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Journal articles on the topic 'Group homomorphism'

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Published: 4 June 2021

Last updated: 19 February 2022

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1

DAY, MATTHEW B. "EXTENSIONS OF JOHNSON'S AND MORITA'S HOMOMORPHISMS THAT MAP TO FINITELY GENERATED ABELIAN GROUPS." Journal of Topology and Analysis 05, no. 01 (March 2013): 57–85. http://dx.doi.org/10.1142/s1793525313500027.

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We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improves on the author's previous results [5]. To prove the first result, we express the higher Johnson homomorphisms as coboundary maps in group cohomology. Our methods involve the topology of nilpotent homogeneous spaces and lattices in nilpotent Lie algebras. In particular, we develop a notion of the "polynomial straightening" of a singular homology chain in a nilpotent homogeneous space.

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2

Kepert, Andrew G. "The Range of Group Algebra Homomorphisms." Canadian Mathematical Bulletin 40, no. 2 (June 1, 1997): 183–92. http://dx.doi.org/10.4153/cmb-1997-022-6.

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AbstractA characterisation of the range of a homomorphism between two commutative group algebras is presented which implies, among other things, that this range is closed. The work relies mainly on the characterisation of such homomorphisms achieved by P. J. Cohen.

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3

CABRER, LEONARDO, and DANIELE MUNDICI. "RATIONAL POLYHEDRA AND PROJECTIVE LATTICE-ORDERED ABELIAN GROUPS WITH ORDER UNIT." Communications in Contemporary Mathematics 14, no. 03 (June 2012): 1250017. http://dx.doi.org/10.1142/s0219199712500174.

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An ℓ-groupG is an abelian group equipped with a translation invariant lattice-order. Baker and Beynon proved that G is finitely generated projective if and only if it is finitely presented. A unital ℓ-group is an ℓ-group G with a distinguished order unit, i.e. an element 0 ≤ u ∈ G whose positive integer multiples eventually dominate every element of G. Unital ℓ-homomorphisms between unital ℓ-groups are group homomorphisms that also preserve the order unit and the lattice structure. A unital ℓ-group (G, u) is projective if whenever ψ : (A, a) → (B, b) is a surjective unital ℓ-homomorphism and ϕ : (G, u) → (B, b) is a unital ℓ-homomorphism, there is a unital ℓ-homomorphism θ : (G, u) → (A, a) such that ϕ = ψ ◦ θ. While every finitely generated projective unital ℓ-group is finitely presented, the converse does not hold in general. Classical algebraic topology (à la Whitehead) is combined in this paper with the Włodarczyk–Morelli solution of the weak Oda conjecture for toric varieties, to describe finitely generated projective unital ℓ-groups.

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4

MORDESON, J. N., and P. S. NAIR. "FUZZY MEALY MACHINES: HOMOMORPHISMS, ADMISSIBLE RELATIONS AND MINIMAL MACHINES." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 04, no. 01 (February 1996): 27–43. http://dx.doi.org/10.1142/s0218488596000032.

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Homomorphisms and admissible relations of fuzzy Mealy machines are studied. Admissible relations play a role similar to normal subgroups in group theory. The kernel of a homomorphism is shown to be an admissible relation. Conversely, corresponding to an admissible relation, there exists a homomorphism. The fundamental theorem on homomorphisms; and the existence and uniqueness of minimal machines are also presented.

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5

Reyes, Edgar N. "Homomorphisms of ergodic group actions and conjugacy of skew product actions." International Journal of Mathematics and Mathematical Sciences 19, no. 4 (1996): 781–88. http://dx.doi.org/10.1155/s0161171296001081.

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LetGbe a locally compact group acting ergodically onX. We discuss relationships between homomorphisms on the measured groupoidX×G, conjugacy of skew product extensions, and similarity of measured groupoids. To do this, we describe the structure of homomorphisms onX×Gwhose restriction to an extension given by a skew product action is the trivial homomorphism.

