##### Document Text Contents

Page 1

Advanced Studies in Pure Mathematics 32, 2001

Groups and Combinatorics–in memory of Michio Suzuki

pp. 1-39

Michio Suzuki

Koichiro Harada

\S 1 Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

\S 2 The Early Work of Michio Suzuki. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

\S 3 Theory of Exceptional Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

\S 4 The $CA$-paper of Suzuki. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

\S 5 Zassenhaus Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

\S 6 Suzuki’s Simple Groups $Sz(2^{n})\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots 13$

\S 7 $ZT$-groups and Related Classification Theorems . . . . . . . . . . . . . 17

\S 8 Group Theory in Japan before Suzuki. . . . . . . . . . . . . . . . . . . . . . . 29

\S 9 Michio Suzuki, my teacher and my mentor. . . . . . . . . . . . . . . . . . . 31

\S 1. Biographical Sketch

1926, October 2. Born in Chiba, Japan.

1942, April. Entered the Third High School of Japan located at Kyoto

(Noboru Ito, Katsumi Nomizu, Hidehiko Yamabe were his seniors by

one year and Singo Murakami was in the same class).

1945, April. Entered the University of Tokyo. Majored in mathematics.

(Gaishi Takeuchi, Nagayoshi Iwahori, Tsuneo Tamagawa were friends of

this period.)

1948, April. Entered the Graduate School of Tokyo University. Suzuki’s

supervisor was Shokichi Iyanaga. Kenkichi Iwasawa had a profound

influence on Suzuki.

1948-,51. Received a special graduate fellowship from the Government

of Japan.

1951, April to ’52, January. Held a lecturership at Tokyo University of

Education

1952, January to ’52, May. Held a graduate fellowship at University of

Illinois at Urbana-Champaign.

Received April 27, 1999.

Revised May 17, 1999.

Page 2

K. Harada

1952, May. Received the Doctor of Science Degree from the University

of Tokyoin absentia.

1952. Spent two months in the summer at University of Michigan. R.

Brauer was a professor of Mathematics at Michigan. J. Walter, W. Feit

were graduate students there.

1952, September to ’53, May. Held a post-doctoral fellowship at Univer-

sity of Illinois at Urbana-Champaign.

1952, November. Married to a daughter Naoko of Yasuo Akizuki (then

Professor at Kyoto University).

1953, September to ’55, May. Held a research associateship at University

of Illinois.

1955, September. Promoted to an assistant professor at University of

Illinois.

1956, September to ’57, May. Held a research associateship at Harvard

University.

1958, September. Promoted to an associate professor at University of

Illinois.

1959, September. Promoted to a full professor at University of Illinois.

1960, Discovered a new series of finite simple groups $Sz(q)$ .

1960-,61. Held a visiting appointment at the University of Chicago.

1962. Invited to speak at the International Congress of Mathematicians

in Stockholm.

1962-,63. Held a Guggenheim Fellowship.

1962, September to ’63, May. Held a membership at the Institute for

Advanced Study, Princeton.

1967. Discovered a sporadic simple group Suzuki of order 448,345,497,600.

1968-,69. Held a visiting appointment at the Institute for Advanced

Study, Princeton, $NJ$.

1970. Invited to speak at the International Congress of Mathematicians

in Nice, France.

1974. Received the Academy Prize from the Japan Academy.

1987. The conference of group theory and combinatorics for the occasion

of Suzuki’s 60th birthday was held in Kyoto, Japan.

1991. Awarded an honorary doctoral degree from the University of Kiel,

Germany.

1997. The conference of group theory and combinatorics for the occasion

of Suzuki’s 70th birthday was held in Tokyo, Japan.

Page 237

A characterization of $2E_{6}(2)$ 237

order with $|M_{0}|\geq|M|$ . Hence replacing $M$ by $M_{0}$ if necessary, we may

assume $P=F$ . In particular taking $F\leq S\in Syl_{3}(G)$ , $Z=Z(S)\leq F$ .

