The automorphism tower problem II

1998 ◽  
Vol 103 (1) ◽  
pp. 93-109 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Automorphism Tower

scholarly journals Closed maximality principles: implications, separations and combinations

2008 ◽  
Vol 73 (1) ◽  
pp. 276-308 ◽  
Author(s):  
Gunter Fuchs
Keyword(s):  
Equivalence Relation ◽  
Regular Cardinal ◽  
Highly Sensitive ◽  
Souslin Trees ◽  
Maximality Principles ◽  
Automorphism Tower

AbstractI investigate versions of the Maximality Principles for the classes of forcings which are <κ-closed, <κ-directed-closed, or of the form Col(κ, <λ). These principles come in many variants, depending on the parameters which are allowed, I shall write MPΓ (A) for the maximality principle for forcings in Γ, with parameters from A. The main results of this paper are:• The principles have many consequences, such as <κ-closed-generic (Hκ) absoluteness, and imply, e.g., that ◊κ holds. I give an application to the automorphism tower problem, showing that there are Souslin trees which are able to realize any equivalence relation, and hence that there are groups whose automorphism tower is highly sensitive to forcing.• The principles can be separated into a hierarchy which is strict, for many κ.• Some of the principles can be combined, in the sense that they can hold at many different κ simultaneously.The possibilities of combining the principles are limited, though: While it is consistent that MP<κ-closed(Hκ +) holds at all regular κ below any fixed α, the “global” maximality principle, stating that MP<κ-closed (Hκ ∪ {κ} ) holds at every regular κ, is inconsistent. In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal. It is also consistent that every local statement with parameters from Hκ⊦ that's provably <κ-closed-forceably necessary is true, for all regular κ.

scholarly journals The automorphism tower of groups acting on rooted trees

2005 ◽  
Vol 358 (1) ◽  
pp. 329-358 ◽  
Author(s):  
Laurent Bartholdi ◽  
Said N. Sidki
Keyword(s):  
Rooted Trees ◽  
Automorphism Tower

scholarly journals THE GROUPS OF AUTOMORPHISMS ARE COMPLETE FOR FREE BURNSIDE GROUPS OF ODD EXPONENTS n ≥ 1003

2013 ◽  
Vol 23 (06) ◽  
pp. 1485-1496 ◽  
Author(s):  
V. S. ATABEKYAN
Keyword(s):  
Normal Subgroup ◽  
Free Groups ◽  
Group Of Automorphisms ◽  
Burnside Groups ◽  
Free Burnside Group ◽  
Trivial Center ◽  
Groups Of Automorphisms ◽  
Group B ◽  
Inner Automorphisms ◽  
Automorphism Tower

It is proved that the group of automorphisms Aut (B(m, n)) of the free Burnside group B(m, n) is complete for every odd exponent n ≥ 1003 and for any m > 1, that is, it has a trivial center and any automorphism of Aut (B(m, n)) is inner. Thus, the automorphism tower problem for groups B(m, n) is solved and it is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, the group of all inner automorphisms Inn (B(m, n)) is the unique normal subgroup in Aut (B(m, n)) among all its subgroups, which are isomorphic to free Burnside group B(s, n) of some rank s.

The automorphism tower of a centerless group without Choice

2009 ◽  
Vol 48 (8) ◽  
pp. 799-815 ◽  
Author(s):  
Itay Kaplan ◽  
Saharon Shelah
Keyword(s):  
Automorphism Tower

scholarly journals Infinitely generated free nilpotent groups: completeness of the automorphism groups

2009 ◽  
Vol 147 (3) ◽  
pp. 541-566 ◽  
Author(s):  
VLADIMIR TOLSTYKH
Keyword(s):  
Nilpotent Group ◽  
Automorphism Groups ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Automorphism Tower

AbstractWe transfer the results of Dyer, Formanek and Kassabov on the automorphism towers of finitely generated free nilpotent groups to infinitely generated free nilpotent groups. We prove that the automorphism groups of infinitely generated free nilpotent groups are complete. By combining the results of Dyer, Formanek and Kassabov with the results in this paper, one gets that the automorphism tower of any free nilpotent group terminates after finitely many steps.

scholarly journals The Automorphism Tower Problem Revisited

1999 ◽  
Vol 148 (2) ◽  
pp. 243-265 ◽  
Author(s):  
Winfried Just ◽  
Saharon Shelah ◽  
Simon Thomas
Keyword(s):  
Automorphism Tower
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