Permutation IAPS
From Apstheory
Contents |
Definition
Symbol-free definition
The permutation IAPS is an IAPS of groups where the Failed to parse (Can't write to or create math temp directory): n^{th}
member is the symmetric group Failed to parse (Can't write to or create math temp directory): S_n
, and where the block concatenation map Failed to parse (Can't write to or create math temp directory): S_m
× Failed to parse (Can't write to or create math temp directory): S_n
→ S_{m+n} is defined as the permutation that permutes the first Failed to parse (Can't write to or create math temp directory): m
symbols according to the left argument and the next Failed to parse (Can't write to or create math temp directory): n
symbols according to the second argument.
Definition with symbols
The permutation IAPS is an IAPS of groups where the Failed to parse (Can't write to or create math temp directory): n^{th}
member is Failed to parse (Can't write to or create math temp directory): S_n
and the block concatenation map Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}: S_m
× Failed to parse (Can't write to or create math temp directory): S_n
→ Failed to parse (Can't write to or create math temp directory): S_{m+n}
is defined as follows: (fillin)
Property theory
Simplicity
For Failed to parse (Can't write to or create math temp directory): n
≥Failed to parse (Can't write to or create math temp directory): 3
, the group Failed to parse (Can't write to or create math temp directory): S_n
is not a simple group. Thus, the permutation IAPS is not eventually simple. In fact, only one member, namely Failed to parse (Can't write to or create math temp directory): S_2
, is simple.
For Failed to parse (Can't write to or create math temp directory): n
≥ Failed to parse (Can't write to or create math temp directory): 5 the only proper nontrivial normal subgroup of Failed to parse (Can't write to or create math temp directory): S_n is Failed to parse (Can't write to or create math temp directory): A_n
, the alternating group on Failed to parse (Can't write to or create math temp directory): n
elements or the group of even permutations. Hence, the only strongly proper nontrivial normal sub-IAPS of Failed to parse (Can't write to or create math temp directory): S_n is the even permutation IAPS, that is, the IAPS that associates to each Failed to parse (Can't write to or create math temp directory): n the group Failed to parse (Can't write to or create math temp directory): A_n
.
Thus, the permutation IAPS is not p-simple. However, since the even permutation IAPS is not a saturated sub-IAPS, the permutation IAPS is i-simple.
Completeness
For Failed to parse (Can't write to or create math temp directory): n
≥ Failed to parse (Can't write to or create math temp directory): 6
, the group Failed to parse (Can't write to or create math temp directory): S_n
is a complete group, that is, it is centerless and every automorphism of it is inner.
