Permutation IAPS
From Apstheory
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===Simplicity=== | ===Simplicity=== | ||
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| + | For <math>n</math> ≥<math> 3</math>, the group <math>S_n</math> is ''not'' a simple group. Thus, the permutation IAPS is not eventually simple. In fact, only one member, namely <math>S_2</math>, is simple. | ||
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| + | For <math>n</math> ≥ <math>5</math> the only proper nontrivial normal subgroup of <math>S_n</math> is <math>A_n</math>, the alternating group on <math>n</math> elements or the group of even permutations. Hence, the only [[strongly proper sub-APS|strongly proper]] [[nontrivial APS|nontrivial]] [[normal sub-IAPS|normal]] [[sub-IAPS]] of <math>S_n</math> is the [[even permutation IAPS]], that is, the IAPS that associates to each <math>n</math> the group <math>A_n</math>. | ||
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| + | Thus, the permutation IAPS is not [[p-simple IAPS|p-simple]]. However, since the even permutation IAPS is not a [[saturated sub-APS|saturated]] sub-IAPS, the permutation IAPS is [[i-simple IAPS|i-simple]]. | ||
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| + | ===Completeness=== | ||
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| + | For <math>n</math> ≥ <math>6</math>, the group <math>S_n</math> is a [[complete group]], that is, it is centerless and every automorphism of it is inner. | ||
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| + | [[Category: Terminology local to the wiki]] | ||
| + | [[Category: IAPSes of groups]] | ||
Current revision as of 01:14, 30 December 2006
Contents |
[edit] Definition
[edit] 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.
[edit] 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)
[edit] Property theory
[edit] 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.
[edit] 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.
