数理解析研究所講究録 1214 巻 2001 年 27-36

27

On P. Hall’s Relations in Finite groups
Yugen Takegahara(竹ヶ原 裕元)

Muroran Institute of Technology(室蘭工業大学)

1. INTRODUCTION
For afinite group $H$ and for an automorphism 0of $H$ whose order divides , wedefine$n$$L_{n}(H, \theta)=\{x\in H|x\cdot x^{\theta}\cdot x^{\theta^{2}}\cdots x^{\theta^{n-1}}=1\}$,

denotes the effect of 0on . The following theorem is due to P. Hall [8,whereTheorem 1.6].
$x^{\theta}$$x$Hall’s theorem Under the notation above,$\# L_{n}(H, \theta)\equiv 0$

mod$\mathrm{g}\mathrm{c}\mathrm{d}(n, |H|)$.

We can prove directly the assertion of this theorem in the case where is aprime ,but we need to prove ageneralized assertion in an arbitrary case (Theorem 3.1).
Hall’s theorem has various applications. Especially, it is applicable to Frobenius
in Hall’s theorem, thenconjecture as below. If we take 0as the identity
the result is due to Frobenius (see, e.g., [4, \S 37]).
$n$

$p$

$\epsilon\in \mathrm{A}\mathrm{u}\mathrm{t}H$

Frobenius theorem The number of elements x of a finite group G that satisfy the.
equation $x^{n}=1$ is a multiple of$\mathrm{g}\mathrm{c}\mathrm{d}(n, |G|)$

Relating to this theorem, the following theorem was conjectured by Frobenius andwas shown to be true by Iiyori-Yamaki on the basis of the classification theorem offinite simple groups.
Theorem 1.1 (Frobenius conjecture, [9]) If
elements ofform a subgroup of $H$ .

$\# L_{n}(H, \epsilon)=\mathrm{g}\mathrm{c}\mathrm{d}(n, |H|)$, then the$L_{n}(H, \epsilon)$Hall’s theorem is useful for reducing the Frobenius conjecture to the case where $H$ isasimple group [23].
In this report, denotes aprime and denotes anonnegative integer. As for thegeneralization of Frobenius conjecture, Sylow’s theorem yields the following
$p$

$u$

28
Proposition 1.2 ([19]) Suppose that 0is an automorphism of a finite group $H$, then the elements ofwhose rder divides . If
$H$form a subgroup of .
$L_{p^{u}}(H, \theta)$$\# L_{p^{u}}(H, \theta)=\mathrm{g}\mathrm{c}\mathrm{d}(p^{u}, H)$$p^{u}$$\mathit{0}$According to [13], the exceptional pgroups are the cyclic groups if $p>2$ , andthe exceptional 2-groups are the cyclic, dihedral, generalized quaternion, and semidihedral groups; the last three types of finite 2-groups are defined by$D_{2^{\ell}}=\langle x, y|x^{2^{\ell-1}}=y^{2}=1, y^{-1}xy=x^{-1}\rangle$$Q_{2^{\ell}}=\langle x, y|x^{2^{l-2}}=y^{2}, y^{-1}xy=x^{-1}\rangle$$y|x^{2^{p-1}}=y^{2}=1$$S_{2^{p}}=(x,$,,$\ell\geq 2$,$\ell\geq 3$$y^{-1}xy=x^{-1+2^{\ell-2}}\rangle$,$\ell\geq 4$,,,

respectively. The four group is exceptional in this report, even though it is not in[13]. In the proof of Theorem 1.1, the following theorem plays an important role.
Theorem 1.3 ([13]) Let be a finite group with a Sylow subgroup of order .
is the highest power of that divides a positive integer and if
For $0<m<\ell$ , if
the number of elements of that satisfy the equation $x^{n}=1$ is not a multiple of, then $P$ is either cyclic or non-abelian exceptional.
$G$

$P$

$p$

$p^{m}$$p^{\ell}$

$n$

$p$

$G$

$x$

$\mathrm{g}\mathrm{c}\mathrm{d}(pn, |G|)$

Now, we emphasis that the following generalization of Theorem 1.3 holds.

