Abstract. We give a solution of De Bruijn's problem s to which homomorphic images of an infinite Symmetrie group S () are embeddable into S (). We prove (in ZF -f V = L) that $()/$() can be embedded into S () if and only if < cf (). We also discuss F.Clare's problem whether S(K)/SK () is a universal group. We prove that it is consistent with ZFC that 8()/8() is not universal. However S(K)/SK(K) is almost universal, since the group SK+ (+) of all bounded permutations of + embeds into it when is regul r. 1991 Mathematics Subject Classification: 20B30; 20A15, 03E05. 1. Introduction In its long history of more than 150 years the theory of finite permutation groups has reached a high level of perfection. During this rather long period only few authors also studied infinite permutation groups. A reason for this apparent disproportion may be that the rieh machinery of finite combinatorics and number theory was at band to all mathematicians while the powerful methods of infinite combinatorics (set theory) remained unknown to most of them. Without exaggeration one might say that the use of set theory consisted in the use of the bracked
Forum Mathematicum – de Gruyter
Published: Jan 1, 1993
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