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Centralizers Of Finite Subgroups In Simple Locally Finite GroupsErsoy, Kivanc 01 August 2009 (has links) (PDF)
A group G is called locally finite if every finitely generated subgroup of G is finite. In this thesis we study the centralizers of subgroups in simple locally finite groups. Hartley proved that in a linear simple locally finite group, the fixed point of every semisimple automorphism contains infinitely many elements of
distinct prime orders. In the first part of this thesis, centralizers of finite abelian subgroups of linear simple locally finite groups are studied and the following result is proved: If G is a linear simple locally finite group and A is a finite d-abelian
subgroup consisting of semisimple elements of G, then C_G(A) has an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes pi.
Hartley asked the following question: Let G be a non-linear simple locally finite group and F be any subgroup of G. Is CG(F) necessarily infinite? In the second part of this thesis, the following problem is studied: Determine the nonlinear
simple locally finite groups G and their finite subgroups F such that C_G(F) contains an infinite abelian subgroup which is isomorphic to the direct product of cyclic groups of order pi for infinitely many distinct primes p_i. We prove the following: Let G be a non-linear simple locally finite group with a split Kegel cover K and F be any finite subgroup consisting of K-semisimple elements of G. Then the centralizer C_G(F) contains an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes
p_i.
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A k-Conjugacy Class ProblemRoberts, Collin 15 August 2007 (has links)
In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k.
When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc.
In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that:
(G has finitely many k-conjugacy classes) implies (G is finite)?
Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have.
We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.
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A k-Conjugacy Class ProblemRoberts, Collin 15 August 2007 (has links)
In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k.
When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc.
In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that:
(G has finitely many k-conjugacy classes) implies (G is finite)?
Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have.
We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.
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