Page 1

No. 10]

Proc. Japan Acad., 52 (1976)

555

150.

On the Jordan.Hblder Theorem

By Zensiro GOSEKI

Gunma University

(Communicated by Kenjiro SHOD/k, M. ,T./k., Dec. 13, 1976)

Let {A,f} be a family of groupsA and homomorphisms

-A_, defined for all n e Z (Z--{0, +_1, +_2, ...}).

fn+lA f>An

is exact, then we denote it by (A, :f,) and we say (A, :f,) to be well

defined.

HSlder Theorem in group theory have been given in some papers (or

example, [2] and [3]).

The purpose of this note is also to give those

theorems or

sequence (A, :f,).

1.

Isomorphism Theorem.

(B g) be well defined. A translation {} of (A :f) into (B g) is

the set of homomorphisms

:A--.B such that ,_f=g, for all

n e Z.

Moreover, if each

is an isomorphism,, we say that (A fD

is isomorphic to (B :g).

i.e., A>B,, and f=g on B, then we denote (B :g) by (B :f).

In this case, we call (B:f.0a subsequence of (A :f) and write it in

the notation (A f)>(B f).

we call.(B :f)a normal subsequence of (A:f0 and write it in the

notation: (A f)>(B

It is easy to prove the ollowing

Lemma 1.

Let (A f) be well defined.

be a subgroupof A.

Then (M :f) is welldefined iff.f(M)=f(A)

M_forall n e Z.

By Lemma 1 and the same way as in proofs of [1, Lemma 2] and

[1, Lemma 3], we can prove the following

Lemma 2.

Let (A f)>/(P f).

>P.

eachfis a mapping which is naturally induced byf.

Theorem 1.

Let {} (A f)--.(B g) be a translation.

(a(A): g) is well defined iff(Ker (a):f) is well defined.

case, (A/Ker ():f)is also well definedand isomorphic to (a(A)

gD, whereforeach n e Z,fis a mapping which is naturally induced

by f,.

Proof.

The first assertion follows from routine arguments and

the remainder follows from Lemma 2.

If a sequence

f-

>An+l

>

?Z-

--1

>

Generalizations o Isomorphism Theorem and the Jordan-

In this section,

let (An:f,)and

If or each n e Z,B is a subgroup o A,

Moreover., if A.B for all n e Z,

For each n e Z, letM

For each n e Z, let A>M

Then (M f) is welldefined iff(M/P :f) is welldefinedwhere

Then

In this

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556

Z. GOSEKI

[Vol. 52,

Theorem 2o

(MH f) is welldefinedi(M H" f) is welldefined.

(MH/M"f) and (H/MH"f) are welldefinedand mutually iso-

morphic, where for each n e Z, f andfare mappings which are

naturally induced byf.

Proof.

By Lemma 2, (A/M"f) is well defined.

the translation {}" (H" fn)--*(A/M"f) where each a is a natural

homomorphism.

By Theorem 1, (MH/M’f) is well defined iff

(M H"f)is well defined.

Hence the first assertion follows from

Lemma 2. A proof of the remainder is obvious.

2.

]ordan.HSlder Theorem.

that is, we write G* instead of (G "f).

If (A B f), (AnB f) and (G/M"f) are well defined where for

each n e Z,fis a mapping which is naturally induced by f, then we

write A* VIB*, A’B* and G*/M* instead of those and say that A* IB*,

A’B* and G*/M* are well defined, respectively.

{K*} such that G*--Ko*>K*>...>K*=A*, A* is said to be subnormal

in G*, G*>>A*.

Let G*>A*. We say that A* has the I-property

in G* if for every subnormal subsequence B* of G*, A* VIB* is well

defined.

LetG*--Ko*>K*>...>K*--A*.

normal series if eachK*has the/-property in G*.

From the definition, we have easily the following

Proposition 1.

LetG*-Ko*>K* >. .>K*--A*.

I-normal seriesiffeach K*/ has the I-property inK.

Let G*/>A*.

of Gn, then A* is said to be a proper subsequence of G*.

G* is I-simple if no proper normal subsequence of G* has the/-property

in G*.

Furthermore, an/-normal series G*--Ko*>K*>...>K*--A*

is called an I-composition series from G* to A* if each K*/ is a proper

subsequence ofK*such thatK*/K*/is/-simple.

Proposition 2.

Let G*>M* and suppose M* has the I-property

in G*.

Then G*/M* is I-simpleif forevery H* having the I-property

in G*, G*>H*>M* implies H*---G* or H*----M*.

Proof’

If part"

Let G*/M*>X* and suppose X* has the I-

property in G*/M*.

