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Universal completions of the cyclic amalgams of the same type

Atapattu Arachchille, Kanchana Chamila (2010)
M.Phil. thesis, University of Birmingham.

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Abstract

Automorphisms of Z/nZ is isomorphic to (Z/Z)×. If G is a finite abelian group, which is isomorphic to direct product of m cyclic groups of order q where q = p\(^n\) for some prime p. Then Aut(G) is isomorphic to the set of m×m matrices with determinant coprime to p, GL\(_m\) (e\(_q\)) Also Aut(G)=p\(^{(n-1)}\) \(^{(m2)}\) GL\(_m\) (Z\(_q\)). If \(\alpha\)is an automorphism of S\(_n\)and t is a transposition of S\(_n\)for n \(\not\) 6, then \(\alpha\)(t)is a transposition. If \(\alpha\) maps transposition to a transposition, then \(\alpha\) is an inner automorphism. Then AUT (S\(_n)\) \(\simeq\) S \(_n\) \(\not\) 6. Furthermore, there exists an outer automorphism of S\(_6\)and OUT (S\(_6\)) \(\simeq\) \(\frac{z}{zz}\). Thus OUT (S\(_g\))=2. Coset enumeration is one of the basic methods for investigating finitely generated subgroups in finitely presented.. Information are gradually added to a coset, a relation, a subgroup tables and once they are filled in, all cosets have been enumerated, the algorithm terminates. Goldschmidt’s Lemma on the number of isomorphism classes of amalgams having fixed type, verify that there is one isomorphism class of amalgam of type A=(S\(_n\)S\(_n\)S_(n-1), \(\phi\)\(_1\), \(\phi\)\(_2\)) where \(\phi\) is an identity map from S_(n-1) to (S\(_n\)for i=1, 2 and n\(\not\) 2,3,6,7. When n=2,7 we have two isomorphic class of amalgam of type A. Finally, i A and A’ are cyclic amalgams of the same type then there universal completions are isospectral.

Type of Work:M.Phil. thesis.
Supervisor(s):Parker, Christopher W
School/Faculty:Colleges (2008 onwards) > College of Engineering & Physical Sciences
Department:School of Mathematics
Subjects:QA Mathematics
Institution:University of Birmingham
ID Code:559
This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder.
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