Theorem 1.4 ([19]) Suppose that is an automorphism of a finite group $H$ whose. Let $P$ a Sylow -subgroup of $H$ , andorder divides and that divides
is notlet ’be the highest power of that divides a positive integer . If
, then is exceptional.
multiple of$\theta$$n$$\mathrm{g}\mathrm{c}\mathrm{d}(n, |H|)$

$p$

$p$

$\# L_{n}(H, \theta)$

$n$

$p$

$p^{\iota}$

$a$

$P$

$\mathrm{g}\mathrm{c}\mathrm{d}(pn, |H|)$

The proof of Theorem 1.4 is due to Murai [14] in the case where $H$ is a-group andis apower of . Remarkably, in an arbitrary case, it runs parallel with that of Hall’s
theorem (see Section 3). In this report, we will sketch out the proof of Hall’s theorem
and Theorem 1.4. Also, we will present related results with them.
$n$

$p$

2. THE CASE OF$\mu \mathrm{G}\mathrm{R}\mathrm{O}\mathrm{U}\mathrm{P}\mathrm{S}$For afinite group $H$ and afinite abelian group that acts on $H$ , let $CH$ denotethe semidirect product of and $H$ , and let $z(C, H)$ be the number of complementsof $H$ in $CH$ , i.e.,
$C$

$C$

$z(C, H)=\#\{D\leq CH|DH=CH, D\cap H=\{1\}\}$ .

Let be afinite -group, and let be an automorphism of whose order divides
is afinite cyclic group, where is apositive integer. Suppose thatfor all $x\in P$ , andgenerated by and is of order . Then acts on by$z(C, P)=\# L_{p^{u}}(P, \theta)$ , because$P$$p^{u}$$P$$\theta$$p$$C=\langle c\rangle$

$u$

$c$

$p^{u}$$C$$P$$x^{\mathrm{c}}=x^{\theta}$$z(C, P)=\#\{x\in P|(cx)^{p^{u}}=1\}=\#\{x\in P|(xc^{-1})^{p^{u}}=1\}$

29$x\cdot x^{\theta}\cdot x^{\theta^{2}}\cdots$$x^{\theta^{p^{u}-1}}=xc^{-1}xcc^{-2}xc^{2}\cdots c^{-(p^{u}-1)}xc^{p^{u}-1}c^{-p^{u}}=(xc^{-1})^{p^{u}}$for all $x\in P$ . Thus Hall’s theorem with $H=P$ and $n=p^{u}$ is equivalent to thefollowing results which is due to Asai and Yoshida [3, Proposition 3.3].
Proposition 2.1 Under the assumptions above,$z(C, P)\equiv \mathrm{O}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(p^{u}, |P|)$.

This proposition seems to have various applications. We will present some of themin Sections 4and 5.
In connection with Probenius theorem, Kulakoff proved the following [11, Satz 2].
Kulakoff’s theorem Suppose that $p>2$ and that $P$ is a finite non-cyclic groupof order . Then, for $0<m<\ell$, the number of elements of $P$ that satisfy theequation $x^{p^{m}}=1$ is a multiple of $p^{m+1}$ .
$p^{\ell}$$x$

The following generalization of this theorem relates to Hall’s theorem.
Theorem 2.2 ([2, 19]) Suppose that $p>2$ and that
. Then $|P|\geq p^{u+1}$ and $P$ is cyclic.