Let $ U=\langle Z^{M}\rangle$ , so that $Z\cong E_{3^{n}}$ for some $n$ . Then $C_{M}(U)\leq C_{M}(Z)$ ,

and $C_{M}(Z)$ is a 3-group by 7.15.5. Therefore $C_{M}(U)\leq O_{3}(M)=F$ .

Hence $|M|\leq|F|N_{n}$ , where $N_{n}$ is the maximal order of a subgroup $X$ of

odd order in $GL_{n}(3)$ with $O_{3}(X)=1$ .

By 7.15.6, $n\leq 5$ , so $|M|_{3’}$ divides 5 $\cdot 11\cdot 13$ . Indeed if 11 divides

$|M|$ then $n=5$ , so $U=J(S)$ for $S\in Syl_{3}(G)$ by 7.15.6, whereas by the

last remark in 7.15.6, 11 does not divide the order of $N_{G}(J(S))$ . So 11

does not divide the order of $M$ . Further by 7.15.4, $G$ has no subgroup

of order 13 $\cdot 5$ , so by Hall’s Theorem, (cf. 18.5 in [FGT]) $|M|_{3’}=1,5$ ,

or 13. But $|G|_{3}=3^{9}$ and $3^{9}$ . $5<10^{5}>3^{8}\cdot 13$ , so we are left with the

case $|M|=3^{9}\cdot 13$ .

By 7.15.1, if $Y$ is of order 13 in $M$ then $C_{F}(Y)=1$ and $|N_{M}(Y)|=1$

or 3. Therefore $|F|=3^{3k}$ for some $k$ and hence $F\in Syl_{3}(G)$ , contra-

dicting 7.15.1. Q.E.D.

\S 8. Groups of type ${}^{2}E_{6}(2)$ are isomorphic to ${}^{2}E_{6}(2)$

In this section we assume the hypotheses and notation of section 6.

In particular $G$ is of type $2E_{6}(2)$ , $z$ is a 2-central involution in $G$ , $H=$

$C_{G}(z)$ , etc. Further let $G_{0}=2E_{6}(2)$ and $z_{0}$ a long root involution of $G_{0}$ .

By 7.1, $G_{0}$ is of type $2E_{6}(2)$ with $z_{0}2$-central in $G_{0}$ . Let $H_{0}=C_{G_{0}}(z_{0})$ ,

$Q_{0}=O_{2}(H_{0})$ , etc.

(8.1) $\tilde{H}\cong H_{0}/\langle z_{0}\rangle$ .

Proof. First $Q_{0}\cong Q$ , so we may identify the two groups. Further

by 6.2, the representation of $H_{0}^{*}$ on $\tilde{Q}_{0}$ is quasiequivalent to that of $H^{*}$

on $\tilde{Q}$ , so $\tilde{H}\cong\tilde{H}_{0}$ by 3.1. Q.E.D.

By 8.1, $\tilde{H}_{0}\cong\tilde{H}$ , so by 7.8 there is $h\in H-C_{H}(\tilde{t})$ with $t^{h}\in E$ .

Let $k=gh$ , $ V_{3}=\langle z, t, z^{k}\rangle$ , $U_{3}=Q\cap Q^{g}\cap Q^{k}$ , $ X_{3}=\langle Q, Q^{g}, Q^{k}\rangle$ ,

$R_{3}=C_{X_{3}}(V_{3})$ ,

$S_{3}=(Q\cap Q^{g})(Q\cap Q^{k})(Q^{g}\cap Q^{k})$ ,

and $P_{3}=N_{G}(V_{3})$ . By 8.16 in [SG],

$R_{3}=C_{Q}(V_{3})C_{Q^{g}}(V_{3})C_{Q^{k}}(V_{3})=O_{2}(X_{3})$ ,

$X_{3}/R_{3}=GL(V_{3})\cong L_{3}(2)$ , $[X_{3}, U_{3}]\leq V_{3}$ , $\Phi(U_{3})=1$ , $P_{3}=X_{3}C_{H}(V_{3})$ ,

and $P_{3}/R_{3}=X_{3}/R_{3}\times C_{H}(V_{3})/R_{3}$ .

