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Free Groups and Free Products [PDF]
, 1993 Among the basic properties of free groups established in Chapter 2 was the fact that every group is isomorphic with a quotient group of a free group, a fundamental result that demonstrates clearly the significance of free groups. Thus the quotient groups of free groups account essentially for all groups.
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Products of Commutators in Free Products
International Journal of Algebra and Computation, 1997The genus of an element in the commutator subgroup of a group \(G\) is the minimal number of commutators of which the element is a product. It has been shown previously that in a free group each element of genus \(n\) can be obtained by permutation and suitable substitution on one of a finite number of words called orientable forms of genus \(n\).
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A relation in free products [PDF]
, 1974 A comparative study which contrasts the theory of free groups with the theory of free products has some general interest. One area where comparison can sometimes be made in an elementary way concerns relations, and their consequences: for example, the consequences of the relation XY = YX .
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Linear and Multilinear Algebra, 1995
See the review in Zbl 0824.15016 of the author's article with the same title [Linear Multilinear Algebra 38, No. 3, 215-223 (1995)].
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See the review in Zbl 0824.15016 of the author's article with the same title [Linear Multilinear Algebra 38, No. 3, 215-223 (1995)].
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The free product is not associative
Fuzzy Sets and Systems, 1993The paper gives a counterexample to a theorem by \textit{S. Ray} [cf. Theorem 2.4 in Fuzzy Sets Syst. 50, 225-235 (1992; Zbl 0785.20033)]. Therefore the considered free product of fuzzy subsets of a group does not form a semigroup.
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International Journal of Algebra and Computation, 1995
Topological methods are used to show that for certain subgroups S of a free product F, if w∈S is a commutator in F, then w is a commutator in S.
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Topological methods are used to show that for certain subgroups S of a free product F, if w∈S is a commutator in F, then w is a commutator in S.
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1995
The notion of generators and relations can be extended from abelian groups to arbitrary groups once we have a nonabelian analogue of free abelian groups. We use the property appearing in Theorem 10.11 as our starting point.
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The notion of generators and relations can be extended from abelian groups to arbitrary groups once we have a nonabelian analogue of free abelian groups. We use the property appearing in Theorem 10.11 as our starting point.
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1997
This chapter discusses the production of free radicals while focusing on oxygen free radicals. A free radical is defined in very general terms as an organizational state of a molecule or atom having a free electron associated with it. Almost all atoms or molecules in biological systems exist such that their electrons are paired. Thus the existence of a
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This chapter discusses the production of free radicals while focusing on oxygen free radicals. A free radical is defined in very general terms as an organizational state of a molecule or atom having a free electron associated with it. Almost all atoms or molecules in biological systems exist such that their electrons are paired. Thus the existence of a
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Multidisciplinary standards of care and recent progress in pancreatic ductal adenocarcinoma
Ca-A Cancer Journal for Clinicians, 2020Aaron J Grossberg +2 more
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Methylammonium-free wide-bandgap metal halide perovskites for tandem photovoltaics
Nature Reviews Materials, 2023Alexandra J Ramadan +2 more
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