数理解析研究所講究録
991 巻 1997 年 171-179

171

The number of homomorphisms from a finite
abelian group to a finite group
Yugen Takegahara

竹ケ原

裕元

Muroran Institute of Technology

室蘭工業大学
1

Generating functions

Let be a finitely generated group, and let
be the set of all subgroups $B$ suchthat the factor groups $A/B$ are finite groups. Let
be a finite group, and letbethe symmetric group on letters. For the wreath product $GlS_{n}$ , put$A$$\mathcal{F}_{A}$$G$$S_{n}$$n$$|\mathrm{H}\mathrm{o}\mathrm{m}(A, G\iota s_{n})|$$h_{n}(A;G)=\{$For each subgroup $B$ of

Theorem 1. 1 Let

$A$

$A$

, let

1

$h(B, G)=|\mathrm{H}\mathrm{o}\mathrm{m}(B, G)|$

if $n\geq 1$ ,if $n=0$ .

.

be a finitely generated group, and let $G$ be a finite group. Then,$1+ \sum_{n=1}^{\infty}\frac{h_{n}(A,G)}{|G|^{n}n!}.$. $X= \exp n(\sum_{B\in F_{A}}\frac{h(B,G)}{|G|\cdot|A.B|}.\cdot X|A:B|)$ .

This theorem is a consequence of the following fact that is proved in [6] when
where is the identity element.

$G=\{\epsilon\}$$\epsilon$

Let

$A$

be a finitely generated group, and let$\frac{h_{n}(A,G)}{|G|^{n}(n-1)!}.=.\sum_{\mathit{1}^{A}\cdot B|\leq n}\frac{h(B,G)}{|G|}$

where the summation isover. all subgroups

.

$B$

$G$

be a finite group. Then,$\frac{h_{n-|A.B|(A,G)}}{|G|^{n-|\cdot|}A.B(n-|A.B|)!}.\cdot.$

of

$A$

’

such that $|A:B|$ are less than .
$n$Sketch of proof. Let
denote the direct product of copies of . Thennaturally acts on, and the wreath product $GlS_{n}$ is the semidirect product ofand . Define the action of $GlS_{n}$ on the set $G\cross[n]$ where $[n]=\{1,2, \ldots, n\}$ by$G^{(n)}$

$n$

$G$

$G^{(n)}$$G$$(g_{1},g_{2}, \ldots,g_{n})\sigma.(g, i)=(g_{\sigma()g}i, \sigma(i))$$(g_{1},g_{2}, \ldots,g_{n})\in G^{(n)}$,$\in G\cross[n]$$\sigma\in S_{n}$,,$(g,i)\in G\cross[n]$.

$S_{n}$$S_{n}$

172

This action is semiregular, and hence, $GlS_{n}$ is embedded in the symmetric group. Let us define the following:
on $|G|n$ letters. Let$\varphi\in \mathrm{H}\mathrm{o}\mathrm{m}(A, clS_{n})$$S_{|G|n}$

is a positive integer such that $r\leq n$ ;$B$
is a subgroup of such that $|A$ : $B|=r$ ;;is a mapping from the set of all cosets of $B$ in to $[n]$ such that $\pi(B)=1$ ,;whereis an element of

$r$

$A$

$\kappa\in \mathrm{H}\mathrm{o}\mathrm{m}(B, c)$

$A$

$\pi$

$G^{(r)}$$(y_{1}, y_{2}, \ldots , y_{r})$$\psi\in \mathrm{H}_{0}\mathrm{m}(A, G\iota S_{n}-r)$Let$\mu_{n}$$y_{1}=\epsilon$.

be the homomorphism from $GlS_{n}$ to$S_{n}$

defined by: $c\iota sn\ni(g1,g_{2,\ldots,g)\sigma}narrow\sigma\in S_{n}$ .