Page 238

238 M. Aschbacher

By 7.8, $C_{H}(V_{3})/R_{3}\cong A_{5}$ , so $P_{3}/R_{3}\cong L_{3}(2)\times A_{5}$ . Again by 7.8,

$m(U_{3})=6$ , so by 8.16 in [SG], $S_{3}/U_{3}$ is the sum of 4 copies of the dual $V_{3}^{*}$

of $V_{3}$ as an $X_{3}/R_{3}$ module and $R_{3}/S_{3}$ is the sum of 4 copies of $V_{3}$ as an

$X_{3}/R_{3}$ module By 7.8, $C_{H}(V_{3})$ has chief series $0<\tilde{V}<\tilde{V}_{3}<\tilde{U}_{3}<\tilde{E}$

on $\tilde{E}$ with $E/U_{3}$ the $\Omega_{4}^{-}(2)$ module and $U_{3}/V_{3}$ the $L_{2}(4)$ module for

$C_{H}(V_{3})$ . Finally by 7.8, $C_{H}(V_{3})$ has four $L_{2}(4)$ -sections and three $\Omega_{4}^{-}(2)-$

sections on $R_{3}$ . We summarize all this as:

(8.2) (1) $P_{3}/R_{3}=X_{3}/R_{3}\times C_{H}(V_{3})/R_{3}$ with $X_{3}/R_{3}\cong L_{3}(2)$ and

$C_{H}(V_{3})/R_{3}\cong A_{5}$ .

(2) $R_{3}$ has chief series

$0<V_{3}<U_{3}<S_{3}<R_{3}$

with $V_{3}$ the natural module for $X_{3}/R_{3}$ , $[X_{3}, U_{3}]\leq V_{3}$ and $U_{3}/V_{3}$ is the

$L_{2}(4)$ module for $C_{H}(V_{3})/R_{3}$ , $S_{3}/U_{3}$ is the tensor product of the dual

of $V_{3}$ as an $X_{3}/R_{3}$ -module with the $\Omega_{4}^{-}(2)$ module for $C_{H}(V_{3})/R_{3}$ , and

$R_{3}/S_{3}$ is the tensor product of $V_{3}$ as an $X_{3}/R_{3}$ -module with the $L_{2}(4)-$

module for $C_{H}(V_{3})/R_{3}$ .

(8.3) There exists $s\in z^{G}$ with $sz$ of order 3, $C_{G}(\langle s, z\rangle)\cong U_{6}(2)$ ,

and $N_{G}(\langle sz\rangle)=\langle s, z\rangle\times C_{G}(\langle s, z\rangle)$ .

Proof Let $ X_{2}=\langle Q, Q^{g}\rangle$ . Then $X_{2}\leq X_{3}$ so there is $x$ of order

3 in $X_{2}$ fused to $y\in X_{3}\cap H$ . Notice $y^{*}$ is inverted by a transvection

in $H^{*}$ as $\tilde{H}_{0}\cong\tilde{H}$ and the remark holds in $H_{0}^{*}$ since $y$ is inverted by

some conjugate $c\in Q^{g}$ of $z$ in $H_{0}$ and $c^{*}$ is a transvection in $H_{0}^{*}$ by

7.2 and 7.3.2. Therefore $C_{Q}(y)\cong D_{8}^{4}$ and $C_{H}(y)/C_{Q}(y)\langle y\rangle\cong U_{4}(2)$ .

Let $T_{y}\in Syl_{2}(C_{H}(y))$ ; then $\langle z\rangle=Z(T_{y})$ and $T_{y}$ is of order $2^{15}$ . As

$\langle z\rangle=Z(T_{y})$ , $T_{y}\in Syl_{2}(C_{G}(y))$ .

