APS of groups
From Apstheory
This article gives a basic definition in the APS theory. It is strictly local to the wiki
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[edit] Definition
An APS of groups is an APS over the category of groups. More specifically, an APS Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
of groups is the following data:
- For each natural number Failed to parse (Can't write to or create math temp directory): n
, a group, denoted Failed to parse (Can't write to or create math temp directory): G_n .
- For each ordered pair Failed to parse (Can't write to or create math temp directory): (m,n)
of natural numbers, a homomorphism Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}:G_m X G_n
→ Failed to parse (Can't write to or create math temp directory): G_{m+n}
.
Satisfying the following compatibility conditions:
For Failed to parse (Can't write to or create math temp directory): g, h, k
in Failed to parse (Can't write to or create math temp directory): G_m, G_n, G_p respectively,
Failed to parse (Can't write to or create math temp directory): \\Phi_{m+n,p} (\\Phi_{m,n}(g,h),k) = \\Phi_{m,n+p} (g, \\Phi_{n,p}(h,k)) .
The above condition is termed an associativity condition.
We may assume Failed to parse (Can't write to or create math temp directory): G_0
as the trivial group and define Failed to parse (Can't write to or create math temp directory): \\Phi_{m,0}
and Failed to parse (Can't write to or create math temp directory): \\Phi_{0,n}
as trivial paddings.
[edit] Terminology
[edit] Members and elements
For an APS Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
the member Failed to parse (Can't write to or create math temp directory): G_n
is termed the Failed to parse (Can't write to or create math temp directory): n^{th}
member of the APS. A member of the APS is an object that is the Failed to parse (Can't write to or create math temp directory): n^{th}
member for some Failed to parse (Can't write to or create math temp directory): n
.
An element of the APS is an element of some member of the APS.
The home of an element of the APS is the member in which it lies. The index of a member is the Failed to parse (Can't write to or create math temp directory): n
for which it is the Failed to parse (Can't write to or create math temp directory): n^{th}
member.
[edit] Block concatenation map
The maps Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}
are termed block concatenation maps.
[edit] Ground member
The ground member of an APS is its first member.
[edit] Other notions
[edit] Homomorphism of APSes
{{further|APS homomorphism}
Given APSes Failed to parse (Can't write to or create math temp directory): (G,\\Phi)
and Failed to parse (Can't write to or create math temp directory): (H,\\Psi)
, a homomorphism Failed to parse (Can't write to or create math temp directory): h: G
→ Failed to parse (Can't write to or create math temp directory): H associates, to each natural number Failed to parse (Can't write to or create math temp directory): n
, a map Failed to parse (Can't write to or create math temp directory): h_n: G_n
→ Failed to parse (Can't write to or create math temp directory): H_n
, such that:
Failed to parse (Can't write to or create math temp directory): \\Psi_{m,n}(h_m(a),h_n(b)) = h_{m+n}(\\Phi_{m,n}(a,b))
[edit] Sub-APS notion
- Further information: sub-APS
Given an APS Failed to parse (Can't write to or create math temp directory): (G,\\Phi) , a sub-APS Failed to parse (Can't write to or create math temp directory): H
associates, to each Failed to parse (Can't write to or create math temp directory): n
, a subgroup Failed to parse (Can't write to or create math temp directory): H_n
of Failed to parse (Can't write to or create math temp directory): G_n
, such that the image of Failed to parse (Can't write to or create math temp directory): H_m X H_n
under Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}
lies inside Failed to parse (Can't write to or create math temp directory): H_{m+n}
.
When the APS of groups is injective, any sub-APS is also injective.
[edit] Quotient APS notion
- Further information: quotient APS
A quotient APS is the image of an APS in a homomorphism that is surjective at each member.
[edit] Normal sub-APSes, kernels and images
Given a homomorphism of APSes of groups, the kernels of the individual homomorphisms for a sub-APS of the domain APS, and the images of the individual homomorphism form a sub-APS of the range APS. The image is thus a quotient APS.
Further, we have the following result: a sub-APS of an APS of groups occurs as the kernel of an APS homomorphism if and only if every member of it is a normal subgroup of the corresponding member of the whole APS. A sub-APS satisfying either of these equivalent conditions is termed a normal sub-APS.
This parallels the group theory result that a subgroup of a group occurs as the kernel of a group homomorphism if and only if it is normal.
[edit] Properties
[edit] Injectivity
An APS of groups is termed injective, or an IAPS of groups, if every block concatenation map is injective. For an IAPS of groups, we usually also assume the condition of refinability.
[edit] Commutativity
Very few APSes of groups are commutative. Note that a commutative APS cannot also be injective.
[edit] Padding-injectivity
Most APSes of groups that we encounter satisfy the condition of being padding-injective.