$\mu n$Let be the subset of consisting of all elements ofand let $r=|A:B|$ . Define the homomorphism from
$A$

$B$

$A$

$a$

$\kappa$$B$such thatto by$\mu_{n^{\circ}}\varphi(a)(1)=1$,$G$$\varphi(b).(\epsilon, 1)=(\kappa(b), 1)$

where $b\in B$ . Let$a_{1}B,$ $a_{2}B,$$\ldots,$$a_{r}B$, where$a_{1}=\epsilon$, be all cosets of$A=a_{1}B\cup a_{2}B\cup\cdots\cup a_{r}B$

Define an element $(y_{1}, y2, \ldots, yr)$ of$\{a_{1}B, a_{2}B, \ldots, a_{r}B\}$ toby$G^{(r)}$

in

$B$

$A$

, i.e.,.

and a mapping$\pi$

from the set of all cosets$[n]$$\varphi(a_{j}).(\epsilon, 1)=(yj,\pi(ajB))$for each . In particular,
be the subset of $[n]$ such that$j$$y_{1}=\epsilon$. Let$\{k_{1}, k_{2}, \ldots, k_{n-r}\}$where $k_{1}<k_{2}<\cdots<k_{n-r}$$[n]=\{\pi(a_{1}B), \pi(a_{2}B), \ldots,\pi(a_{r}B)\}\cup\{k_{1}, k_{2}, \ldots, k_{n-r}\}$We define a homomorphism l ノ from
$\nu$

Put

:

$\psi=\nu\circ\varphi$$\varphi(A)$.

to $Gls_{n-r}$ by$\varphi(A)\ni(g_{1},g2, \ldots,gn)\sigmaarrow(g_{k_{1}},g_{k^{\wedge}}2’\ldots,gkn-r)\sigma\in GlS_{n}-r$. Thus, we get$r,$ $B,$$\kappa,$$\pi,$$(y_{1},y2, \ldots, yr)$.

and . Then, the correspondence$\psi$$\varphiarrow\{r, B, \kappa, \pi, (y1, y2, \ldots, yr), \psi\}$

is a bijection. Therefore, we have that$h_{n}(A;G)=.

\sum h(B|A\cdot B|\leq n’ G)\frac{(n-1)!}{(n-|A\cdot B|)!}.|G|^{1}A:B|-1h_{n}-|A:B|(A;G)$

.

173

The result follows from this.

$\square$Suppose thatis a finitely generated abelian group. We denote bytheintersection of all maximal subgroups of index 2 in . The wreath product $GlA_{n}$ isa subgroup of $GlS_{n}$ . For each subgroup of containing
, let
$A$

$\Phi_{2}(A)$

$A$

$C$

$A$

$\Phi_{2}(A)$$h_{n}^{+}(A : C;G)=^{\beta}\{\varphi\in \mathrm{H}\mathrm{o}\mathrm{m}(A, GlS_{n})|\varphi(C)\subset GlA_{n}\}$.

In particular, $h_{n}^{+}(A:\Phi 2(A);G)=h_{n}(A;c)$ , and $h_{n}^{+}(A:A, G)=h(A;G[A_{n})$ .
Define the subgroup $Al_{n}(G)$ of $GlS_{n}$ by$Al_{n}(G)=\{(g_{1},g2, \ldots,g_{n})\sigma\in c[s_{n}|_{0}\mathrm{r}\mathrm{d}_{2}(|g1g_{2gn}\ldots|)<\mathrm{o}\mathrm{r}\mathrm{d}_{2}(|G|)\}$

,

)
where
is the largest integer such that
divides for each nonzero$|G|$
integer . If 2 divides
and if a Sylow 2-subgroup of is not a cyclic group, then$Al_{n}(G)=GlS_{n}$ . If 2 does not divide $|G|$ , let $Al_{n}(G)=GlA_{n}$ . As was mentioned
earlier, $GlS_{n}$ is embedded in. Then, $Al_{n}(G)$ is identified with $A_{|G|n}\cap GlS_{n}$ .
$W(D_{n})$$Al_{n}(C_{2})=W(D_{n})$
Thus,
where
is the Weyl group. For each subgroup $C$ of, letcontaining$2^{\mathrm{o}\mathrm{r}\mathrm{d}_{2(x}}$$x$$\mathrm{o}\mathrm{r}\mathrm{d}_{2}(x)$

