scholarly journals A note on central group extensions

1973 ◽  
Vol 15 (4) ◽  
pp. 428-429 ◽  
Author(s):  
G. J. Hauptfleisch
Keyword(s):  
Abelian Group ◽  
Homomorphic Image ◽  
Free Group ◽  
Central Extension ◽  
Dihedral Group ◽  
Group Extension ◽  
Central Group ◽  
Group Extensions

If A, B, H, K are abelian group and φ: A → H and ψ: B → K are epimorphisms, then a given central group extension G of H by K is not necessarily a homomorphic image of a group extension of A by B. Take for instance A = Z(2), B = Z ⊕ Z, H = Z(2), K = V4 (Klein's fourgroup). Then the dihedral group D8 is a central extension of H by K but it is not a homomorphic image of Z ⊕ Z ⊕ Z(2), the only group extension of A by the free group B.

scholarly journals A note on group extensions

1974 ◽  
Vol 18 (4) ◽  
pp. 509-510 ◽  
Author(s):  
L. R. Vermani
Keyword(s):  
Homomorphic Image ◽  
Present Note ◽  
Abelian Groups ◽  
Group Extension ◽  
Central Group ◽  
Group Extensions ◽  
Central Loop

In [2] Hauptfleisch proved that if A, B, H, K are Abelian groups, φ:A → H and ψ:B → K are epimorphisms, then every central group extension G of H by K is homomorphic image of a central loop extension L of A by B. The aim of the present note is to prove (using almost the same argument as in [2])

Constructing pure injective hulls

1980 ◽  
Vol 45 (3) ◽  
pp. 544-548 ◽  
Author(s):  
Wilfrid Hodges
Keyword(s):  
Abelian Group ◽  
Homomorphic Image ◽  
Injective Hull ◽  
Pure Extension ◽  
The One ◽  
Injective Hulls

Let A be an abelian group and B a pure injective pure extension of A. Then there is a homomorphic image C of B over A which is a pure injective hull of A; C can be constructed by using Zorn's lemma to find a suitable congruence on B. In a paper [4] which greatly generalises this and related facts about pure injectives, Walter Taylor asks (Problem 1.5) whether one can find a “construction” of C which is more concrete than the one mentioned above; he asks also whether the points of C can be explicitly described. In this note I return the answer No.

scholarly journals Abelian groups with small cotorsion images

1991 ◽  
Vol 50 (2) ◽  
pp. 243-247 ◽  
Author(s):  
Rüdiger Göbel
Keyword(s):  
Homomorphic Image ◽  
Free Group ◽  
Abelian Groups ◽  
Torsion Free Group ◽  
Compact Abelian Groups ◽  
Torsion Free

AbstractEpimorphic images of compact (algebraically compact) abelian groups are called cotorsion groups after Harrison. In a recent paper, Ph. Schultz raised the question whether “cotorsion” is a property which can be recognized by its small cotorsion epimorphic images: If G is a torsion-free group such that every torsion-free reduced homomorphic image of cardinality is cotorsion, is G necessarily cortorsion? In this note we will give some counterexamples to this problem. In fact, there is no cardinal k which is large enough to test cotorsion.

On Graded Categorical Groups and Equivariant Group Extensions

2002 ◽  
Vol 54 (5) ◽  
pp. 970-997 ◽  
Author(s):  
A. M. Cegarra ◽  
J. M. Garćia-Calcines ◽  
J. A. Ortega
Keyword(s):  
Group Cohomology ◽  
Group Extension ◽  
Extension Problem ◽  
Homotopy Classification ◽  
Group Extensions ◽  
Categorical Groups

AbstractIn this article we state and prove precise theorems on the homotopy classification of graded categorical groups and their homomorphisms. The results use equivariant group cohomology, and they are applied to show a treatment of the general equivariant group extension problem.

scholarly journals Planar pure braids on six strands

2020 ◽  
Vol 29 (01) ◽  
pp. 1950097
Author(s):  
Jacob Mostovoy ◽  
Christopher Roque-Márquez
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Free Product ◽  
Free Group ◽  
Configuration Space ◽  
Braid Group ◽  
Free Abelian Group ◽  
Braid Groups ◽  
Pure Braid Group

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.

On the homology theory of central group extensions: I—The commutator map and stem extensions

1972 ◽  
Vol 47 (1) ◽  
pp. 102-122 ◽  
Author(s):  
Beno Eckmann ◽  
Peter J. Hilton ◽  
Urs Stammbach
Keyword(s):  
Homology Theory ◽  
Central Group ◽  
Group Extensions

scholarly journals The co-surface graph and the geometry of hyperbolic free group extensions

2017 ◽  
Vol 10 (2) ◽  
pp. 447-482 ◽  
Author(s):  
Spencer Dowdall ◽  
Samuel J. Taylor
Keyword(s):  
Free Group ◽  
Group Extensions ◽  
Surface Graph

scholarly journals Source Algebras of p-Central Group Extensions

2001 ◽  
Vol 235 (1) ◽  
pp. 359-398 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Central Group ◽  
Group Extensions

scholarly journals p-Grup pada Grup Dihedral

2019 ◽  
Author(s):  
Muhammad Irfan Hidayat
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Prime Number ◽  
Dihedral Group ◽  
Binary Operations ◽  
Interesting Part ◽  
Over Time

Group theory is an interesting part of algebra. The group theory is often researched and developed over time. The group is defined as a set with binary operations and fulfills several other conditions. One interesting and often discussed group is the Dihedral Group. The Dihedral group denoted by D_2n is the set of regular n-aspect symmetries, ∀nϵN, n≥3 with the composition operation "◦" which satisfies the axioms of the group and does not belong to the abelian group (commutative) while the form of the group is D_2n = {e, a, a ^ 2, ..., a ^ (n-1), b, ab, a ^ 2 b, .., a ^ (n-1) b} with n≥3. From a D_2n group subgroups can be formed which can also be viewed as another group. This research will examine several subgroups of group D_2n which are p-groups. p-Group is a group with the order p where p is a prime number. Previously all forms of subgroups had been obtained from the Dihedral group (D_2n). Based on that, this research will look for the D_2n subgroup that forms the p-group by identifying orders from the dihedral group.

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