An Extension of a Theorem of Remak

Annals of Mathematics

An Extension of a Theorem of RemakAuthor(s): Charles HopkinsSource: Annals of Mathematics, Second Series, Vol. 40, No. 3 (Jul., 1939), pp. 636-638Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1968948 .

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ANNALS OF MATHEMATICS

Vol. 40, No. 3, July, 1939

AN EXTENSION OF A THEOREM OF REMAK

BY CHARLES HOPKINS

(Received October 16, 1938)

Let G denote a group and let Q denote a set of (proper) automorphisms of G which contains at least all the inner automorphisms of G. By the term admissible subgroup (a.s.) we shall mean any subgroup H of G such that He = H for all 0 in U. Evidently an admissible subgroup is a normal subgroup of G. Regarding G we assume only that the minimal condition holds for admissible subgroups. From this assumption it follows that the set of all admissible subgroups different from the identity subgroup E must contain at least one minimal admissible subgroup (m.a.s.). Let S denote the union of all m.a. subgroups.

THEOREM 1. The union S is the direct product of a finite number of m.a. subgroups-i.e. S = M1 X M2 X ... X Mn I where each Mi is a m.a.s.

PROOF. We know that G contains at least one m.a. subgroup M1 different from E. If M1 = S our theorem is trivially valid. So we assume M1 C S.

Let M2 be any fixed m.a.s. not contained in M1 and denote the union (M1 , M2) by U2. We write U, for M1 and assume that Uj has been defined for i = 1, 2, ... k - 1. Then we define Uk to be (Uk-i, Mk), where Mk is any fixed m.a.s. not contained in Ukl .

LEMMA 1. For k FZ. 1, Uk is the direct product

M1 X M2 X... X Mk.

This assertion is clearly true for k = 2, since the cross-cut of the two m.a. subgroups M1 and M2 is itself a m.a.s. and a proper subgroup of both M1 and M2 . Let us therefore assume that our lemma is valid for Uj, i = 1, 2, ... k - 1. By hypothesis, the m.a.s. Mk is not contained in Uki . Hence the cross-cut [Uk-i, Mkj is the identity subgroup E. So Uk = (M1 X M2 X ... X Mki , Mk) - M1 X M2 X ... X Mk , and our lemma is established.

We now construct the ascending chain

C1: U1 C U2 C ... C Uk C*

Let S, denote the sum of the subgroups Uj in Cl-i.e. Si is the set of all elements of G which occur in at least one U with finite subscript.

Let S2 denote the sum of the terms in the chain

C2: M2 C M2 X M3 C ... C M2 X ... X Mk C.--

By Si we shall mean the sum of the terms in the chain CQ, where C2 is obtained from Cjl by deleting Mjl from each term in Csi1.

Now each Si is an admissible subgroup, since each term in the chain C, is the direct product of admissible subgroups. Furthermore, we see that Si-, is

636



AN EXTENSION OF A THEOREM OF REMAK 637

contained properly in Si, since Si contains Mi, while Si-1 does not. From the minimal condition for admissible subgroups we know that the descending chain Sl D S2 D ... must terminate in an S with finite subscript-in S., say. But this is possible only if the ascending chain C1 above terminates in a U with finite subscript. Let Un denote the last term in C1 . Then S, = U., and from Lemma 1 we have S, = M1 X ... X M,. Since U, must contain every m.a.s. of G-else the chain C1 could be continued to Un+1 , at least-it follows that S, is the sum of all m.a. subgroups of G. This completes the proof of Theorem 1.

In Theorem 1 we have established the existence of at least one direct decomposition of S into the product of m.a. subgroups. The following result gives a partial answer to the "uniqueness-question."

THEOREM 2. Let (1) S = M1 X ...X Mn and (2) S = M X - X M be any two representations of S as the direct product of m.a. subgroups. Then n = n' and the subscripts in the decomposition (2) can be so chosen that Mi and Mi are centrally-isomorphic.

This theorem is a special case of a theorem due to Korlnek.1 It also can be obtained by modifying one of Fitting's proofs.2

From this point on we shall be concerned with the structure of a m.a.s. M of G, where M 5$ E. By the term normal subgroup of M we shall mean a subgroup of M which is normal with respect to M (but not necessarily with respect to G).