$G$

$x$

$S_{|G|n}$$A$$\Phi_{2}(A)$$h_{n}^{-}(A:C;G)=\#\{\varphi\in \mathrm{H}\mathrm{o}\mathrm{m}(A, c\iota Sn)|\varphi(C)\subset Al_{n}(G)\}$.

In particular, $h_{n}^{+}(A:\Phi 2(A);G)=h_{n}(A;c)$ , and $h_{n}^{+}(A:A;^{c_{2})}=h(A;W(D_{n}))$ .
We shall present the generating functions for $h_{n}^{+}(A:c;G)$ and $h_{n}^{-}(A:c;G)$ . For
each subgroup $D$ of containing
and for each, define$A$$\Phi_{2}(A)$if a Sylow 2-subgroup of $A/B$ is a cyclic groupthat is notand that $A=DB$ ,otherwise,$-h(B, G)$$f_{A}^{D}(B, G)=\{$$\{\epsilon\}$$h(B, G)$

$h_{A}^{D}(B, G)=\#${$B\in \mathcal{F}_{A}$$\kappa\in \mathrm{H}\mathrm{o}\mathrm{m}(B,$$G)|\mathrm{o}\mathrm{r}\mathrm{d}_{2}(|\kappa(a^{||}A:B)|)=\mathrm{o}\mathrm{r}\mathrm{d}_{2}(|G|)$$h(B, G)-2hA(DB, G)$$g_{A}^{D}(B, G)=\{$

for some $a\in D$ },if either a Sylow 2-subgroup of $|G|$ isa cyclic group that is notorotherwise.
$\{\epsilon\}$$h(B, G)$For each subgroup $D$ of$A$

containing$\Phi_{2}(A)$, let$E_{A}^{D}(+, G;x)$$= \exp(_{B\in}\sum_{FA}\frac{f_{A}^{D}.(B,G)}{|G||A.B|}.\cdot x|A:B|)$,$E_{A}^{D}(-, G;x)$$= \exp(_{B\in}\sum_{\mathcal{F}A}\frac{g_{A}^{D}.(B,G)}{|G||A.B|}.\cdot x|A:B|)$.

It follows from definition that$f_{A}^{\Phi_{2}(}A)(B, c)=g_{A}(\Phi_{2}A)(B, G)=h(B, c)$where$B\in \mathcal{F}_{A}$. Put $E_{A}(G;x)=E_{A}^{\Phi_{2}()}A(+, G;x)=E_{A}^{\Phi_{2}()}A(-, G;x)$ .

$B\in \mathcal{I}_{A}^{D}$

,

174

Theorem 1. 2 ([4]) Let be a finitely generated abelian group and $C$ a subgroup of
the set of all subgroups $D$ of $C$ that contain. We denote bycontaining
as a subgroup of index 2. Let be a finite group. Then,
$A$

$A$

$\Phi_{2}(A)$$\mathcal{K}_{A}^{C}$$G$$\Phi_{2}(A)$$1+ \sum_{n=1}^{\infty}\frac{h_{n}^{+}(A.C,G)}{|G|^{n}n!}..$.

$X^{n}=.

.

\frac{1}{|C\cdot\Phi_{2}(A)|}\{E_{A}(G;x)+\sum_{D\in \mathcal{K}^{C}A}E_{A}^{D}(+, G;X)\}$

Suppose that a Sylow 2-subgroup of$G$

is a cyclic group that is not$\{\epsilon\}$. Then.