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6

BRZDȨK, JANUSZ. "CONTINUITY OF MEASURABLE HOMOMORPHISMS." Bulletin of the Australian Mathematical Society 78, no. 1 (August 2008): 171–76. http://dx.doi.org/10.1017/s0004972708000610.

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AbstractWe give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

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7

Selvachandran, Ganeshsree, and Abdul Razak Salleh. "On Normalistic Vague Soft Groups and Normalistic Vague Soft Group Homomorphism." Advances in Fuzzy Systems 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/592813.

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We further develop the theory of vague soft groups by establishing the concept of normalistic vague soft groups and normalistic vague soft group homomorphism as a continuation to the notion of vague soft groups and vague soft homomorphism. The properties and structural characteristics of these concepts as well as the structures that are preserved under the normalistic vague soft group homomorphism are studied and discussed.

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8

Hirshon, Ron, and David Meier. "Groups with a quotient that contains the original group as a direct factor." Bulletin of the Australian Mathematical Society 45, no. 3 (June 1992): 513–20. http://dx.doi.org/10.1017/s0004972700030422.

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We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

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9

CHILDERS, LEAH R. "SIMPLY INTERSECTING PAIR MAPS IN THE MAPPING CLASS GROUP." Journal of Knot Theory and Its Ramifications 21, no. 11 (August 27, 2012): 1250107. http://dx.doi.org/10.1142/s0218216512501076.

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The Torelli group, [Formula: see text], is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying [Formula: see text]: bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on [Formula: see text], while SIP-maps have received relatively little attention until recently, due to an infinite presentation of [Formula: see text] introduced by Putman that uses all three types of elements. We will give a topological characterization of the image of an SIP-map under the Johnson homomorphism and Birman–Craggs–Johnson homomorphism. We will also classify which SIP-maps are in the kernel of these homomorphisms. Then we will look at the subgroup generated by all SIP-maps, SIP (Sg), and show it is an infinite index subgroup of [Formula: see text].

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10

Sunderrajan, K., M. Suresh, and R. Muthuraj. "Homomorphism and Anti Homomorphism on Multi L-Fuzzy Quotient Group of a Group." International Journal of Mathematics Trends and Technology 23, no. 1 (July 25, 2015): 33–39. http://dx.doi.org/10.14445/22315373/ijmtt-v23p505.

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11

Shen, Liang. "ℵ0-Injective group rings." Journal of Algebra and Its Applications 19, no. 03 (February 22, 2019): 2050042. http://dx.doi.org/10.1142/s0219498820500425.

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Recall that a ring [Formula: see text] is called right [Formula: see text]-injective if every homomorphism from a countably generated right ideal of [Formula: see text] to [Formula: see text] can be extended to a homomorphism from [Formula: see text] to [Formula: see text]. These rings are not only a natural generalization of self-injective rings but also strongly connected with regularities of rings. Let [Formula: see text] be the group ring of a group [Formula: see text] over a ring [Formula: see text]. It is proved that [Formula: see text] is right [Formula: see text]-injective if and only if (i) [Formula: see text] is right [Formula: see text]-injective; (ii) [Formula: see text] is finite; (iii) for each countably generated right ideal [Formula: see text] of [Formula: see text], any right [Formula: see text]-homomorphism from [Formula: see text] to [Formula: see text] can be extended to a right [Formula: see text]-homomorphism from [Formula: see text] to [Formula: see text].

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12

GABBAY, MURDOCH J., and PETER H. KROPHOLLER. "Imaginary groups: lazy monoids and reversible computation." Mathematical Structures in Computer Science 23, no. 5 (May 15, 2013): 1002–31. http://dx.doi.org/10.1017/s0960129512000849.

