scholarly journals An Extension Theorem for Terraces

10.37236/2161 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Matt Ollis ◽  
Devin Willmott
Keyword(s):  
Cyclic Subgroup ◽  
Dihedral Group ◽  
Abelian Groups ◽  
Extension Theorem ◽  
Group V ◽  
Index 2

We generalise an extension theorem for terraces for abelian groups to apply to non-abelian groups with a central subgroup isomorphic to the Klein 4-group $V$.  We also give terraces for three of the non-abelian groups of order a multiple of 8 that have a cyclic subgroup of index 2 that may be used in the extension theorem.  These results imply the existence of terraces for many groups that were not previously known to be terraced, including 27 non-abelian groups of order 64 and all groups of the form $V^s \times D_{8k}$ for all $s$ and all $k > 1$ where $D_{8k}$ is the dihedral group of order $8k$.  

On the Morava K-theory of some finite 2-groups

1997 ◽  
Vol 121 (1) ◽  
pp. 7-13 ◽  
Author(s):  
BJÖRN SCHUSTER
Keyword(s):  
Finite Group ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Wreath Products ◽  
Classifying Space ◽  
Generalized Cohomology ◽  
K Theory ◽  
A Finite Group ◽  
Generalized Quaternion

For any fixed prime p and any non-negative integer n there is a 2(pn − 1)-periodic generalized cohomology theory K(n)*, the nth Morava K-theory. Let G be a finite group and BG its classifying space. For some time now it has been conjectured that K(n)*(BG) is concentrated in even dimensions. Standard transfer arguments show that a finite group enjoys this property whenever its p-Sylow subgroup does, so one is reduced to verifying the conjecture for p-groups. It is easy to see that it holds for abelian groups, and it has been proved for some non-abelian groups as well, namely groups of order p3 ([7]) and certain wreath products ([3], [2]). In this note we consider finite (non-abelian) 2-groups with maximal normal cyclic subgroup, i.e. dihedral, semidihedral, quasidihedral and generalized quaternion groups of order a power of two.

On the automorphism group of the integral group ring of the infinite dihedral group

1988 ◽  
Vol 108 (1-2) ◽  
pp. 1-10
Author(s):  
D. A. R. Wallace
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Group Ring ◽  
Polynomial Ring ◽  
Dihedral Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Index 2 ◽  
Skolem Theorem ◽  
Inner Automorphisms

SynopsisLet ℤ(G) be the integral group ring of the infinite dihedral groupG; the aim is to calculate Autℤ(G), the group of Z-linear automorphisms of ℤ(G). It is shown that Aut ℤ(G), the subgroup of Aut *ℤ(G) consisting of those automorphisms that preserve elementwise the centre of ℤ(G), is a normal subgroup of Aut *ℤ(G) of index 2 and that Aut*ℤ(G) may be embedded monomorphically intoM2(ℤ[t]), the ring of 2 × 2 matrices over a polynomial ring ℤ[t]From this embedding and by the Noether—Skolem Theorem it is shown that Inn (G), the group of inner automorphisms of ℤ(G) induced by the units of ℤ(G), is a normal subgroup of Aut*ℤ(G)/ such that Aut*ℤ(G)/Inn (G) is isomorphic to the Klein four-group.

Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups

1999 ◽  
Vol 42 (3) ◽  
pp. 335-343 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Sufficient Condition ◽  
Residual Finiteness ◽  
Necessary And Sufficient Condition ◽  
Residual Properties ◽  
Hnn Extensions ◽  
Subgroup Separability ◽  
Necessary And Sufficient

AbstractWe derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties ofHNN-extensions of nilpotent groups with cyclic associated subgroups.

scholarly journals EXTENSION OF AUTOMORPHISMS OF SUBGROUPS

2012 ◽  
Vol 54 (2) ◽  
pp. 371-380
Author(s):  
G. G. BASTOS ◽  
E. JESPERS ◽  
S. O. JURIAANS ◽  
A. DE A. E SILVA
Keyword(s):  
Abelian Group ◽  
Finite Groups ◽  
Dihedral Group ◽  
Abelian Groups ◽  
Alternating Group ◽  
Finite Abelian Groups ◽  
Quaternion Group ◽  
Icosahedral Group ◽  
Odd Order ◽  
Even Order

AbstractLet G be a group such that, for any subgroup H of G, every automorphism of H can be extended to an automorphism of G. Such a group G is said to be of injective type. The finite abelian groups of injective type are precisely the quasi-injective groups. We prove that a finite non-abelian group G of injective type has even order. If, furthermore, G is also quasi-injective, then we prove that G = K × B, with B a quasi-injective abelian group of odd order and either K = Q8 (the quaternion group of order 8) or K = Dih(A), a dihedral group on a quasi-injective abelian group A of odd order coprime with the order of B. We give a description of the supersoluble finite groups of injective type whose Sylow 2-subgroup are abelian showing that these groups are, in general, not quasi-injective. In particular, the characterisation of such groups is reduced to that of finite 2-groups that are of injective type. We give several restrictions on the latter. We also show that the alternating group A5 is of injective type but that the binary icosahedral group SL(2, 5) is not.

scholarly journals On finite groups whose 2-Sylow subgroups have cyclic subgroups of index 2

1964 ◽  
Vol 4 (1) ◽  
pp. 90-112 ◽  
Author(s):  
W. J. Wong
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Sylow Subgroups ◽  
Cyclic Subgroups ◽  
Index 2 ◽  
Generalized Quaternion

If the finite group G has a 2-Sylow subgroup S of order 2a+1, containing a cyclic subgroup of index 2, then in general S may be one of the following six types [8]:(i) cyclic; (ii) Abelian of type (a, 1), a > 1; (iii) dihedral1; (iv) generalized quaternion; (v) {α, β}, α2a = β2, α2a−1+1, a ≧ 3;(vi) {α, β}, α2a = β2, α2a−1+1, a ≧ 3.

scholarly journals On polygonal products of finitely generated abelian groups

1992 ◽  
Vol 45 (3) ◽  
pp. 453-462 ◽  
Author(s):  
Goansu Kim
Keyword(s):  
Finite Groups ◽  
Cyclic Subgroup ◽  
Abelian Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Residually Finite ◽  
Necessary And Sufficient

We prove that a polygonal product of polycyclic-by-finite groups amalgamating subgroups, with trivial intersections, is cyclic subgroup separable (hence, it is residually finite) if the amalgamated subgroups are contained in the centres of the vertex groups containing them. Hence a polygonal product of finitely generated abelian groups, amalgamating any subgroups with trivial intersections, is cyclic subgroup separable. Unlike this result, most polygonal products of four finitely generated abelian groups, with trivial intersections, are not subgroup separable (LERF). We find necessary and sufficient conditions for certain polygonal products of four groups to be subgroup separable.

scholarly journals On the structure of a real crossed group algebra

1990 ◽  
Vol 41 (1) ◽  
pp. 113-115
Author(s):  
P.P. Pálfy
Keyword(s):  
Left Ideal ◽  
Group Algebra ◽  
Finite Index ◽  
Cyclic Subgroup ◽  
Dihedral Group ◽  
Real Field ◽  
Finitely Generated ◽  
The Real ◽  
Torsion Free

We prove that the crossed group algebra A of the infinite dihedral group over the real field defined by the generators a and b, relations b−1 ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ is a principal left ideal ring. This corrects a result of Buzási and provides the missing step towards the classification of finitely generated torsion-free RG-modules for groups G which contain an infinite cyclic subgroup of finite index.

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