$1+ \sum_{n=1}^{\infty}\frac{h_{n}^{-}(A\cdot C\cdot c)}{|G|^{n}n!}.,.x^{n}=.\frac{1}{|C\cdot\Phi_{2}(A)|}\{E_{A}(G;x)+\sum D\in \mathcal{K}^{C}AED(A-, c;X)\}$

Remark. In the paper [2], N. Chigira proved this theorem in the case where
cyclic group.
Example.

$A$

Let $A=C_{2}^{(t)}$ . For each subgroup $B$ of $A,$ $h(B, C_{2})=|B|$ , and hence.

$E_{A}(C_{2}; X)= \exp(\sum_{B\leq A}\frac{|B|}{2|A.B|}.\cdot X|A:B|)$

For each cyclic subgroup $D$ of order 2 in$-|B|$$f_{A}^{D}(B, C2)$

$=\{$
$|B|$

$2^{t-1}$$h_{A}^{D}(B, C_{2})$

$=\{$
$0$

$A$

and for each subgroup $B$ of$E_{A}^{D}($-,$C_{2}$; $x)$

if $B=A$ ,otherwise.

$1+ \sum_{n=1}^{\infty}\frac{h(C_{2}^{()}tW(D_{n}))}{2^{n}n!},$.

$X^{n}$$A$,$=E_{A}(C_{2}; x)\exp(-2^{2t3}-X^{2})$$=E_{A}(c_{2}; X)\exp(-2^{t-1}x)$

Since the number of cyclic subgroups of order 2 in
$1+ \sum_{n=1}^{\infty}\frac{h(C_{2}^{(t)},c_{2}1A)n}{n!}\cdot X^{n}$,

if $|B|=2^{t-1}$ and if $A=DB$ ,otherwise,

Therefore, for any cyclic subgroup $D$ of order 2 in
$E_{A}^{D}(+, \mathit{0}_{2}; X)$

$A$

$A$

,

.

is $2^{t}-1$ , it follows that$= \frac{1}{2^{t}}E_{C_{2}^{()}}t(o2;X)\{1+(2^{t}-1)\exp(-2^{2t-}\mathrm{s}x^{2})\}$$= \frac{1}{2^{t}}E_{C_{2}^{(t)}}(C_{2}; x)\{1+(2^{t}-1)\exp(-2^{t-1}x)\}$

.

,

is a

175

2

The number of homomorphisms from a cyclic p–group to a symmetric

group
Let

$A$

be a finite abelian group. It follows from [7] that$|\mathrm{H}\mathrm{o}\mathrm{m}(A, G)|\underline{=}0$

for any finite group

$G$

mod

$\mathrm{g}\mathrm{c}\mathrm{d}(|A|, |G|)$. This result is a generalization of the theorem of Frobenius:$\#\{x\in G|x^{d}=1\}\equiv 0$

mod$\mathrm{g}\mathrm{c}\mathrm{d}(d, |G|)$Let. Let us study$m_{A}(d)$denote by
the number of subgroups of index$h_{n}(A)=|\mathrm{H}\mathrm{o}\mathrm{m}(A, S_{n})|$$\mathrm{o}\mathrm{r}\mathrm{d}_{p}(h_{n}(A))$$d$.

where is a prime integer. We
in . Put$p$$A$$E_{A}(x)= \exp(^{1}\sum_{d=1}^{A|}\frac{m_{A}(d)}{d}\cdot Xd)$.

Then, it follows from Theorem 1.1 that
$E_{A}(x)=1+ \sum_{n=1}\frac{h_{n}(A)}{n!}\cdot X^{n}\infty$.