$\# L_{p^{u}}(P, \theta)$

is not a multiple of$\mathrm{g}\mathrm{c}\mathrm{d}(p^{u+1}, |P|)$, then the assertion is the same as that of Kulakoff’s theorem.
In this theorem, if].
This theorem is also aspecial case of [8, Theorem
Murai proved the following theorem containing the results in the case where $p=2$
and Kulakoff’s theorem, which yields Theorem 1.3.
$\theta=\epsilon$$1(\mathrm{i}\mathrm{i}\mathrm{i})$Theorem 2.3 ([13]) Suppose that $P$ is a finite group of order . For $0<m<\ell$ ,the number of elements of $P$ that satisfy the equation $x^{p^{m}}=1$ is not a multiple of$p^{m+1}$
if and only if either $P$ is a cyclic group, or else $p=2$ , $0<m<\ell-1$ , and $P$ is
non-abelian exceptional.
$p^{\ell}$$x$

The assertion with $p=2$ and $m=1$ in this theorem is also seen in [12, TheOrem 6.2(Th0mps0n) ; see also [10, pp. 52-53]. The following theorem is also due to
Murai.
$]$

Theorem 2.4 ([14]) Suppose that $p=2$ and thatis not a multiple of. Then $|P|\geq 2^{u+1}$ and $P$ is exceptional. Moreover, if $P$ is a non-cyclic, then $u=1$ , $P=D_{4}$ (the four group), and2-group of order.
$\# L_{2^{u}}(P, \theta)$$\mathrm{g}\mathrm{c}\mathrm{d}(2^{u+1}, |P|)$$2^{u+1}$$\langle\theta\rangle P=D_{8}$Murai’s proof of this theorem says that the assertion of the theorem is roughly aconsequence of Theorem 2.3 and the following theorem proved in [2].

Theorem 2.5 ([2, 19]) Suppose that $p=2$ and that $u>1$ . If
, ? is not$|P|\geq2^{u+1}$, thena multiple ofand every -invariant abelian normal
subgroup of is cyclic.
$\# L_{2^{u}}(P$$\mathrm{g}\mathrm{c}\mathrm{d}(2^{u+1}, |P|)$$P$$\theta$$)$30
In this theorem, if$\theta=\epsilon$, then we get the following corollary.

Corollary 2.6 ([2]) Suppose that $|P|=2^{\ell}$ . For 1 $<m<\ell$ , if the number of, thenelements of $P$ that satisfy the equation $x^{2^{m}}=1$ is not a multiple of$P$$P$
every abelian normal subgroup of is cyclic, and consequently, either is cyclic, orelse $1<m<\ell-1$ and $P$ is non-abelian exceptional.
$2^{m+1}$$x$

contains a
The first part of the assertion of Corollary 2.6 fails for $m=1$ , because
is not a
non-cyclic abelian normal subgroup, though the number of involutions ofmultiple of 4. The last part of the assertion of Corollary 2.6 is aconsequence of [15,Chapter 4, (4.3) and is also aspecial case of Theorem 2.3.
$D_{8}$$D_{8}$$]$

3. THE PROOF OF HALL’S THEOREM
In this section, we will make asketch of the proof of Hall’s theorem. If 0is an
automorphism of afinite group $H$ whose order divides , then$n$$x\cdot x^{\theta}\cdot x^{\theta^{2}}\cdots x^{\theta^{n-1}}=(x\theta^{-1})^{n}$for all

$x$

on

$H$

(see Section 2), and hence$L_{n}(H, \theta)=\{h\in H|(\theta h)^{n}=1\}$,

where is regarded as an element of the semidirect productit follows that$\theta$$L_{n}(H, \theta)=\{x\in H\theta H|x^{n}=1\}$

Throughout this section, letand apositive integer. Set$G$$\langle\theta\rangle H$. Since $HflH=\theta H$ ,.

be afinite group, $H$ asubgroup, $z\in C_{G}(H)$ , $y\in G$ ,$n$

$X_{n}(HyH, z)=\{x\in HyH|x^{n}=z\}$.

It seems that acertain generality is necessary for proving Hall’s theorem. We can get
Hall’s theorem as aspecial case of the following theorem which is also due to Hall [8].
Theorem 3.1 We have $\# X_{n}(HyH, z)$$\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(n, |H|)$.

