TY - JOUR
T1 - The Existence of Large ω1-Homogeneous But Not ω-Homogeneous Permutation Groups is Consistent with Zfc+Gch
AU - Shelah, S.
AU - Soukup, L.
PY - 1993/10
Y1 - 1993/10
N2 - Denote by Perm (λ) the group of all permutations of a cardinal λ. A subgroup G of Perm (λ) is called K-homogeneous if and only if for all X, Y € [λ]K there is a g € G with g″X = Y. We show that if either (i) ⋄+ holds and we add ω1 Cohen reals to the ground model, or (ii) we add 2ω1 Cohen reals to the ground model, then in the generic extension for each λ ≥ ω2 there is an ω1-homogeneous subgroup of Perm (λ) which is not ω-homogeneous.
AB - Denote by Perm (λ) the group of all permutations of a cardinal λ. A subgroup G of Perm (λ) is called K-homogeneous if and only if for all X, Y € [λ]K there is a g € G with g″X = Y. We show that if either (i) ⋄+ holds and we add ω1 Cohen reals to the ground model, or (ii) we add 2ω1 Cohen reals to the ground model, then in the generic extension for each λ ≥ ω2 there is an ω1-homogeneous subgroup of Perm (λ) which is not ω-homogeneous.
UR - http://www.scopus.com/inward/record.url?scp=84963063065&partnerID=8YFLogxK
U2 - 10.1112/jlms/s2-48.2.193
DO - 10.1112/jlms/s2-48.2.193
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AN - SCOPUS:84963063065
SN - 0024-6107
VL - s2-48
SP - 193
EP - 203
JO - Journal of the London Mathematical Society
JF - Journal of the London Mathematical Society
IS - 2
ER -