Difference between revisions of "User:Richard Pinch/sandbox-WP"
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| + | =Baer–Specker group= | ||
| + | An example of an infinite [[Abelian group]] which is a building block in the structure theory of such groups. | ||
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| + | ==Definition== | ||
| + | The Baer-Specker group is the group ''B'' = '''Z'''<sup>'''N'''</sup> of all integer sequences with componentwise addition, that is, the [[direct product]] of [[countably infinite|countably]] many copies of '''Z'''. <!-- For each ''n'' in '''N''', let ''e''<sub>''n''</sub> be the sequence with ''n''-th term equal to 1 and all other terms 0.--> | ||
| + | |||
| + | ==Properties== | ||
| + | [[Reinhold Baer]] proved in 1937 that this group is ''not'' [[Free abelian group|free abelian]]; Specker proved in 1950 that every countable subgroup of ''B'' is free abelian. | ||
| + | |||
| + | ==See also== | ||
| + | * [[Slender group]] | ||
| + | |||
| + | ==References== | ||
| + | * {{cite book | author=Phillip A. Griffith | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 | pages=1, 111-112}} | ||
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=Essential subgroup= | =Essential subgroup= | ||
Revision as of 18:22, 25 August 2013
Baer–Specker group
An example of an infinite Abelian group which is a building block in the structure theory of such groups.
Definition
The Baer-Specker group is the group B = ZN of all integer sequences with componentwise addition, that is, the direct product of countably many copies of Z.
Properties
Reinhold Baer proved in 1937 that this group is not free abelian; Specker proved in 1950 that every countable subgroup of B is free abelian.
See also
References
Essential subgroup
A subgroup that determines much of the structure of its containing group. The concept may be generalized to essential submodules.
Definition
A subgroup \(S\) of a (typically abelian) group \(G\) is said to be essential if whenever H is a non-trivial subgroup of G, the intersection of S and H is non-trivial: here "non-trivial" means "containing an element other than the identity".
References
Pinch point
A pinch point or cuspidal point is a type of singular point on an algebraic surface. It is one of the three types of ordinary singularity of a surface.
The equation for the surface near a pinch point may be put in the form
\[ f(u,v,w) = u^2 - vw^2 + [4] \, \]
where [4] denotes terms of degree 4 or more.
References
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
Richard Pinch/sandbox-WP. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-WP&oldid=30235