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Let G Be an Abelian Group. a Subgroup of G Is Fully Invariant (Characteristic) If It Admits Every Endomorphism (Automorphism) of G

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Let G Be an Abelian Group. a Subgroup of G Is Fully Invariant (Characteristic) If It Admits Every Endomorphism (Automorphism) of G 538 MA THEMA TICS: I. KAPLA NSK Y PROC. N. A. S. SOME RESULTS ON ABELIAN GROUPS' By IRVING KAPLANSKY DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF CHICAGO Communicated by A. A. Albert, April 25, 1952 1. Introduction.-The author is engaged in the preparation of a mono- graph on infinite abelian groups. Besides exposition of known theory, it is planned to include some new material. Since publication will pre- sumably be somewhat delayed, it seems appropriate to make this brief announcement of the new results. 2. Definitions.-Let G be an abelian group. A subgroup of G is fully invariant (characteristic) if it admits every endomorphism (automorphism) of G. A subgroup H of G is divisible if nH = H for all n. G is reduced if its only divisible subgroup is 0. For most questions it suffices to con- sider reduced groups. G is primary (for the prime p) if every element has order a power of p. For a reduced primary group G we define a trans- finite series as follows: G.+1 = PGa for every ordinal a, and Ga is the intersection of the preceding subgroups if a is a limit ordinal. If a non- zero element x lies in Ga but not in Ga+i, we write h(x) = a. We write h(O) co, with the agreement that co exceeds any ordinal. Let P be the subgroup of elements of order p. The dimension of (P n Ga)/(P n G,a+,), regarded as a vector space over the integers mod p, is the ath Ulm invariant of G, written f(a). If G is countable, it is characterized by its Ulm invariants. 3. The Ring of Endomorphisms.-THEOREM 1. Let G and H be primary abelian groups, E and F their rings of endomorphisms. Then any iso- morphism of E and F is induced by a group isomorphism of G and H. In particular, every automorphism of E is inner. This theorem was proved by Baer,2 for groups of bounded order having at least three linearly independent elements of maximum order, with the aid of his theory of the lattice of subgroups. The author's proof uses ring-theoretic techniques, reminiscent of the classical treatment of auto- morphisms of matrix rings. 4. Characteristic Subgroups.-In Theorems 2 and 3 we confine ourselves to a reduced primary group G, but the theorems are easily extended to the non-reduced case. Let aj be a monotone increasing sequence of ordinals; it is allowed to be co from some point on. The set of x satisfying h(ptx) > a, for all i is manifestly a fully invariant subgroup of G.3 THEOREM 2. Let G be a reduced primary group such that every two elements can be embedded in a countable direct summand (this is in particular true if G itself is countable, or if it has no elements of infinite height). Then every fully invariant subgroup of G has the form just described. The lattice of Downloaded by guest on September 29, 2021 VOL. 38, 1952 MA THEMA TICS: I. KAPLA NSK Y 539 fully invariant subgroups is distributive and even satisfies the infinite dis- tributive law A n ( U B,) = u (A n Be); the dual infinite law is, however, satisfied if and only if G has no elements of infinite height. Characteristic subgroups are fully invariant, with an exceptional case generalizing that discovered by Shoda4 and Baer.5 THEOREM 3. The hypothesis on G is the same as in Theorem 2. Then thefollowing two statements are equivalent: (a) G has a characteristic subgroup which is not fully invariant, (b) p = 2, and there exist ordinals a, ,B with > a + 1 such that the Ulm invariants f(a) and f((3) are both 1. 5. Complete Modules.-In this final section we are concerned with modules over the p-adic integers (any complete discrete valuation ring would do in place of the p-adic integers). If M is such a module, we say it has no elements of infinite height if the intersection of pIM is 0. These submodules may then be taken as neighborhoods of 0 for a topology, and we say that M is complete if it is complete in this topology. THEOREM 4. A complete module over the p-adic integers is the completion of a direct sum of cyclic modules. The cardinal numbers giving the number of cyclic summands of each- order are a complete set of invariants.6 From Theorem 4 it is possible to deduce the answer to a question raised by the author in reference 7. THEOREM 5. Any module M over the p-adic integers has a direct summand of rank one. In particular, M is indecomposable if and only if it has rank one. Theorem 4 can also be applied to the problem of determining which abelian groups can be compact. Let G be a compact abelian group. It is known that its component C of the identity is its maximal divisible sub- group. Hence C is (algebraically, not topologically!) a direct summand of G. It is also known that the totally disconnected group GIC is the complete direct product of groups H, each of which is a complete module over the p-adic integers. Thus G belongs to a class of abelian groups which is completely characterized by a set of cardinal numbers. The author has not fully solved the problem of determining which sets of cardinals are eligible to be those of a compact group, but this appears to be of lesser interest. Indeed it would seem to be a good idea to call a group G "algebraically compact" if it has the form G = C E H, where C is divisible and H is a complete direct sum of complete modules over the p-adic integers. Then at any rate negative theorems carry greater force when they assert that a group is not algebraically compact. Sample theorem: an abelian group with an infinite cyclic direct summand is not algebraically compact. I This work was supported in part by the Office of Naval Research. 2 Ann. Math., 44, 192-227 (1943). Downloaded by guest on September 29, 2021 540 PATHOLOGY: A. B. SABIN PROC. N. A. S. 3 This description of fully invariant subgroups is a convenient modification of that of Shiffman, Duke Math. J., 6, 579-597 (1940). 4 Math. Zeit., 31, 611-624 (1930). 5 Proc. London Math. Soc., 39, 481-514 (1935). 6 The torsion-free case of Theorem 4 was discovered by Isidore B. Fleischer and appears in his dissertation. Trans. Am. Math. Soc., 72, 327-340 (1952). NA TURE OF INHERITED RESISTANCE TO VIRUSES AFFECTING THE NER VOUS SYSTEM By ALBERT B. SABIN THE CHILDREN'S HOSPITAL RESEARCH FOUNDATION, UNIVERSITY OF CINCINNATI COLLEGE OF MEDICINE, CINCINNATI, OHIO Read before the Academy, April 28, 1952 Introduction.-The vast majority of viruses which attack the nervous system of human beings and animals produce. recognizable disease and death in only a small proportion of ,infected individuals. A variety of factors pertaining to the virus, the host, or both, may influence the out- come of infection. The studies to be presented here were designed to elucidate the intricate biological mechanisms which form the basis of this important phenomenon. Previous studies by other investigators have established that plants and animals may possess an inherited resistance to various infectious and noxious agents, including viruses. The studies of Lynch and Hughes' with the virus of yellow fever and those of Webster2 with the viruses of louping ill and St. Louis encephalitis provided the first experimental evi- dence that the genetic constitution of the host can determine the outcome of mammalian viral infections. A most important contribution to this subject was Webster's demonstration that the inherent resistance or sus- ceptibility of mice to the virus of St. Louis encephalitis was correlated with the level of viral multiplication in the brain, not only in the intact animal3 but also in simple cultures containing the minced brain tissue.4 Earlier attempts to establish the manner in which such resistance is inherited by animals, and to segregate the factors which are genetically affected, were complicated by the fact that no uniformly resistant animals were available. Experimental Results.-In 1944, the author accidentally discovered that the albino mice which had been bred for over 25 years at the Rockefeller In- stitute at Princeton, N. J., were 100 per cent resistant to the 17 D strain of yellow fever virus that is widely used for human vaccination. Swiss mice, intracerebrally inoculated with this virus, invariably die after exhibiting Downloaded by guest on September 29, 2021.
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