As a special case, we obtain$E_{C_{p^{l}}}(X)= \exp(_{k=0}\sum^{l}\frac{1}{p^{k}}\cdot x^{p^{k}}\mathrm{I}$where$C_{p^{l}}$is a cyclic p–group of order$p^{l}$,. The p–adic power series$E_{p}(x)= \exp(_{k=0}\sum^{\infty}\frac{1}{p^{k}}\cdot x)pk$$E_{p}(x)$is defined by,which is called the Artin-Hasse exponential. It is well known that,
where
is the ring of p–adic integers. Put $E_{p}(x)=\Sigma_{n=0^{a}n}\infty Xn$ . Then, this fact yields
thatfor any . If $n<p^{l+1}$ , then $a_{n}=h_{n}(C_{p^{\iota)}}/n!$ and then$E_{p}(x)\in \mathrm{Z}_{p}[[x]]$$\mathrm{Z}_{p}$$\mathrm{o}\mathrm{r}\mathrm{d}_{p}(a_{n})\geq 0$$n$,$\mathrm{o}\mathrm{r}\mathrm{d}_{p}(hn(cl)\mathrm{P})\geq \mathrm{o}\mathrm{r}\mathrm{d}_{p}(n!)$

where$\mathrm{o}\mathrm{r}\mathrm{d}_{p}(n!)=\sum_{j=1}^{\infty}[\frac{n}{p},]$.

Furthermore, we have the following.
Theorem 2. 1 ([5]) For each positive integer ,$n$$\mathrm{o}\mathrm{r}\mathrm{d}(\mathrm{P}h_{n}(cp\iota))\geq\sum j=1l[\frac{n}{p^{\check{J}}}]-l[\frac{n}{p^{l+1}}]$and equality holds if $n\equiv 0$ mod$p^{l+1}$

.

,

176

To prove this theorem, we use the decomposition:$E_{C_{p^{l+}}}(1x)= \exp(\frac{1}{p^{l+1}}\cdot x^{p^{l+1}})Ec_{p}t(x)$.

Example. For each positive integer ,$n$

mod 4,$[ \frac{n}{2}]-[\frac{n}{4}]+1$

if$[ \frac{n}{2}]-[\frac{n}{4}]$otherwise.

$n\equiv 3$$\mathrm{o}\mathrm{r}\mathrm{d}_{2}(hn(C_{2}))=\{$

3

The riumber of subgroups of a finite abelian

Let $P$ be a finite abelian p–group. The partition$l\succ$group$\lambda=(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{r}, \ldots)$

where$\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{r}\geq\lambda r+1=\lambda_{r}+2=\ldots=0$

is called the type of $P$ if $P$ is isomorphic to the direct product of cyclic groups$C_{p^{\lambda_{1}}}\cross C_{p^{\lambda}2}\cross\cdots \mathrm{x}c_{p}\lambda_{r}$.

We write
if is the type of a finite abelian -group of order . For the partition, letdenote the number of subgroups of order in a finite abelian $r$ groupof type , which is a polynomial in with nonnegative coefficients and depends only
on and . It is well known thatif
. It follows from [3]that for each partition , if
, then$|\lambda|=s$$\lambda$$\lambda$$p^{s}$$I\succ$$p^{i}$$\alpha_{\lambda}(i;p)$$\lambda$$p$$\alpha_{\lambda}(i;p)=\alpha_{\lambda}(s-i;p)$$i$$\lambda$$\lambda$$|\lambda|=s$$|\lambda|=s$$\alpha_{\lambda}(i;p)-\alpha_{\lambda}(i-1;p)=p\alpha i\hat{\lambda}(i;p)-p^{s}-i+1(_{S}\alpha-i+1;p\hat{\lambda})$where$\hat{\lambda}=(\lambda_{2}, \ldots, \lambda_{r}, \ldots)$. Using this fact, we have the following theorem in [1]:Let be a partition. Let
nonnegative coefficients; moreover$\lambda$,$|\lambda|=s$

and $t=s-\lambda_{1}$ . Then$\alpha_{\lambda}(i;p)\equiv\alpha_{\lambda}(i-1;p)+p^{i}$$\alpha_{\lambda}(i;p)=\alpha_{\lambda}(i-1;p)$mod$p^{i+1}$

if $0 \leq i\leq\min\{t,$if$t<i \leq[\frac{s}{2}]$Example. Let $\lambda=(1,1,1,1,1,1)$ that is the type ofThen, we get the coefficients
as follows.
$a_{i,j}$$\alpha_{\lambda}(i;p)-\alpha_{\lambda}(i-1;p)$$C_{p}^{(6)}$. Let$[ \frac{s}{2}]\}$

has

,

.
$\alpha_{\lambda}(i;p)=\sum_{j=0}a_{i,j}\oint$

.