Next let $T_{x}\in Syl_{2}(C_{P_{2}}(x))$ . From the structure of $P_{2}$ described in

6.1,

$C_{P_{2}}(x)/\langle x\rangle\cong L_{3}(4)/E_{2^{9}}$

with $O_{2}(C_{P_{2}}(x))$ quasiequivalent to the Todd module for $C_{P_{2}}(x)/O_{2}(C_{P_{2}}$

$(x))\langle x\rangle$ . In particular $T_{x}$ is of order $2^{15}$ and hence as $x$ and $y$ are conju-

gate, the previous paragraph says that $T_{x}\in Syl_{2}(C_{G}(x))$ and $Z(T_{x})$ is

generated by a conjugate of $z$ . Now the hypotheses of Theorem 30.1 in

$[3T]$ are satisfied, so by that Theorem, $C_{G}(x)/\langle x\rangle\cong C_{G}(y)/\langle y\rangle\cong U_{6}(2)$ .

Next $x$ is inverted by an involution $u\in Q$ with $[C_{P_{2}}(x), u]=$

$\langle x\rangle$ , so $u$ induces an automorphism of $C_{G}(x)/\langle x\rangle\cong U_{6}(2)$ centralizing

Page 473

$|Hom(A, G)|$ (III) 473

Then we have

$X_{G}(t)$ $=$ $\sum_{n=0}^{\infty}\sum_{r=0}^{n}(-1)^{r}p^{(_{2}^{r})}s(C_{p}^{r}, G)(-t)^{n}$

$=$ $(1+t)^{-1}\sum_{r=0}^{\infty}p^{(_{2}^{r})}s(C_{p}^{r}, G)t^{r}$

$=$ $(1+t)^{-1}S_{G,f}^{E_{p}}(t)$ .

Thus the transition identity (16) gives

(22) $X_{G}(t)=\prod_{n=0}^{\infty}(1+p^{-n}t)^{-1}\cdot H_{G,f}^{E_{p}}(t)$ .

Similarly, if we view $X_{G}(t)$ and $H_{G,f}^{E_{p}}(t)$ as $p$-adic power series (18), we

have

(23) $X_{G}(t)=\prod_{n=1}^{\infty}(1+p^{n}t)\cdot H_{G,f}^{E_{p}}(t)$ .

These formula gives a transition formula between $\{h_{n}\}$ and $\{\chi_{n}’\}$ :

(24) $h_{n}=\sum_{r=0}^{n}(-1)^{r}p^{n-r}$ $\left\{\begin{array}{l}n\\r\end{array}\right\}$

$p\chi_{r}’$

.

By (21) and (24), if $p^{n}$ divides $|G|$ , then

(25) $\chi_{n}’\equiv 0$ $(mod p^{n})$ $\Leftrightarrow$ $h_{n}\equiv 0$ $(mod p^{n})$ .

The right hand side of this statement is valid by [Yo 93]. Thus we again

have Brown’s cohomological Sylow theorem ([Yo 96]):

(26) $\chi(S_{p}(G))\equiv 1$ $(mod |G|_{p})$ .

References

[St 97] R.P. Stanley, Enumerative Combinatorics, Volume I, Cambridge Uni-

versity Press, 1997.

[Su 82] M. Suzuki, Group Theory, I, II, Springer, 1982.

[Yo 92] T. Yoshida, P.Hall’s strange formula for abelian $p$-groups, Osaka

Math. J., 29 (1992), 421-431.

[Yo 93] T. Yoshida, $|(A, G)|$ , J. Algebra, 156 (1993), 125-156.

Page 474

474 T. Yoshida

[Yo 96] T. Yoshida, Classical problems in group theory (I): Enumerating sub-

groups and homomorphisms, Sugaku Expositions, 9 (1996),169-

184.

