5, we have

$$\begin{aligned} \big |G \big | \le ({\widetilde{\beta }}+1)\cdot \big | C \big | \cdot \big ( \big | C \big |-1 \big ) \end{aligned}$$

and hence, if ${\widetilde{\beta }}=0$, we are done. Let $x \not \in \{1 \} \cup {\mathsf {Inv}}(G)$ be a strongly real element of G such that $\beta (x)={\widetilde{\beta }}$, let $D=C_{G}(x)$ and $u \in {\mathsf {Inv}}(G)$ be such that $x^{u}=x^{-1}$, that is $C_{G}^{*}(x)=D\langle u \rangle =D^{*}$. We can assume $| D | \ge | C |$, as otherwise $|G| \le (|C|+1)\cdot |C| \cdot (|C| -1) < |C|^{3}$.

We must distinguish two cases.

$\bullet$ |D| is even.

In this case, $D \langle u \rangle$ has at least two conjugacy classes of involutions, so, from Lemma 3,

$$\begin{aligned} 2 \cdot | D | \le \Big | \max \big \{ C_{D^{*}}(v) \; \big | \; v \in {\mathsf {Inv}}(D^{*}) \big \} \Big |^{3} \le \big | C \big |^{3}. \end{aligned}$$

Since $\beta (x) \le |D|$, we obtain

$$\begin{aligned} \big | G \big | \le \big |D \big | \cdot \big |C \big |^{2} \le \frac{1}{2} \big | C \big |^{5} < \big | C \big |^{6}, \end{aligned}$$

(6)

and we are done.

$\bullet$ |D| is odd.

Let $v \not =u$ be an involution commuting with u and let $H=D \cap D^{v}$. By Lemma 6(d), $\beta (x)= | D : C_{D}(u) |$ so

$$\begin{aligned} \big |G \big | \le |C|^{2}\cdot \frac{|D|}{|C_{D}(u)|} \end{aligned}$$

and

$$\begin{aligned} \big |G: D \big | \le \big | C \big |^{2} \cdot \big | C_{D}(u) \big |^{-1}. \end{aligned}$$

Since $[u,v]=1$, H is $\langle u,v \rangle$-invariant. By Lemma 7, we obtain

$$\begin{aligned} \big | G \big |< & {} \frac{|D|}{|C_{D}(u)|} \cdot \big |C \big |^{2} \le \frac{|H|}{|C_{D}(u)|^{2}} \cdot \big |C \big |^{4} \nonumber \\\le & {} \frac{|C_{H}(u)| \cdot |C_{H}(v) | \cdot | C_{H}(uv)|}{|C_{D}(u)|^{2} \cdot | C_{H}(\langle u,v \rangle ) |^{2}} \cdot \big |C \big |^{4} \nonumber \\\le & {} \big | C_{H}(v) \big | \cdot \big | C_{H}(uv) \big |\cdot \big | C \big |^{4} < \big | C \big |^{6} \end{aligned}$$

(7)

and the claim is proved. $\square$

Proof of Theorem 2

Let $t \in {\mathsf {Inv}}(G)$ and $C=C_{G}(t)$. Arguing as in the proof of Theorem 1, we can suppose that G has a unique conjugacy class of involution and that there is a strongly real element $x \not \in \{1\} \cup {\mathsf {Inv}}(G)$ such that $\beta (x)={\widetilde{\beta }}$ (with the notation of Lemma 5) and $|G| \le (\beta (x)+1) \cdot |C| \cdot \big (|C|-1 \big )$. Let $D=C_{G}(x)$, we can assume, without loss of generality, $x^{t}=1$, so $C^{*}_{G}(x)=D\langle t \rangle$.

If $|x|=2\ell$ is even, then $x^{\ell }$ is an involution and $D \le C_{G}(x^{\ell })$. In this case, $|G| \le |C|^{3}$ and we are done. Assume that |x| is odd. Since the centralizers of involutions are (Sylow) 2-subgroups of G, |D| is odd and $C_{D}(t)=\{1\}$. Hence D is abelian and t induces on D the inversion.

Let $g \not \in N_{G}(D)$, and assume $D \cap D^{g} \not =\{1\}$. Let $1 \not = y \in D \cap D^{g}$, then $y^{t}=y^{-1}$ and $\langle D, D^{g} \rangle \le C_{G}(y)$, against the maximality of $\beta (x)$.

