Difference between revisions of "User:Richard Pinch/sandbox-WP"
From Encyclopedia of Mathematics
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(Start article: Residual property) |
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| + | =Residual property= | ||
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| + | In the [[mathematics|mathematical]] field of [[group theory]], a group is '''residually ''X''''' (where ''X'' is some property of groups) if it "can be recovered from groups with property ''X''". | ||
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| + | Formally, a group ''G'' is residually ''X'' if for every non-trivial element ''g'' there is a [[Group homomorphism|homomorphism]] ''h'' from ''G'' to a group with property ''X'' such that <math>h(g)\neq e</math>. | ||
| + | |||
| + | More categorically, a group is residually ''X'' if it embeds into its pro-''X'' completion (see [[profinite group]], [[pro-p group]]), that is, the [[inverse limit]] of <math>\phi\colon G \to H</math> where ''H'' is a group with property ''X''. | ||
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| + | ==Examples== | ||
| + | Important examples include: | ||
| + | * [[Residually finite group|Residually finite]] | ||
| + | * Residually [[nilpotent group|nilpotent]] | ||
| + | * Residually [[solvable group|solvable]] | ||
| + | * Residually [[free group|free]] | ||
| + | |||
| + | ==References== | ||
| + | * {{cite book | title=The theory of groups | author=Marshall Hall Jr | authorlink=Marshall Hall (mathematician) | location=New York | publisher=Macmillan | year=1959 | page=16 }} | ||
=Stably free module= | =Stably free module= | ||
Revision as of 18:13, 25 August 2013
Residual property
In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X".
Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that \(h(g)\neq e\).
More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of \(\phi\colon G \to H\) where H is a group with property X.
Examples
Important examples include:
- Residually finite
- Residually nilpotent
- Residually solvable
- Residually free
References
Stably free module
A module which is close to being free.
Definition
A module M over a ring R is stably free if there exist free modules F and G over R such that
\[ M \oplus F = G . \, \]
Properties
- A projective module is stably free if and only if it possesses a finite free resolution.
See also
References
How to Cite This Entry:
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30231
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30231