Department of Mathematics

Hokkaido University

Sapporo 060-0810, Japan

e-mail: [email protected]

Advanced Studies in Pure Mathematics 32, 2001

Groups and Combinatorics–in memory of Michio Suzuki

pp. 1-39

Michio Suzuki

Koichiro Harada

\S 1 Biographical Sketch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

\S 2 The Early Work of Michio Suzuki. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

\S 3 Theory of Exceptional Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

\S 4 The $CA$-paper of Suzuki. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

\S 5 Zassenhaus Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

\S 6 Suzuki’s Simple Groups $Sz(2^{n})\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots 13$

\S 7 $ZT$-groups and Related Classification Theorems . . . . . . . . . . . . . 17

\S 8 Group Theory in Japan before Suzuki. . . . . . . . . . . . . . . . . . . . . . . 29

\S 9 Michio Suzuki, my teacher and my mentor. . . . . . . . . . . . . . . . . . . 31

\S 1. Biographical Sketch

1926, October 2. Born in Chiba, Japan.

1942, April. Entered the Third High School of Japan located at Kyoto

(Noboru Ito, Katsumi Nomizu, Hidehiko Yamabe were his seniors by

one year and Singo Murakami was in the same class).

1945, April. Entered the University of Tokyo. Majored in mathematics.

(Gaishi Takeuchi, Nagayoshi Iwahori, Tsuneo Tamagawa were friends of

this period.)

1948, April. Entered the Graduate School of Tokyo University. Suzuki’s

supervisor was Shokichi Iyanaga. Kenkichi Iwasawa had a profound

influence on Suzuki.

1948-,51. Received a special graduate fellowship from the Government

of Japan.

1951, April to ’52, January. Held a lecturership at Tokyo University of

Education

1952, January to ’52, May. Held a graduate fellowship at University of

Illinois at Urbana-Champaign.

Received April 27, 1999.

Revised May 17, 1999.

Page 2

K. Harada

1952, May. Received the Doctor of Science Degree from the University

of Tokyoin absentia.

1952. Spent two months in the summer at University of Michigan. R.

Brauer was a professor of Mathematics at Michigan. J. Walter, W. Feit

were graduate students there.

1952, September to ’53, May. Held a post-doctoral fellowship at Univer-

sity of Illinois at Urbana-Champaign.

1952, November. Married to a daughter Naoko of Yasuo Akizuki (then

Professor at Kyoto University).

1953, September to ’55, May. Held a research associateship at University

of Illinois.

1955, September. Promoted to an assistant professor at University of

Illinois.

1956, September to ’57, May. Held a research associateship at Harvard

University.

1958, September. Promoted to an associate professor at University of

Illinois.

1959, September. Promoted to a full professor at University of Illinois.

1960, Discovered a new series of finite simple groups $Sz(q)$ .

1960-,61. Held a visiting appointment at the University of Chicago.

1962. Invited to speak at the International Congress of Mathematicians

in Stockholm.

1962-,63. Held a Guggenheim Fellowship.

1962, September to ’63, May. Held a membership at the Institute for

Advanced Study, Princeton.

1967. Discovered a sporadic simple group Suzuki of order 448,345,497,600.

1968-,69. Held a visiting appointment at the Institute for Advanced

Study, Princeton, $NJ$.

1970. Invited to speak at the International Congress of Mathematicians

in Nice, France.

1974. Received the Academy Prize from the Japan Academy.

1987. The conference of group theory and combinatorics for the occasion

of Suzuki’s 60th birthday was held in Kyoto, Japan.

1991. Awarded an honorary doctoral degree from the University of Kiel,

Germany.

1997. The conference of group theory and combinatorics for the occasion

of Suzuki’s 70th birthday was held in Tokyo, Japan.

Page 237

A characterization of $2E_{6}(2)$ 237

order with $|M_{0}|\geq|M|$ . Hence replacing $M$ by $M_{0}$ if necessary, we may

assume $P=F$ . In particular taking $F\leq S\in Syl_{3}(G)$ , $Z=Z(S)\leq F$ .