Hence $D \cap D^{g}=1$ for every $g \not \in N_{G}(D)$ and

$$\begin{aligned} \big |G \big | =\big |G : D \cap D^{g} \big | \le \big | G :D \big |^{2} \le \big | C \big |^{4}, \end{aligned}$$

in particular,

$$\begin{aligned} \big |D \big | =\big |D : D \cap D^{g} \big | \le \big | G :D \big | \le \big | C \big |^{2}, \end{aligned}$$

(8)

Let $N=N_{G}(D)$. Since $t \in Z(N/D)$, we can conclude that $N=DK$ is a Frobenius group with complement $K \le C$. Moreover, since K contains a unique involution [5, Theorem 10.3.1] and, by hypothesis, the rank of C is at least 2, we obtain $|K| < |C|$. From the double coset formula [7, Theorem 1.7.1], we have

$$\begin{aligned} G=\bigcup _{k \in K} kD \cup \bigcup _{i=1}^{r} Dg_{i}D, \end{aligned}$$

where

$$\begin{aligned} r=\frac{|G|-|K|}{|D|^{2}} \end{aligned}$$

and $|Dg_{i}D|=|D|^{2}$ for every $1 \le i \le r$.

Let $g_{i}$ be such that ${\mathsf {Inv}}(Dg_{i}D) \not = \emptyset$ and let u be an involution of $Dg_{i}D$, then $Dg_{i}D=DuD$ and we can assume $g_{i}=u$. It is clear that $\{ u^{d} \mid d \in D \}$ is a subset of order |D| of ${\mathsf {Inv}}(DuD)$. Let $w\in DuD$ be an involution, then $w=d_{1}ud_{2}$ with $d_{1}, d_{2} \in D$ and $1=w^{2}=d_{1}ud_{1}d_{2}ud_{2}$, that is $(d_{1}d_{2})^{u}=(d_{1}d_{2})^{-1}$ and, since $D \cap D^{u} =\{ 1 \}$, $d_{1}=d_{2}^{-1}$, so $| {\mathsf {Inv}}(DuD) |=|D|$.

Since $N=DK$ is a Frobenius group with complement K, we also have $tD \subseteq {\mathsf {Inv}}(G)$ and $kD \cap {\mathsf {Inv}}(G)=\emptyset$ if $k \in K {\setminus } \{t\}$. We can now evaluate the number of involutions in G:

$$\begin{aligned} \big | {\mathsf {Inv}}(G) \big | =\frac{|G|}{|C|} \le \big |D \big | + r\cdot \big |D \big | = \big | D \big |+ \frac{|G|-|K|}{|D|}. \end{aligned}$$

(9)

From (9), we obtain

$$\begin{aligned} \big |G \big | \left( \big |D \big |- \big |C \big | \right) \le \left( \big | D \big |^{2} -\big |K \big | \right) \cdot \big |C \big | \end{aligned}$$

or

$$\begin{aligned} \big | G \big | \le \frac{|D|^{2}-|K|}{|D|-|C|} \cdot |C|. \end{aligned}$$

(10)

Since $|G| \le |D||C|^{2}$, if $|D| \le 2 \cdot |C|$, we are done. If $|D| > 2\cdot |C|$, then, taking into account that $|K| \le |C|$, we can write

$$\begin{aligned} \frac{|D|^{2}-|K|}{|D|-|C|} \le 2 \cdot \big | D \big |. \end{aligned}$$

(11)

From (8), (10), and (11), we deduce

$$\begin{aligned} \big | G \big | \le 2 \cdot \big | D \big | \cdot \big | C \big | < 2 \cdot \big | C \big |^{3} \end{aligned}$$

and the proof is complete. $\square$

In order to prove the best bound (3) for the order of G, we will adopt the same notation as in the proof of Theorem 1. In addition, we will denote by S a Sylow 2-subgroup of G and we choose S so that $t \in Z(S) \le S \le C=C_{G}(t)$. We will therefore improve inequalities (6) and (7) to obtain the desired result.

We divide the proof into four steps.

(A) We can assume $| S | >4$.