Hall [8] showed more general theorem under some additional conditions (see also [23]).
However we have proved only Theorem 3.1 [19].
The following theorem contains Theorem 3.1 and ageneralization of Theorem 1.4.
Theorem 3.2 ([19]) Suppose th\‘at $n=puq$ where
Sylow -subgroup of H. Then the following conditions hold.

$\mathrm{g}\mathrm{c}\mathrm{d}(p, q)=1$

$p$

(1)

$\# X_{n}(HyH, z)\equiv \mathrm{O}$

mod (p, $|P|$ ).

and that

$P$

is

$a$

31
(2) If p divides
then y CE $N_{G}(H)$ , |P|$\mathrm{g}\mathrm{c}\mathrm{d}(\mathrm{v}\mathrm{z}, |H|)$$\ovalbox{\tt\small REJECT}$andz) is not a multiple of mod$(p^{u+1}, |P|)$ ,, and P is exceptional.

$i\ovalbox{\tt\small REJECT}$$p^{n+r}$$\# X_{n}(HyH,$

We devote the rest of this section to the sketch of the proof of Theorem 3.2. The
details of the proof will be shown in [19]. According to [19], we may assume that $H$
is apgroup and $n=p^{u}$ . For each proper subgroup $K$ of $H$ , set$X_{n}(HyH, x;K)=\{x\in HyH|x^{n}=z, H\cap H^{x}=K\}$ .

The proof of (1) is as follows. We use induction on $|H|$ . Suppose that $K$ isaproper subgroup of $H$ . For each $w\in G$ , the inductive assumption implies thatprovided $KwK\cap X_{n}(HyH, z;K)\neq\emptyset$ . Then$\# X_{n}(KwK, z)\equiv \mathrm{O}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(n, |K|)$.

$hC_{H}(K/C_{H}(K)\mathrm{I}\# X_{n}(HyH, z;K^{h})\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(|H : K|n, |H|)$

Now, if $H\neq H^{y}$ , then $H\cap H^{x}\neq H$ for all $x\in HyH$ , which, together with the fact
above, yields %Xn(HyH, ). On the other hand, if $H=H^{y}$ , thenthe assertion (1) follows from Proposition 2.1.
The proof of (2) runs parallel with that of (1). Suppose that $K$ is aproper
subgroup of $H$ . For each $w\in G$ , the assertion (1) implies that $\# X_{n}(KwK, z)\equiv$
provided $KwK\cap X_{n}(HyH, z;K)\neq\emptyset$ . Then$z$$\equiv \mathrm{O}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(n, |H|)$$0\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(n, |K|)$$\sum_{hC_{H}(K)\in H/C_{H}(K)}\# X_{n}(HyH, z;K^{h})\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(pn, |H|)$.

If $H\neq H^{y}$ , then $H\cap H^{x}\neq H$ for all $x\in HyH$ , which, together with the fact above,
yields %Xn(HyH, ). Now, if $\# X_{n}(HyH, z)$ is not amultiple of, then $H=H^{y}$ and, by Theorems 2.2 and 2.4, $P$ is exceptional.
$z$$\equiv \mathrm{O}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(pn, |H|)$$\mathrm{m}\mathrm{o}\mathrm{d} (pn, |P|)$

4. FROBENIUS NUMBERS
Let $H$ be afinite group and $C$ afinite abelian group that acts on $H$ . We consider
the condition$\mathrm{I}(C, H)$:$z(C, H)\equiv \mathrm{O}\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(|C|, |H|)$.