177

4

A decomposition of $E_{P}(x)$Let be a finite abelian pgroup of type
Then $\alpha_{\lambda}(s-k;p)=m_{P}(p^{k})$ , and$P$$\lambda=$$(\lambda_{1}, \lambda_{2}, \ldots , \lambda_{r}, \ldots)$$E_{P}(x)= \exp(\sum_{k=0}^{s}\frac{\alpha_{\lambda}(s-k,p)}{p^{k}}.X)p^{k}$Let$\alpha_{\lambda}(i;p)=\Sigma_{j}a_{i_{d}}p^{t}$. Define the integers$l(\lambda)$

and$m(\lambda)$$= \max\{\lambda_{1},$ $[ \frac{s+1}{2}]\}$$l(\lambda)$$=s-l(\lambda)$$m(\lambda)$

where $|\lambda|=s$ .

.
by

,

.

To simplify the notation, write $l=l(\lambda)$ and $m=m(\lambda)$ .

Definition 4. 1 For each pair $(v,u)$ of nonnegative integers$b_{u,v}-bu-1,v$$c_{u,v}=\{$$a_{u,v}-a_{u-1,v}$$a_{v,u}$$\mathit{8}uch$

that

let

$v\leq s_{f}$if $0\leq v\leq m$ and if $0\leq u\leq s-v$ ,if $m<v\leq s$ and if $0\leq u\leq s-v$ ,if $s-v<u$where $b_{u,v}=a_{u,v}-a_{u-1,-}v1$ .

We have a decomposition of the series $E_{P}(x)$ as follows.

Theorem 4. 1 ([5]) For any

$u$

and

$v,$

$c_{u,v}\geq 0$, and$E_{P}(x)=$$F_{P}(x) \cdot\prod_{v=0u=S}\prod_{-v+}^{\infty}\mathrm{e}s1\mathrm{x}\mathrm{p}(px)^{\mathrm{C}_{u,v}}u+v-sp^{S}-v$$F_{P}(x)=$$\{\prod_{v=0}^{m}\prod_{u=0}^{v}ECuc_{\mathrm{p}}m-v(\mathrm{x}x^{p})cu.v\}\mathrm{p}^{t_{-}}v$.

,$\{_{v=m+}\prod_{1u=}^{s}\prod^{v}E_{C}\mathrm{p}S-0s-u-v(X^{p^{v}})^{c_{\mathrm{u}}},v\}$

.

178

In the proof of this theorem, we use the preceding results that relate to the numberof subgroups.
Example. Let$P=C_{\mathrm{p}}^{(6)}$. Then $l=m=3$ , and.

$F_{P}(x)=E_{c_{\mathrm{p}}}3\cross c_{p^{3}}(x)E_{C_{\mathrm{p}^{2}}}\mathrm{x}C_{p}(X^{p^{2}})E_{C_{p}}\mathrm{x}cp(x^{p})E_{C}(px2p^{4})\exp(X^{\mathrm{p}^{4}})\exp(X)p^{5}$

Remark. Let be an integer, and let $m$ be an integer such that $l\geq m$ . Then,$l$$\alpha_{(l,m})(i;p)=\{$$1+p+\cdots+p^{i}$$0\leq i<m$ ,$1+p+\cdots+p^{m}$$m\leq i\leq l$$1+p+\cdots+p^{l-}+mi$$l<i\leq l+m$ .

,Using this fact, we have that
$E_{C_{p^{t^{\mathrm{X}}}}C\mathrm{p}^{m}}(X)=E_{C\iota,p}+m(x)E_{c_{\mathrm{p}}m-}(l+2X)\cdots E_{c_{p^{l-m}}}(x)$5.

