Normal sub-APS
From Apstheory
This article gives a basic definition in the APS theory. It is strictly local to the wiki
This article describes a property that can be evaluated for a sub-APS of an APS of groups
Contents |
[edit] Definition
A sub-APS Failed to parse (Can't write to or create math temp directory): H
of an APS of groups Failed to parse (Can't write to or create math temp directory): G is termed normal if it satisfies the following equivalent conditions:
- For every 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): H_n
is a normal sub-APS of Failed to parse (Can't write to or create math temp directory): G_n
- There is a surjective APS homomorphism Failed to parse (Can't write to or create math temp directory): \\sigma
from Failed to parse (Can't write to or create math temp directory): (G,\\Phi) to some APS Failed to parse (Can't write to or create math temp directory): (K,\\Gamma) such that the kernel of each homomorphism Failed to parse (Can't write to or create math temp directory): \\sigma_n is Failed to parse (Can't write to or create math temp directory): H_n
.
[edit] Equivalence of definitions
To prove that these definitions are equivalent, we need to show that if for each 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): H_n
is normal in Failed to parse (Can't write to or create math temp directory): G_n
, then the quotient groups Failed to parse (Can't write to or create math temp directory): K_n
naturally form an APS (termed a quotient APS).
[edit] Facts
[edit] Quotient being an IAPS
In general, the quotient of an APS of groups by a normal sub-APS need not be an IAPS of groups. In fact, it is an IAPS if and only if the normal sub-APS is a saturated sub-APS, viz if whenever a block concatenation of two elements of the APS is in the sub-APS, the two elements themselves are in the sub-APS.
