TY - JOUR
T1 - On distinguishing quotients of symmetric groups
AU - Shelah, S.
AU - Truss, J. K.
PY - 1999/3/21
Y1 - 1999/3/21
N2 - A study of the elementary theory of quotients of symmetric groups is carried out in a similar spirit to Shelah (1973). Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(μ) on an infinite cardinal μ are all of the form Sκ(μ)= the subgroup consisting of elements whose support has cardinality < κ, for some κ≤μ. A many-sorted structure Mκλμ is defined which, it is shown, encapsulates the first order properties of the group Sλ(μ)/Sκ(μ). Specifically, these two structures are (uniformly) bi-interpretable, where the interpretation of Mκλμ in Sλ(μ)/Sκ(μ) is in the usual sense, but in the other direction is in a weaker sense, which is nevertheless sufficient to transfer elementary equivalence. By considering separately the cases cf(κ)>2Ν0, cf(κ)≤2Ν0 <κ, Ν0 <κ <2Ν0, and κ = Ν0, we make a further analysis of the first order theory of Sλ(μ)/Sκ(μ), introducing many-sorted second order structures script N2 κλμ, all of whose sorts have cardinality at most 2Ν0, and in terms of which we can completely characterize the elementary theory of the groups Sλ(μ)/Sκ(μ).
AB - A study of the elementary theory of quotients of symmetric groups is carried out in a similar spirit to Shelah (1973). Apart from the trivial and alternating subgroups, the normal subgroups of the full symmetric group S(μ) on an infinite cardinal μ are all of the form Sκ(μ)= the subgroup consisting of elements whose support has cardinality < κ, for some κ≤μ. A many-sorted structure Mκλμ is defined which, it is shown, encapsulates the first order properties of the group Sλ(μ)/Sκ(μ). Specifically, these two structures are (uniformly) bi-interpretable, where the interpretation of Mκλμ in Sλ(μ)/Sκ(μ) is in the usual sense, but in the other direction is in a weaker sense, which is nevertheless sufficient to transfer elementary equivalence. By considering separately the cases cf(κ)>2Ν0, cf(κ)≤2Ν0 <κ, Ν0 <κ <2Ν0, and κ = Ν0, we make a further analysis of the first order theory of Sλ(μ)/Sκ(μ), introducing many-sorted second order structures script N2 κλμ, all of whose sorts have cardinality at most 2Ν0, and in terms of which we can completely characterize the elementary theory of the groups Sλ(μ)/Sκ(μ).
KW - Elementary theory
KW - Infinite symmetric group
KW - Many sorted structure
KW - Quotient
UR - http://www.scopus.com/inward/record.url?scp=0033590736&partnerID=8YFLogxK
U2 - 10.1016/S0168-0072(98)00023-2
DO - 10.1016/S0168-0072(98)00023-2
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:0033590736
SN - 0168-0072
VL - 97
SP - 47
EP - 83
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 1-3
ER -