The number of homomorphisms from a finite abelian group to a symmetric group

Using Theorem 4.1, we have the following.
Theorem 5. 1 ([5]) Let-subgroup of is
$A$

$p$

$A$

be a finite abelian group such that the type of a Sylow
where $|\lambda|=s$ . Let$\lambda=(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{r}, \ldots)$$l( \lambda)=\max\{\lambda_{1},$ $[ \frac{s+1}{2}]\}$

Then for each positive integer.

$n_{\mathrm{Z}}$$\mathrm{o}\mathrm{r}\mathrm{d}_{p}(h_{n}(A))\geq\sum_{1j=}^{l()}\lambda[\frac{n}{p?}]-(2l-S)[\frac{n}{p^{l(\lambda)+1}}]$and the equality holds if $n\equiv 0$ mod$2l(\lambda)=s\geq 2$

integer ,$p^{l(\lambda)+1}$,, except for the cases where $p=2$ and. Suppose that $p=2$ and that $2l(\lambda)=s\geq 2$ . Then for each$po\mathit{8}itive$$n$$\mathrm{o}\mathrm{r}\mathrm{d}_{2}(h_{n}(A))\geq\sum_{1j=}^{l()}\lambda[\frac{n}{\mu}.]+[\frac{n}{2^{l(\lambda_{1})+2}}]-[\frac{n}{2^{l(\lambda_{1})+3}}]$and the equality holds if$n\equiv 0$

mod$2^{l(\lambda}$) $+1$and if$n\not\equiv 2^{l(\lambda}$,) $+1+2^{l(\lambda)+2}$

mod$2^{l(\lambda}$) $+3$.

179

Corollary 5. 1 ([5]) Let-subgroup of is
$A$

$p$

be a finite abelian group
where
. Let

$A$

$\mathit{8}uchth,at$$\lambda=(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{r}, .

$l( \lambda)=\max\{\lambda_{1},$Then the -adic power$p$

the type of a Sylow$|\lambda|=s$r\cdot)$$[ \frac{s+1}{2}]\}$.

$\mathit{8}e7^{\cdot}ies$$E_{A}(_{X})= \sum_{n=0}.\frac{h_{n}(A)}{n!}\infty$.

$x^{n}$

converges for$\mathrm{o}\mathrm{r}\mathrm{d}_{2}(X)>$$1- \sum_{i=1}^{l()}\frac{1}{2^{i}}-\lambda\frac{1}{2^{l(\lambda)+}2}+\frac{1}{2^{l(\lambda)+}3}$$\mathrm{o}\mathrm{r}\mathrm{d}_{\mathrm{p}}(x)>$$\frac{1}{p-1}-\sum_{i=1}^{l(\lambda)}\frac{1}{p^{i}}+\frac{2l(\lambda)-s}{p^{l(\lambda)+1}}$

if $p=2$ and if$2l(\lambda)=s$,

otherwise.

References[1]L. M. Butler, A unimodality result in the enumeration of subgroups of a finite
abelian group, Amer. Math. Soc., 101 (1987), 771-775.

[2]

N. Chigira, The solution of $x^{d}=1$ in finite group, J. Algebra, 180 (1996),

653-661.

[3]

T. Stehling, On computing the number of subgroups of a finite abelian group,
Combinatorica, 12 (1992), 475-479.

[4]

Y. Takegahara, Generating function for the number of homomorphisms from afinitely generated abelian group to an alternating group, preprint.

[5]

Y. Takegahara, The number of homomorphisms ffom a finite abelian group toa symmetric group, preprint.

[6]

K. Wohlfahrt, \"Uber einen Satz von Dey und die Modulgruppe, Arch. Math., 29(1977), 455-457.

[7]

T. Yoshida,$|\mathrm{H}\mathrm{o}\mathrm{m}(A, G)|$, J. Algebra, 156 (1993), 125-156.