# zbMATH — the first resource for mathematics

Existentially closed L$${\mathfrak X}$$-groups. (English) Zbl 0598.20036
If $${\mathfrak Y}$$ is a class of groups, then a $${\mathfrak Y}$$-group G is existentially closed if every system of finitely many equations and inequations with coefficients from G, which has a solution in some $${\mathfrak Y}$$-supergroup of G, can already be solved in G. Properties of such groups when $${\mathfrak Y}$$ is the class of all locally finite groups follow from the work of P. Hall [J. Lond. Math. Soc. 34, 305-319 (1959; Zbl 0088.023)] and when $${\mathfrak Y}$$ is the class of all locally finite p-groups from the work of B. Maier [Arch. Math. 37, 113-128 (1981; Zbl 0474.20015)]. In this paper, the author considers classes $${\mathfrak Y}$$ of the form L$${\mathfrak X}$$ (the class of all locally $${\mathfrak X}$$-groups) when $${\mathfrak X}$$ is closed under the taking of subgroups, extensions, quotient groups and cartesian powers of finitely generated subgroups. (Not all these properties are required in every case however.)
A group is said to be verbally complete if every element of the group is an instance of every non-trivial word (taken from some fixed countable free group). It is shown that e.c. L$${\mathfrak X}$$-groups are verbally complete in many cases; for example when the group is periodic or when the class $${\mathfrak X}$$ contains a non-trivial p-group for each prime p. The general question, however, remains open. More particularly, if $${\mathfrak X}$$ is closed under taking subgroups, extensions, quotient groups and cartesian powers of finitely generated subgroups, is an e.c. L$${\mathfrak X}$$-group verbally complete ?
The next section considers characteristic subgroups of e.c. L$${\mathfrak X}$$- groups. Again, it is shown that in many cases these must be proper or trivial but the general case is not fully understood. In particular, the author asks under what conditions an e.c. L$${\mathfrak X}$$-group ($${\mathfrak X}$$ locally finite) is characteristically simple.
The normal subgroup structure of countable e.c. L$${\mathfrak X}$$-groups is then considered and it is shown that such groups have a unique chief series (i.e. the normal subgroups are totally ordered by inclusion). This does not require all the restrictions on $${\mathfrak X}$$ but using all these restrictions, it can be shown that the order type of the set of normal subgroups is that of the rationals.
The automorphisms of e.c. L$${\mathfrak X}$$-groups are considered next. Finally it is shown that there are $$2^{\aleph_ 0}$$ isomorphism types of e.c. locally-(soluble with torsion elements all $$\pi$$-elements) groups whenever $$\pi$$ is an infinite class of primes. This is in contrast to the case for L$${\mathfrak X}$$ the class of locally finite groups or locally finite p-groups where there is a unique countable e.c. L$${\mathfrak X}$$- group.
Reviewer: J.R.J.Groves

##### MSC:
 20E25 Local properties of groups 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20F50 Periodic groups; locally finite groups 20E15 Chains and lattices of subgroups, subnormal subgroups 20F14 Derived series, central series, and generalizations for groups 20E07 Subgroup theorems; subgroup growth 20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Full Text:
