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Centralizers of subgroups in direct limits of symmetric groups with strictly diagonal embedding. (English) Zbl 1330.20055

This paper describes a simple construction which produces locally finite permutation groups and allows construction of the centralizers of their finite subgroups.
The construction is as follows. Given a permutation \(\alpha\in S_n\) and a positive integer \(p\), then the homogeneous \(p\)-spreading of \(\alpha\) is the permutation in \(S_{pn}\) defined by \(d^p(\alpha)\colon kn+i\mapsto kn+i^\alpha\) (\(k=0,\ldots,p-1\) and \(i=1,\ldots,n\)); clearly \(d^p\) is an embedding of \(S_n\) into \(S_{np}\). Let \(\xi=(p_1,p_2,\ldots)\) be an infinite sequence of (not necessarily distinct) primes and put \(n_i=\prod_{j=1}^ip_j\). Then the locally finite group \(S(\xi)\leq\text{Sym}(\mathbb N)\) is defined to be the direct limit of the system \(\{S_{n_0}=1\to S_{n_1}\to S_{n_2}\to\cdots\}\) with the embeddings \(d^{p_i}\colon S_{n_{i-1}}\to S_{n_i}\). The character \(\text{Char}(\xi)\) is defined to be the formal product \(2^{s_2}3^{s_3}5^{s_5}\cdots\) where \(s_p\) is the number of times that the prime \(p\) occurs in \(\xi\) (so \(0\leq s_p\leq\infty\)); it is known that \(S(\xi)\) is a simple, non-linear locally finite group if \(r_2=\infty\) [O. H. Kegel and B. A. F. Wehrfritz, Locally finite groups. North-Holland Mathematical Library. Vol. 3. Amsterdam-London: North-Holland Publishing Comp. (1973; Zbl 0259.20001)].
Suppose that \(F\) is a finite subgroup of \(S(\xi)\). Let \(\Omega_1,\ldots,\Omega_k\) be representatives of the (images of the) transitive representations of \(F\), that \(n_{j_i}\) is the first level in the direct system where the representation \(\Omega_i\) appears, and that it appears with multiplicity \(r_i\). Then the authors show that the centralizer \(C_{S(\xi)}(F)\) is isomorphic to the direct product of the groups \(C_{\text{Sym}(\Omega_i)}(F|_{\Omega_i})(C_{\text{Sym}(\Omega_i)}(F|_{\Omega_i})\overline\wr S(\xi_i))\) where \(\text{Char}(\xi_i)=\text{Char}(\xi)r_i/n_{j_i}\). The construction of \(S(\xi)\) can be generalized by replacing \(n_1\) by an arbitrary cardinal \(\kappa\) and \(S_{n_1}\) by the finitary symmetric group \(\text{FSym}(\kappa)\). For each natural number \(p\geq 1\), \(\kappa p\) is an ordinal number and the homogeneous \(p\)-spreading of \(\alpha\in\text{FSym}(\kappa)\) is defined to be the permutation \(d^p(\alpha)\) of \(\text{FSym}(\kappa p)\) given by \(\kappa m+i\mapsto\kappa m+i^\alpha\) (\(m=0,\ldots,p-1\) and \(i\in\kappa\)). The authors show that many of the results which are true when \(\kappa\) is finite can be extended to the more general case.

MSC:

20F50 Periodic groups; locally finite groups
20B07 General theory for infinite permutation groups
20B30 Symmetric groups
20B35 Subgroups of symmetric groups
20E32 Simple groups
20E07 Subgroup theorems; subgroup growth
20E25 Local properties of groups
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References:

[1] DOI: 10.2307/1970080 · Zbl 0067.01004
[2] Güven Ü. B., J. SFU, Math. Physics. 6 (4) pp 437– (2014)
[3] DOI: 10.1112/jlms/s1-34.3.305 · Zbl 0088.02301
[4] Finite and Locally Finite Groups (1995)
[5] DOI: 10.1112/plms/s3-62.2.301 · Zbl 0682.20020
[6] Kargapolov , M. I. Merzljakov , Ju. I. ( 1979 ). Fundamentals of the Theory of Groups. Graduate Text in Mathematics, Vol. 62. Springer-Verlag . · Zbl 0549.20001
[7] Kegel O. H., Locally Finite Groups (1973) · Zbl 0259.20001
[8] DOI: 10.1007/s000130050249 · Zbl 0929.20031
[9] Lavrenyuk Ya. V., Algebra Discrete Math. 10 (1) pp 67– (2010)
[10] Lavrenyuk Ya. V., Algebra Discrete Math. 2 pp 104– (2007)
[11] Lavrenyuk Ya. V., Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 9 pp 24– (2005)
[12] Lavrenyuk Ya. V., Algebra Discrete Math. 4 pp 33– (2003)
[13] Kuzucuoğlu M., Int. J. Group Theory 2 (1) pp 1– (2013)
[14] Meldrum , J. D. P. ( 1995 ). Wreath products of groups and semigroups.Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 74. Harlow: Longman . · Zbl 0833.20001
[15] Rabinovich , E. B. ( 1981 ). Inductive limits of symmetric groups and universal groups.Visti Akad. Navuk. Belerussion SSR (Ser. Fiz. Math. Navuk No 5, Russian), pp. 39–42 . · Zbl 0497.20011
[16] DOI: 10.1016/0021-8693(76)90150-2 · Zbl 0363.20032
[17] Zalesskii A. E., Leningrad Math. J. 2 pp 1287– (1990)
[18] DOI: 10.1007/978-94-011-0329-9_9
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