Let $ U=\langle Z^{M}\rangle$ , so that $Z\cong E_{3^{n}}$ for some $n$ . Then $C_{M}(U)\leq C_{M}(Z)$ ,

and $C_{M}(Z)$ is a 3-group by 7.15.5. Therefore $C_{M}(U)\leq O_{3}(M)=F$ .

Hence $|M|\leq|F|N_{n}$ , where $N_{n}$ is the maximal order of a subgroup $X$ of

odd order in $GL_{n}(3)$ with $O_{3}(X)=1$ .

By 7.15.6, $n\leq 5$ , so $|M|_{3’}$ divides 5 $\cdot 11\cdot 13$ . Indeed if 11 divides

$|M|$ then $n=5$ , so $U=J(S)$ for $S\in Syl_{3}(G)$ by 7.15.6, whereas by the

last remark in 7.15.6, 11 does not divide the order of $N_{G}(J(S))$ . So 11

does not divide the order of $M$ . Further by 7.15.4, $G$ has no subgroup

of order 13 $\cdot 5$ , so by Hall’s Theorem, (cf. 18.5 in [FGT]) $|M|_{3’}=1,5$ ,

or 13. But $|G|_{3}=3^{9}$ and $3^{9}$ . $5<10^{5}>3^{8}\cdot 13$ , so we are left with the

case $|M|=3^{9}\cdot 13$ .

By 7.15.1, if $Y$ is of order 13 in $M$ then $C_{F}(Y)=1$ and $|N_{M}(Y)|=1$

or 3. Therefore $|F|=3^{3k}$ for some $k$ and hence $F\in Syl_{3}(G)$ , contra-

dicting 7.15.1. Q.E.D.

\S 8. Groups of type ${}^{2}E_{6}(2)$ are isomorphic to ${}^{2}E_{6}(2)$

In this section we assume the hypotheses and notation of section 6.

In particular $G$ is of type $2E_{6}(2)$ , $z$ is a 2-central involution in $G$ , $H=$

$C_{G}(z)$ , etc. Further let $G_{0}=2E_{6}(2)$ and $z_{0}$ a long root involution of $G_{0}$ .

By 7.1, $G_{0}$ is of type $2E_{6}(2)$ with $z_{0}2$-central in $G_{0}$ . Let $H_{0}=C_{G_{0}}(z_{0})$ ,

$Q_{0}=O_{2}(H_{0})$ , etc.

(8.1) $\tilde{H}\cong H_{0}/\langle z_{0}\rangle$ .

Proof. First $Q_{0}\cong Q$ , so we may identify the two groups. Further

by 6.2, the representation of $H_{0}^{*}$ on $\tilde{Q}_{0}$ is quasiequivalent to that of $H^{*}$

on $\tilde{Q}$ , so $\tilde{H}\cong\tilde{H}_{0}$ by 3.1. Q.E.D.

By 8.1, $\tilde{H}_{0}\cong\tilde{H}$ , so by 7.8 there is $h\in H-C_{H}(\tilde{t})$ with $t^{h}\in E$ .

Let $k=gh$ , $ V_{3}=\langle z, t, z^{k}\rangle$ , $U_{3}=Q\cap Q^{g}\cap Q^{k}$ , $ X_{3}=\langle Q, Q^{g}, Q^{k}\rangle$ ,

$R_{3}=C_{X_{3}}(V_{3})$ ,

$S_{3}=(Q\cap Q^{g})(Q\cap Q^{k})(Q^{g}\cap Q^{k})$ ,

and $P_{3}=N_{G}(V_{3})$ . By 8.16 in [SG],

$R_{3}=C_{Q}(V_{3})C_{Q^{g}}(V_{3})C_{Q^{k}}(V_{3})=O_{2}(X_{3})$ ,

$X_{3}/R_{3}=GL(V_{3})\cong L_{3}(2)$ , $[X_{3}, U_{3}]\leq V_{3}$ , $\Phi(U_{3})=1$ , $P_{3}=X_{3}C_{H}(V_{3})$ ,

and $P_{3}/R_{3}=X_{3}/R_{3}\times C_{H}(V_{3})/R_{3}$ .