Hall’s theorem states that the condition
holds provided $C$ is acyclic group(see Section 3). The following conjecture was introduced in [3].
$\mathrm{I}(C, H)$

Conjecture IIf $H$ and $C$ are -groups, then the condition$\mathrm{I}(C, H)$

holds.

holds in some special cases as below. Suppose that $H$ andare pgroups. The following proposition plays an important role in the proof of
Theorem 4.4 below.
The condition$C$$\mathrm{I}(C, H)$

32
Proposition 4.1 ([1, 2]) If $H$ is abelian, then the condition$\mathrm{I}(C, H)$

holds.

The following theorem is ageneralization of Proposition 2.1.
Theorem 4.2 ([1, 3]) If is the direct product of a cyclic -group and an elements
tary abelian -group, then the condition
holds.
$C$$p$$\mathrm{I}(C, H)$$p$Now, we state atheorem, that closely relates to Theorem 1.4.

is the direct product of a cyclic
Theorem 4.3 ([2]) Suppose that is odd. If-group and a cyclic -group of order at most , then the condition
holds.
$C$$p$$p^{2}$$p$$p$$\mathrm{I}(C, H)$

Akey result to this theorem yields Theorem 2.2. Theorem 4.3 comes out of the facts
below. Define $C_{2}(G)=[G, G]$ and $C_{\dot{l}}(G)=[C_{\dot{l}}-1(G), G]$ for $i\geq 3$ . The following
theorem is due to Hall (see also [15, Chapter 4, \S 3]).
Theorem 4.4 ([7]) For elements and, $2\leq i\leq n$ , such thatexist .
$x$$\mathrm{q}$

$y$

of

$G$

and for a positive integer , there$n$$\in C_{\dot{l}}(G)$$x^{n}y^{n}=(xy)^{n}c_{2^{2}}^{e}\cdots c_{n}^{e_{n}}$

,

where
$e:=(\begin{array}{l}ni\end{array})$$= \frac{n(n-1)\cdots(n-i+1)}{i!}$.

We actually use the following corollary to Theorem 4.4.
Corollary 4.5 ([2]) Assume that $\exp C_{i}(G)\leq p^{u-:+2}$ for each with $2\leq i\leq u+2$ .
, then $\Omega_{u}(G)=\{x\in G|x^{p^{u}}=1\}$ .
If either $p>2$ or$i$$\exp C_{2}(G)\leq p^{\mathrm{u}-1}$

Another useful result for Theorem 4.3 is atheorem which is also due to Hall (see,.e.g.,[15, Chapter 4, Theorem 4.22]):
Theorem 4.6 If $p>2$ and if every characteristic abelian subgroup of a finite p-groupis cyclic, then is the central product of a cyclic group and $E$ , where is either{1} or an extraspecial -group of exponent .
$P$

$P$

$E$

$p$

$p$

denote the number of hom0Let and be finite groups, and letmorphisms from to . Such anumber is called the Probenius number of with. As
respect to , because, if is acyclic group of order ,ageneralization of Probenius theorem, Yoshida proved the following [22].
$G$

$A$

$|\mathrm{H}\mathrm{o}\mathrm{m}(A, G)|$

$A$

$A$

Yoshida’s theorem

$G$

$G$

$A$

If A is abelian,$n$$|\mathrm{H}\mathrm{o}\mathrm{m}(A, G)|\equiv \mathrm{O}$$|\mathrm{H}\mathrm{o}\mathrm{m}(A, G)|=mathrm{L}\mathrm{n}(\mathrm{G}, \epsilon)$mod$\mathrm{g}\mathrm{c}\mathrm{d}(|A|, |G|)$.

We consider the condition$\mathrm{H}(A,$G) :$|\mathrm{H}\mathrm{o}\mathrm{m}(A, G)|\equiv \mathrm{O}$mod$\mathrm{g}\mathrm{c}\mathrm{d}(|A/A’|, |G|)$,

where $A’$ denotes the commutator subgroup of A. The following conjecture was also
introduced in [3]33Conjecture

For any finite groups$\mathrm{H}$

$A$

and

$G$

, the condition$\mathrm{H}(A, G)$

holds.

