Complete universal locally finite groups
Author:
Ken Hickin
Journal:
Trans. Amer. Math. Soc. 239 (1978), 213227
MSC:
Primary 20E25; Secondary 20F50
MathSciNet review:
0480750
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Abstract: This paper will partly strengthen a recent application of model theory to the construction of sets of pairwise nonembeddable universal locally finite groups [8]. Our result is Theorem. There is a set of universal locally finite groups of order with the following properties: 0.1. If and A and B are uncountable sugroups of U and V, then A and B are not isomorphic. Let A be an uncountable subgroup of . 0.2. A does not belong to any proper variety of groups, and 0.3. A is not isomorphic to any of its proper subgroups. 0.4. Every is a complete group (every automorphism of U is inner).
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 G. Baumslag, Lecture notes on nilpotent groups, Regional Conf. Ser. in Math., no. 2, Amer. Math. Soc., Providence, R. I., 1971. MR 44 #315. MR 0283082 (44:315)
 [2]
 G. Fodor, On stationary sets and regressive function, Acta Sci. Math. (Szeged) 27 (1966), 105110. MR 34 #66. MR 0200167 (34:66)
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 Philip Hall, Some constructions for locally finite groups, J. London Math. Soc. 34 (1959), 305319. MR 0162845 (29:149)
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 K. Hickin, Countable type local theorems in algebra, J. Algebra 27 (1973), 523537. MR 49 #5179. MR 0340424 (49:5179)
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 B. Jónsson, Homogeneous universal relational systems, Math. Scand. 8 (1960), 137142. MR 23 #A2328. MR 0125021 (23:A2328)
 [6]
 O. Kegel and B. Wehrfritz, Locally finite groups, NorthHolland, Amsterdam, 1973. MR 0470081 (57:9848)
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 A. Karrass and D. Solitar, The subgroups of a free product of two groups with an amalgamated subgroup, Trans. Amer. Math. Soc. 150 (1970), 227255. MR 41 #5499. MR 0260879 (41:5499)
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 A. Macintyre and S. Shelah, Uncountable universal locally finite groups (to appear). MR 0439625 (55:12511)
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 A. Macintyre, Existentially closed structures and Gentzen's principle (to appear).
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 M. Morley and R. L. Vaught, Homogeneous universal models, Math. Scand. 11 (1962), 3757. MR 27 #37. MR 0150032 (27:37)
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 B. H. Neumann, Permutational products of groups, J. Austral. Math. Soc. 1 (1959/60), 299310. MR 23 #A922. MR 0123597 (23:A922)
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 , An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London Ser. A. 246 (1954), 503554. MR 16, 10. MR 0062741 (16:10d)
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 W. R. Scott, Group theory, PrenticeHall, Englewood Cliffs, N. J., 1964. MR 29 #4785. MR 0167513 (29:4785)
 [14]
 W. Sierpiński, Cardinal and ordinal numbers, 2nd rev. ed., Monografie Mat., vol., 34, PWN, Warsaw, 1965. MR 33 #2549.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197804807504
PII:
S 00029947(1978)04807504
Keywords:
Locally finite groups,
universal homogeneous groups,
complete groups,
subgroupincomparability,
regular representation,
group amalgams
Article copyright:
© Copyright 1978
American Mathematical Society