Page 238

238 M. Aschbacher

By 7.8, $C_{H}(V_{3})/R_{3}\cong A_{5}$ , so $P_{3}/R_{3}\cong L_{3}(2)\times A_{5}$ . Again by 7.8,

$m(U_{3})=6$ , so by 8.16 in [SG], $S_{3}/U_{3}$ is the sum of 4 copies of the dual $V_{3}^{*}$

of $V_{3}$ as an $X_{3}/R_{3}$ module and $R_{3}/S_{3}$ is the sum of 4 copies of $V_{3}$ as an

$X_{3}/R_{3}$ module By 7.8, $C_{H}(V_{3})$ has chief series $0<\tilde{V}<\tilde{V}_{3}<\tilde{U}_{3}<\tilde{E}$

on $\tilde{E}$ with $E/U_{3}$ the $\Omega_{4}^{-}(2)$ module and $U_{3}/V_{3}$ the $L_{2}(4)$ module for

$C_{H}(V_{3})$ . Finally by 7.8, $C_{H}(V_{3})$ has four $L_{2}(4)$ -sections and three $\Omega_{4}^{-}(2)-$

sections on $R_{3}$ . We summarize all this as:

(8.2) (1) $P_{3}/R_{3}=X_{3}/R_{3}\times C_{H}(V_{3})/R_{3}$ with $X_{3}/R_{3}\cong L_{3}(2)$ and

$C_{H}(V_{3})/R_{3}\cong A_{5}$ .

(2) $R_{3}$ has chief series

$0<V_{3}<U_{3}<S_{3}<R_{3}$

with $V_{3}$ the natural module for $X_{3}/R_{3}$ , $[X_{3}, U_{3}]\leq V_{3}$ and $U_{3}/V_{3}$ is the

$L_{2}(4)$ module for $C_{H}(V_{3})/R_{3}$ , $S_{3}/U_{3}$ is the tensor product of the dual

of $V_{3}$ as an $X_{3}/R_{3}$ -module with the $\Omega_{4}^{-}(2)$ module for $C_{H}(V_{3})/R_{3}$ , and

$R_{3}/S_{3}$ is the tensor product of $V_{3}$ as an $X_{3}/R_{3}$ -module with the $L_{2}(4)-$

module for $C_{H}(V_{3})/R_{3}$ .

(8.3) There exists $s\in z^{G}$ with $sz$ of order 3, $C_{G}(\langle s, z\rangle)\cong U_{6}(2)$ ,

and $N_{G}(\langle sz\rangle)=\langle s, z\rangle\times C_{G}(\langle s, z\rangle)$ .

Proof Let $ X_{2}=\langle Q, Q^{g}\rangle$ . Then $X_{2}\leq X_{3}$ so there is $x$ of order

3 in $X_{2}$ fused to $y\in X_{3}\cap H$ . Notice $y^{*}$ is inverted by a transvection

in $H^{*}$ as $\tilde{H}_{0}\cong\tilde{H}$ and the remark holds in $H_{0}^{*}$ since $y$ is inverted by

some conjugate $c\in Q^{g}$ of $z$ in $H_{0}$ and $c^{*}$ is a transvection in $H_{0}^{*}$ by

7.2 and 7.3.2. Therefore $C_{Q}(y)\cong D_{8}^{4}$ and $C_{H}(y)/C_{Q}(y)\langle y\rangle\cong U_{4}(2)$ .

Let $T_{y}\in Syl_{2}(C_{H}(y))$ ; then $\langle z\rangle=Z(T_{y})$ and $T_{y}$ is of order $2^{15}$ . As

$\langle z\rangle=Z(T_{y})$ , $T_{y}\in Syl_{2}(C_{G}(y))$ .