Using Proposition 2.1, Asai and Yoshida proved the following.
Theorem 4.7 ([3]) If $A/A’$ is cyclic, then the condition

holds.

$\mathrm{H}(A, G)$

In connection with Theorems 4.2 and 4.3, the following theorems are known.
Theorem 4.8 ([1, 3]) If every Sylow subgroup of $A/A’$ is the direct product of
cyclic group and an elementary abelian group, then the condition
holds.

$a$$\mathrm{H}(A, G)$

Theorem 4.9 ([2]) If is of odd order and if a Sylow subgroup of $A/A’$ is the
direct product of a cyclic group and a cyclic group of order at most for any prime
dividing $|A/A’|$ , then the condition
holds.
$A$$p$$p^{2}$$\mathrm{H}(A, G)$$p$

Conjectures and Iare not still solved. As aconnection of these conjecture, Asaiand Yoshida proved the following.
$\mathrm{H}$Theorem 4.10 ([3]) If Conjecture Iis true, so is Conjecture H.

5. THE NUMBER OF SUBGROUPS OF FINITE GROUPS
Throughout this section,
is afinite group, and $m_{A}(d)$ denotes the number ofsubgroups of index in . Proposition 2.1 is applicable to the following theorem.
$A$

$A$

$d$

Theorem 5.1 ([18]) Let
be an integer withbe the exponent of a Sylow. Then$p^{\lambda_{1}}$$1\leq i\leq\lambda_{1}$$p$subgroup of $A/A’$ . Let$i$$m_{A}(qp^{i-1})\equiv m_{A}(qp^{i})\mathrm{m}\mathrm{o}\mathrm{d} p^{i}$for any positive integer such that$q$$\mathrm{g}\mathrm{c}\mathrm{d}(p, q)=1$.

, where is apositive integer, if the following condiWe say that admits
anypositive integer such thattions hold for:$A$$\mathrm{C}(p^{s})$

$s$

$q$

$\mathrm{g}\mathrm{c}\mathrm{d}(p, q)=1$(1) For any integer with $1\leq i\leq[(s+1)/2]$ , where $[(s+1)/2]$ is the greatest
integer $\leq(s+1)/2$ ,$i$$m_{A}(qp^{i-1})\equiv m_{A}(qp^{i})\mathrm{m}\mathrm{o}\mathrm{d} p^{i}$.

(2) Moreover,$m_{A}(qp^{[(s+1)/2]})\equiv m_{A}(qp^{[(s+1)/2]+1})\mathrm{m}\mathrm{o}\mathrm{d} p^{[s/2]}$.

Also, is said to admits
if the preceding conditions (1) and (2) hold in thecase where $q=1$ . We get the following corollary to Theorem 5.1.
$A$$\mathrm{C}\mathrm{P}(p^{s})$Corollary 5.2 Under the assumptions of Theorem 5.1, if $\lambda_{1}\geq[(s+1)/2]+1$ , then.
admits$A$$\mathrm{C}(p^{s})$34
The preceding conditions appeared in the following proposition which due to Butler.

Proposition 5.3 ([5]) Any finite abelian -group$p$Corollary 5.4 If is abelian, thenof that divides $|A|$ .
$A$

$A$

admits$P$$\mathrm{C}(|A|_{p})$admits$\mathrm{C}\mathrm{P}(|P|)$.

, $where|A|_{p}$ is the highest power$p$

The following proposition is due to Hall.
Proposition 5.5 ([7]) Let$P$

denote a finite -group with $p^{s}=|P:\Phi(P)|$ . Then$p$$m_{P}(p^{\dot{*}})\equiv m_{P/\Phi(P)}(p^{:})$

for any integer with $0\leq i\leq s+1$ , where$i$rood $p^{s-:+1}$$\Phi(P)$denotes the Frattini subgroup of $P$ .