Then, by Lemma 2, there is a subsequence H* of

G* such that G*>H*>M* and X*

L*glM* is well defined and so is L’M* by Theorem 2.

is well defined by Lemma 2 and G*/M*>>L*M*/M*.

L*M*/M* is well defined and so isH*(L*M*)/M*.

--H’L* is well defined by Lemma 2 and so is H*

This shows that H* has the/-property in G*.

--G*.

Therefore G*/M* is/-simple.

Let (A"f)>(M"f) and (A"f)>/(H"f).

Then

In this case,

We consider

Now we simplify our notation,

Let G*>A*, B* and G*>M*.

If there is a family

This series is called an I-

Then this is an

If there is n e Z such thatAis a proper subgroup

We say that

H*/M*.

Now let G*>>L*.

HenceL’M*/M*

Thus H*/M*

Hence H*(L*M*)

L* by Theorem 2.

Hence H*--M* or H*

Only if part" By the same way

Then

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No. 10]

Jordan-HSlder Theorem

557

as in the stated above, the application of Lemma 2 and Theorem 2

gives its proof and so we omit it.

Lemma 3.

Let G*>A* .and G*>/B*.

the 1-property in G*.

Then A’B* is well defined.

G*>A*B*, then A’B* has the 1-property in G*.

Proof.

Let G*>>H*.

Then A*H* is well defined and so is

A’H* by Theorem 2.

Furthermore G*>>A*H* and so B* A*H* is

well defined.

On the other hand, A*

>A*B*.

Hence (A*B*)(B*A*H*) is well defined andso isA*

(B* A*H*).

Thus A*(B* A*H*) and A’B* are well defined by

Theorem

2.

Hence,

simultaneously with A*(B*

A*H*, we obtain that A’B* A*H* is well defined.

Then (A*B*)(A*H*) is well defined and so is (A*B*)H*.

Theorem 2, A’B* H* is well defined.

in G*.

Lemma 4.

Let G*>>A*>B* and let G*>>H*>C*.

A*,B* and C* have the 1-property in G*.

B*(A*H*) are well defined.

Furthermore B*(A*C*) has the I-

property in B*(A*

H*).

Proof.

It is easy to see that B*(A*

defined.

Furthermore G*>>B*(A*

the/-property in G*, those have the/-property in B*(A*

over B*(A* H*)>B*(A*

C*) and B*(A* H*)>B*.

Lemma 3, B*(A*

C*) has the/-property in B*(A*

From Proposition 1, Lemma 4 and the well known results, we have

ollowing

Lemma 5.

Let

(i)

G* K >K >

(ii)

G*

L >L >

be two 1-normal seriesfrom G* to A*.

>/i>1; s>/]>O) and L(L_K*) (=L,; s>]>l r>i>/O) are well

defined.

Furthermore,foreach i, ] (r>i>l; s>]>O), K*, has the I-

property in G* and

( 1 )

K*_ Ko>K>... >K,,

Moreover,foreach i, ] (r>i/>0 s>]>/1), L, has the I-property in G*

and

*

L,o>L,x>...>L,=L.

L_=

Joining the I-normal series (1), respectively (2), together, we obtain

refinements of the 1-normal series (i) and (ii) for which K* /K

++L,_I/L, i8 a one to one correspondenceof their factors such that

correspondingfactorsare isomorphic.

By Lemma 5 and the well known procedure, we have the following

Suppose A* and B* have

Furthermoreif

B* is well defined and B*

A’H*

A’H*)

Let G*>A*B*.

Thus, by

A’B*

Hence A’B* has the/-property

Suppose

Then B*(A*C*) and

C*) and B*(A*

H*).

Since B* and A*

H*) are well

C* have

H*).

Hence,

More-

by

H*).

>K

>L

A*

A*

Then K*(K*_ L) (--K r

K*.

( 2 )

i,-

Page 4

558

Z. G0SEK

[Vol. 52,

Theorem 3 (Jordan.HSlder Theorem).

G*

Ko*>/K >

are two I-composition seriesfromG* to A*, then r=s.

there is a permutation

$

If

>/K

A*

and G*

Lo*>/L >/

Furthermore

of {1,...,r} such that K*_I/K* is isomorphic

to L()_I/L,*(t) foreach i=1, .., r.

References

1

Z. Goseki:

Acad., 50, 576-579 (1974).

O. Tamaschke: A generalization of subnormal subgroup. Arch. Math., 19,

337-347 (1968).

O. Wyler:

Ein Isomorphiesatz. Arch. Math., 14, 13-15 (1963).

On Sylow subgroups and an extension of groups. Proc. Japan

2

3