Next let $T_{x}\in Syl_{2}(C_{P_{2}}(x))$ . From the structure of $P_{2}$ described in

6.1,

$C_{P_{2}}(x)/\langle x\rangle\cong L_{3}(4)/E_{2^{9}}$

with $O_{2}(C_{P_{2}}(x))$ quasiequivalent to the Todd module for $C_{P_{2}}(x)/O_{2}(C_{P_{2}}$

$(x))\langle x\rangle$ . In particular $T_{x}$ is of order $2^{15}$ and hence as $x$ and $y$ are conju-

gate, the previous paragraph says that $T_{x}\in Syl_{2}(C_{G}(x))$ and $Z(T_{x})$ is

generated by a conjugate of $z$ . Now the hypotheses of Theorem 30.1 in

$[3T]$ are satisfied, so by that Theorem, $C_{G}(x)/\langle x\rangle\cong C_{G}(y)/\langle y\rangle\cong U_{6}(2)$ .

Next $x$ is inverted by an involution $u\in Q$ with $[C_{P_{2}}(x), u]=$

$\langle x\rangle$ , so $u$ induces an automorphism of $C_{G}(x)/\langle x\rangle\cong U_{6}(2)$ centralizing

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$|Hom(A, G)|$ (III) 473

Then we have

$X_{G}(t)$ $=$ $\sum_{n=0}^{\infty}\sum_{r=0}^{n}(-1)^{r}p^{(_{2}^{r})}s(C_{p}^{r}, G)(-t)^{n}$

$=$ $(1+t)^{-1}\sum_{r=0}^{\infty}p^{(_{2}^{r})}s(C_{p}^{r}, G)t^{r}$

$=$ $(1+t)^{-1}S_{G,f}^{E_{p}}(t)$ .

Thus the transition identity (16) gives

(22) $X_{G}(t)=\prod_{n=0}^{\infty}(1+p^{-n}t)^{-1}\cdot H_{G,f}^{E_{p}}(t)$ .

Similarly, if we view $X_{G}(t)$ and $H_{G,f}^{E_{p}}(t)$ as $p$-adic power series (18), we

have

(23) $X_{G}(t)=\prod_{n=1}^{\infty}(1+p^{n}t)\cdot H_{G,f}^{E_{p}}(t)$ .

These formula gives a transition formula between $\{h_{n}\}$ and $\{\chi_{n}’\}$ :

(24) $h_{n}=\sum_{r=0}^{n}(-1)^{r}p^{n-r}$ $\left\{\begin{array}{l}n\\r\end{array}\right\}$

$p\chi_{r}’$

.

By (21) and (24), if $p^{n}$ divides $|G|$ , then

(25) $\chi_{n}’\equiv 0$ $(mod p^{n})$ $\Leftrightarrow$ $h_{n}\equiv 0$ $(mod p^{n})$ .

The right hand side of this statement is valid by [Yo 93]. Thus we again

have Brown’s cohomological Sylow theorem ([Yo 96]):

(26) $\chi(S_{p}(G))\equiv 1$ $(mod |G|_{p})$ .

References

[St 97] R.P. Stanley, Enumerative Combinatorics, Volume I, Cambridge Uni-

versity Press, 1997.

[Su 82] M. Suzuki, Group Theory, I, II, Springer, 1982.

[Yo 92] T. Yoshida, P.Hall’s strange formula for abelian $p$-groups, Osaka

Math. J., 29 (1992), 421-431.

[Yo 93] T. Yoshida, $|(A, G)|$ , J. Algebra, 156 (1993), 125-156.

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474 T. Yoshida

[Yo 96] T. Yoshida, Classical problems in group theory (I): Enumerating sub-

groups and homomorphisms, Sugaku Expositions, 9 (1996),169-

184.

Department of Mathematics

Hokkaido University

Sapporo 060-0810, Japan

e-mail: [email protected]