Combining this proposition with Proposition 5.3, we have the following.
Corollary 5.6 Under the assumptions of Proposition 5.5, $P$ admits$\mathrm{C}(p^{s})$.

In [21], Wohlfahrt states that
$1+ \sum_{n=1}^{\infty}\frac{|\mathrm{H}\mathrm{o}\mathrm{m}(A,S_{n})|}{n!}X^{n}=\exp(\sum_{d=1}^{\infty}\frac{m_{A}(d)}{d}X^{d})$,is the symmetric group of degree . Hence the following important prop0where
sition holds.
$n$$S_{n}$

Proposition 5.7 ([17]) If$A$admits$\mathrm{C}(p^{s})$, then$|\mathrm{H}\mathrm{o}\mathrm{m}(A, S_{n})|\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(p^{s}, n!)$Now, in connection with Conjectures$\mathrm{H}$.

and , we present the following conjecture.

Conjecture J Any finite group A admits$\mathrm{I}$$\mathrm{C}(|A/A’|_{p})$.

For this conjecture, the case where A is afinite -group is essential because of thefollowing theorem.
be a normal subgroup of such that the factor group
Theorem 5.8 ([20]) Let$A/B$ is an abelian group of order. Assume that every subgroup of admits.
. Then admits
$A$

$B$

$D$

$p^{s}$$\mathrm{C}\mathrm{P}(|D:D\cap B|)$

$A$

$A$

$\mathrm{C}(p^{s})$The following theorem results from Corollary 5.6 and corresponds Theorem 4.8.
Theorem 5.9 ([20]) Let $B$ be a normal subgroup of such that $A/B$ is the direct product of a cyclic -group and an elementary abelian -group. Then admits
$A$

$p$

$\mathrm{C}(|A/B|)$

.

$p$

$A$

35

.

0 and
Apartition A (Ai, , \ldots ) hs, where
the type of afinite abelian pgroup isomorphic to the direct product$\ovalbox{\tt\small REJECT} \mathrm{X}$$\ovalbox{\tt\small REJECT}$$\mathrm{A}_{2}$$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT})_{2}\ovalbox{\tt\small REJECT}$\cdots$\ovalbox{\tt\small REJECT} \mathrm{p}_{\mathrm{A}_{\ovalbox{\tt\small REJECT}}}\ovalbox{\tt\small REJECT} s$$\ovalbox{\tt\small REJECT}$, is called$C_{p^{\lambda_{1}}}\cross C_{p^{\lambda_{2}}}\cross\cdots$of cyclic pgroups of order$p^{\lambda_{1}}$,$p^{\lambda_{2}}$,\ldots .

We get the following theorems.

Theorem 5.10 ([20]) Let $B$ be a normal subgroup of $P$ such that $P/B$ is of type
, and,
. Assume that $\lambda_{1}\geq[(s+1)/2]$ . If $p>2$ , A2
A.
then $P$ admits$\leq 2$$=(\lambda_{1}, \lambda_{2}, \ldots)\vdash s$$\lambda_{3}\leq 1$$\mathrm{C}\mathrm{P}(p^{s})$Theorem 5.11 ([20]) Let $B$ be a normal subgroup of such that $A/B$ is the direct
product of a cyclic -group and a cyclic -group of order at most . Then admits.
$A$$p^{2}$

$p$

$p$

$A$

$\mathrm{C}(|A/B|)$

Combining this theorem with Proposition 5.7, we have the following.
Corollary 5.12 ([20]) Under the assumptions of Theorem 5.11,.

$|\mathrm{H}\mathrm{o}\mathrm{m}(A, S_{n})|\equiv 0\mathrm{m}\mathrm{o}\mathrm{d} \mathrm{g}\mathrm{c}\mathrm{d}(|A/B|, n!)$This result corresponds to Theorems 4.3 and 4.9. However, the assertion of Corollary 5.12 is true for every prime . So Theorems 4.3 and 4.9 seem to be true even if$p=2$ .
$p$

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