A rather obvious result is the following: LEMMA 2. If N $ E is a normal subgroup of M, then M is generated by the

conjugates N9 of n, where 0 varies over the elements of U. For these conjugates generate an a.s. of G, which must be equal to M, since M

is a m.a.s. Now the minimal condition for admissible subgroups of G gives us relatively

little information concerning M. To obtain significant results it seems necessary to impose further restrictions on M, and this we shall do by assuming the minimal condition for normal subgroups of M. By applying Theorem 1 to M and using Lemma 2, we obtain the following result.

1 V. Kofinek, Sur la decomposition d'un groupe en produit direct des sousgroupes, Cas. mat. fys., 66 (1937), p. 264. A correction by Korinek for Lemma 2.2 may be found in Cas. mat. fys., 67 (1938), pp. 209-210.

2 Fitting, UIber die direkten Produktzerlegungen einer Gruppe in direkt unzerlegbare Faktoren, Math. Zeitschrift, Vol. 39 (1934), p. 21.

Although Fitting assumes both the ascending and descending chain conditions for admissible subgroups, neither chain condition is necessary for the proof of our Theorem; we need only to assume the existence of the decompositions (1) and (2) above. Let x' be any element of M' and write x'-xi1-Xi2 ... Xin, where xis is the component of x4 in (1) above. From the fact that M' and Mi contain no proper admissible subgroups other than E, it is easy to show that the correspondence xi -+ xi is either the "null-homomorphism" or a (proper) isomorphism of M' with Mi. Hence each of Fitting's operators Oj (l.c. p. 22) is either the null-operator or a (proper) isomorphism. The remainder of his proof is applicable to our theorem, since it is only in establishing that at least one Oi is an isomorphism that he uses the double chain condition.



638 CHARLES HOPKINS

THEOREM 3. If the minimal condition holds for the normal subgroups of the m.a.s. M of G, then M is the direct product of a finite number of simple3 groups.

The analogue of Theorem 2 asserts that in any two direct decompositions

(a) M = Ri X ...X Rm

and (b) M = R X ...X R

we must have m = m', while the subscripts in (b) can be so chosen that Ri and R' will be centrally-isomorphic.

From Lemma 2 we can assume that (a) is a direct decomposition in which each Ri is a conjugate of R1 under U. If M is non-abelian, then each Ri is non-abelian. Since the center of M is the direct product of the centers of the Ri , we see that if M is non-abelian, its center is E. In this case, then, the fact that Ri and Ri are centrally-isomorphic implies that they are identical. When M is non-abelian, therefore, we have the following result:

THEOREM 4. If the minimal condition holds for normal subgroups of a non- abelian m.a. subgroup M of G, then M is uniquely representable as the direct product of a finite number of simple groups, which constitute a complete set of conjugates under U.

If M is an abelian group A, then it is necessarily of order pm and type (1, 1, , 1); such a group is uniquely representable as the direct product of simple groups if, and only if, m = 1. From Lemma 2 we know that we can find at least one direct decomposition A = P1 X ... X Pm, where the components Pi, of order p, are all conjugates of P1 under U. Furthermore, the components of every decomposition of A into the direct product of simple groups will be conjugate (under U) to a fixed component if, and only if, U contains the group of automorphisms of M.

In conclusion, we point out that our theorems are for the most part generaliza- tions of results-originally established by Remak4-which appear in the litera- ture mainly in connection with chief composition series. A weaker form of Theorem 1 (and of Theorem 3, also) has recently been obtained by Ore.5

TULANE UNIVERSITY, NEw ORLEANS, LOUISIANA.

3A minimal normal subgroup of M is necessarily simple; for a normal subgroup of any component in the direct-product decomposition would be a normal subgroup of M.

4 R. Remak, Lber minimale invariante Untergruppen in der Theorie der endlichen Gruppen, Journal fur Math., Vol. 162 (1930), pp. 1-16.

60. Ore, Structures and Group Theory II, Duke Mathematical Journal, Vol. 4 (1938), pp. 247-269.



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An Extension of a Theorem of Remak