If $|S|=4$, then since G does not have 2-rank 1, $S=C_{2} \times C_{2}$. By a deep result of Brauer ([3, Theorem 5.A] and [3, Remarks 1 and 5] immediately after it) if G is such that $O_{2'}(G)=1$, then, taking into account that $|C_{G}(S)| \ge 4$,

$$\begin{aligned} \big |G \big | \le 15 \cdot \frac{|C|^{3}}{|C_{G}(S)|^{2}} < \big | C \big |^{3}. \end{aligned}$$

(12)

In this case, we are done since $|C|\ge 4$ and, by (12), we can immediately deduce $|G| < \big ( |C|/2 \big )^{6}$. If $N=O_{2'}(G) \not =1$, then we can consider the action of $S=\langle t,u \rangle$ on N. Since N is odd, $C_{N}(t) \le \frac{1}{4}|C|$ and, by Lemma 7,

$$\begin{aligned} \big | N \big | \le \big |C_{N}(\langle t,u \rangle ) \big |^{-2} \cdot \big | C_{N}(t) \big | \cdot \big | C_{N}(u) \big | \cdot \big | C_{N}(tu) \big | \le \left( \frac{|C|}{4}\right) ^{3}. \end{aligned}$$

(13)

Putting the inequalities (12) and (13) together, we obtain

$$\begin{aligned} \big |G \big | \le \big | C \big |^{3} \cdot \left( \frac{C}{4} \right) ^{3} =\left( \frac{|C|}{2} \right) ^{6}. \end{aligned}$$

In this case, (3) holds. $\Diamond$

(B) We can assume that C is not a 2-group. Hence, in particular, $|C| \ge 24$ and $\nu (C) <\frac{2}{3} \cdot | C|$.

If C is a 2-subgroup of G, then, by Theorem 2, $|G| < 2\cdot |C|^{3}$. By (A), we have $|C|\ge 8$ and

$$\begin{aligned} \big |G \big |< 2\cdot \big |C \big |^{3} \le \frac{2 \cdot |C|^{6}}{8^{3}} <\left( \frac{|C|}{2} \right) ^{6}. \end{aligned}$$

Since C is not a 2-group, then, by Lemma 8(b), $\nu (C) < \frac{2}{3} |C|$, moreover $|S| \ge 8$ and hence $|C| \ge 24$. $\Diamond$

As in the proof of Theorem 1, let $x \in G$ be a strongly real element with $\beta (x)={\widetilde{\beta }}$, $D=C_{G}(x)$, and $D^{*}=C_{G}^{*}(x)=D\langle u \rangle$.

(C) If |D| is even, then we can prove inequality (3).

By Lemma 3 and (B), we deduce

$$\begin{aligned} \big |D^{*} \big |=2\cdot \big | D \big | \le \big |C_{D^{*}}(u) \big |\cdot {\widetilde{\nu }}(D^{*})^{2}< \frac{4}{9} \cdot \big | C \big |^{3} < \frac{1}{2} \cdot \big | C \big |^{3}, \end{aligned}$$

that is $|D| < 4^{-1} \cdot |C|^{3}$. Since $|C| \ge 24$, we can rewrite (6) as

$$\begin{aligned} \big | G \big | \le \big | D \big | \cdot \big | C \big |^{2}< \frac{1}{4} \cdot \big | C \big |^5 \le \frac{1}{96} \cdot \big | C \big |^{6} < \left( \frac{|C|}{2} \right) ^{6}. \end{aligned}$$

In this case, (3) holds. $\Diamond$

(D) If |D| is odd, then we can prove inequality (3).

With the notations of the proof of Theorem 1, let $H=D \cap D^{v}$. Since |H| is odd and $|S| \ge 8$, $|C_{H}(w)| \le |C_{D}(w)| \le 8^{-1} \cdot |C|$ (for $w \in \{u,v,uv\}$). So, we can rewrite (7) as

$$\begin{aligned} \big | G \big | < \big | C_{H}(v) \big | \cdot \big | C_{H}(uv) \big | \cdot \big |C \big |^{4} \le \left( \frac{|C|}{8} \right) ^{2} \cdot \big | C \big |^{4}=\left( \frac{|C|}{2} \right) ^{6}. \end{aligned}$$

And also in this case, (3) holds. The proof is complete. $\square$

Remark 9

In the previous proof of (3), the only not completely elementary step is (A), which requires the theory of modular characters (but that does not use CFSG). It is possible to avoid in (A) the use of [3, Theorem 5.A] with a long and delicate argument, which makes use only of basic group theory. We do not present the proof here so as not to lengthen the paper too much.