##### References:
 [1] R. Grossberg - S. Shelah , On universal locally finite groups , Israe J. Math. , 44 ( 1983 ), pp. 289 - 302 . MR 710234 | Zbl 0525.20025 · Zbl 0525.20025 · doi:10.1007/BF02761988 [2] K.W. Gruenberg , Residual properties of infinite soluble groups , Proc. London Math. Soc . ( 3 ), 7 ( 1957 ), pp. 29 - 62 . MR 87652 | Zbl 0077.02901 · Zbl 0077.02901 · doi:10.1112/plms/s3-7.1.29 [3] P. Hall , Some constructions for locally finite groups , J. London Math. Soc. , 34 ( 1959 ), pp. 305 - 319 . MR 162845 | Zbl 0088.02301 · Zbl 0088.02301 · doi:10.1112/jlms/s1-34.3.305 [4] P. Hall , Wreath powers and characteristically simple groups , Proc. Camb. Phil. Soc. , 58 ( 1962 ), pp. 170 - 184 . MR 139649 | Zbl 0109.01302 · Zbl 0109.01302 [5] P. Hall , On non-strictly simple groups , Proc. Camb. Phil. Soc. , 59 ( 1963 ), pp. 531 - 553 . MR 156886 | Zbl 0118.03601 · Zbl 0118.03601 [6] P. Hall - B. Hartley , The stability group of a series of subgroups , Proc. London Math. Soc . ( 3 ), 16 ( 1966 ), pp. 1 - 39 . MR 191946 | Zbl 0145.24605 · Zbl 0145.24605 · doi:10.1112/plms/s3-16.1.1 [7] G. Higman , Amalgams of p-groups , J. Algebra , 1 ( 1964 ), pp. 301 - 305 . MR 167527 | Zbl 0246.20015 · Zbl 0246.20015 · doi:10.1016/0021-8693(64)90025-0 [8] J. Hirschfeld - W.H. Wheeler , Forcing, arithmetic, division rings , Berlin -Heidelberg -New York , 1975 . MR 389581 | Zbl 0304.02024 · Zbl 0304.02024 [9] C.H. Houghton , On the automorphism groups of certain wreath products , Publ. Math. Debrecen , 9 ( 1962 ), pp. 307 - 313 . MR 150213 | Zbl 0118.26702 · Zbl 0118.26702 [10] B. Huppert , Endliche Gruppen I , Berlin -Heidelberg -New York , 1967 . MR 224703 | Zbl 0217.07201 · Zbl 0217.07201 [11] B. Huppert - N. Blackburn , Finite groups II , Berlin -Heidelberg -New York , 1982 . MR 650245 | Zbl 0477.20001 · Zbl 0477.20001 [12] M.I. Kargapolov , On the \pi -completion of locally nilpotent groups , Algebra i Logika , 1 ( 1962 ), Vol. 1 , pp. 5 - 13 ; (Russian) MR #27-1511. Zbl 0146.03601 · Zbl 0146.03601 [13] O.H. Kegel - B.A.F. Wehrfritz , Locally finite groups , Amsterdam -London , 1973 . MR 470081 | Zbl 0259.20001 · Zbl 0259.20001 [14] H. Kurzweil , Endliche Gruppen , Berlin -Heidelberg -New York , 1977 . MR 486072 | Zbl 0381.20001 · Zbl 0381.20001 [15] F. Leinen , Existenziell abgeschlossene Lx-Gruppen, Dissertation zur Erlangung des Doktorgrades , Albert -Ludwigs- Universität Freiburg i. Br., 1984 . [16] A. Macintyre - S. Shelah , Uncountable universal locally finite groups , J. Algebra , 43 ( 1976 ), pp. 168 - 175 . MR 439625 | Zbl 0363.20032 · Zbl 0363.20032 · doi:10.1016/0021-8693(76)90150-2 [17] B. Maier , Existenziell abgeschlossene lokal endliche p-Gruppen , Arch. Math. , 37 ( 1981 ), pp. 113 - 128 . MR 640796 | Zbl 0474.20015 · Zbl 0474.20015 · doi:10.1007/BF01234334 [18] B.H. Neumann - H. Neumann , Embedding theorems for groups , J. London Math. Soc. , 34 ( 1959 ), pp. 465 - 479 . MR 163968 | Zbl 0102.26401 · Zbl 0102.26401 · doi:10.1112/jlms/s1-34.4.465 [19] H. Neumann - J. Wiegold , Linked products and linked embeddings of groups , Math. Z. , 73 ( 1960 ), pp. 1 - 19 . MR 124386 | Zbl 0105.25801 · Zbl 0105.25801 · doi:10.1007/BF01163265 · eudml:169828 [20] P.M. Neumann , On the structure of standard wreath products of groups , Math. Z. , 84 ( 1964 ), pp. 343 - 373 . MR 188280 | Zbl 0122.02901 · Zbl 0122.02901 · doi:10.1007/BF01109904 · eudml:170273 [21] D.J.S. Robinson , Finiteness conditions and generalized soluble groups , Parts 1 & 2, Berlin -Heidelberg -New York , 1972 . MR 332989 | Zbl 0243.20032 · Zbl 0243.20032 [22] V. Stingl , Radikale und koradikale Regeln , Rend. Sem. Mat. Univ. Padova , 72 ( 1984 ), pp. 343 - 355 . Numdam | MR 778350 | Zbl 0562.20017 · Zbl 0562.20017 · numdam:RSMUP_1984__72__343_0 · eudml:107962 [23] S. Thomas , Complete existentially closed locally finite groups , Arch. Math. 44 ( 1985 ), pp. 97 - 109 . MR 780255 | Zbl 0563.20037 · Zbl 0563.20037 · doi:10.1007/BF01194072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.