Normal subgroups of nonstandard symmetric and alternating groups
Tóm tắt
Let
$${\mathfrak{M}}$$
be a nonstandard model of Peano Arithmetic with domain M and let
$${n \in M}$$
be nonstandard. We study the symmetric and alternating groups S
n
and A
n
of permutations of the set
$${\{0,1,\ldots,n-1\}}$$
internal to
$${\mathfrak{M}}$$
, and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A
n
and S
n
are not split extensions by these normal subgroups, by showing that any such complement if it exists, cannot be a limit of definable sets. We conclude by identifying an
$${\mathbb{R}}$$
-valued metric on
$${\tilde{S}_n = S_n /B_S}$$
and
$${\tilde{A}_n = A_n /B_A}$$
(where B
S
, B
A
are the maximal normal subgroups of S
n
and A
n
identified earlier) making these groups into topological groups, and by showing that if
$${\mathfrak{M}}$$
is
$${\mathfrak\aleph_1}$$
-saturated then
$${\tilde{S}_n}$$
and
$${\tilde{A}_n}$$
are complete with respect to this metric.
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