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We use constructions in monoid and group theory to exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms and the category of partially ordered groups and group homomorphisms such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups.We introduce the new notion of a lazy homomorphism for a function f between partially ordered monoids such that f(m ○ m′) ≤ f(m) ○ f(m′).Every monoid can be endowed with the discrete partial ordering (m ≤ m′ if and only if m=m′), so our constructions provide a way of embedding monoids into groups. A simple counterexample (the two-element monoid with a non-trivial idempotent) and some calculations show that one can never hope for such an embedding to be a monoid homomorphism, so the price paid for injecting a monoid into a group is that we must weaken the notion of a homomorphism to this new notion of a lazy homomorphism.The computational significance of this is that a monoid is an abstract model of computation – or at least of ‘operations’ – and, similarly, a group models reversible computations/operations. With this reading, the adjunction with its injective unit gives a systematic high-level way of faithfully translating an irreversible system into a ‘lazy’ reversible one.Informally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add ‘undo’ (in the sense of ‘control-Z’).

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13

rajan, K. Sunder, A. Senthil kumar, and R. Muthu raj. "Homomorphism and Anti Homomorphism of L-Fuzzy Quotient ℓ-Group." International Journal of Mathematics Trends and Technology 30, no. 1 (February 25, 2016): 39–42. http://dx.doi.org/10.14445/22315373/ijmtt-v30p507.

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14

Steinberg, Benjamin. "Inverse semigroup homomorphisms via partial group actions." Bulletin of the Australian Mathematical Society 64, no. 1 (August 2001): 157–68. http://dx.doi.org/10.1017/s0004972700019778.

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This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.

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15

Moore, Robert C. "Visualizing the Group Homomorphism Theorem." College Mathematics Journal 26, no. 2 (March 1995): 143. http://dx.doi.org/10.2307/2687369.

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16

Moore, Robert C. "Visualizing the Group Homomorphism Theorem." College Mathematics Journal 26, no. 2 (March 1995): 143. http://dx.doi.org/10.1080/07468342.1995.11973686.

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17

Katamadze, R. D. "Homology and cohomology groups of the group homomorphism." Journal of Mathematical Sciences 152, no. 3 (July 2008): 323–29. http://dx.doi.org/10.1007/s10958-008-9071-x.

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18

Tlas, Tamer. "Big free groups are almost free." International Journal of Algebra and Computation 25, no. 05 (August 2015): 855–64. http://dx.doi.org/10.1142/s0218196715500216.

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It is shown that the big free group (the set of countably-long words over a countable alphabet) is almost free, in the sense that any function from the alphabet to a compact topological group factors through a homomorphism. This statement is in fact a simple corollary of the more general result proven below on the extendability of homomorphisms from subgroups (of a certain kind) of the big free group to a compact topological group.

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19

Tkachenko, Mikhail. "Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids." Axioms 9, no. 1 (February 18, 2020): 23. http://dx.doi.org/10.3390/axioms9010023.

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We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = ∏ i ∈ I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous homomorphism f : S → K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every continuous homomorphism f of S to K has a finite type, which means that f admits a continuous factorization through a finite subproduct of D. A similar conclusion is obtained for continuous homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results.

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20

BEN AMOR, MOHAMED AMINE, and KARIM BOULABIAR. "A GEOMETRIC CHARACTERIZATION OF RING HOMOMORPHISMS ON f-RINGS." Journal of Algebra and Its Applications 12, no. 08 (July 31, 2013): 1350042. http://dx.doi.org/10.1142/s0219498813500424.

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Let A be an f-ring with identity u and B be an archimedean f-ring. For every idempotent element w in B, let [Formula: see text] denote the set of all positive group homomorphisms ℌ : A → B with ℌ(u) = w. We prove that [Formula: see text] is a ring homomorphism if and only if ℌ is an extreme point of [Formula: see text]. As a consequence, we derive a characterization of ring homomorphisms in [Formula: see text] in terms of a Gelfand-type transform. Moreover, we show that ring homomorphisms in [Formula: see text] are, up to multiplicative constants, all the basic elements of the ℓ-group of all bounded group homomorphisms from A into ℝ.

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21

Lassueur, Caroline, and Jacques Thévenaz. "On the lifting of the Dade group." Journal of Group Theory 22, no. 3 (May 1, 2019): 441–51. http://dx.doi.org/10.1515/jgth-2018-0145.

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Abstract For the group of endo-permutation modules of a finite p-group, there is a surjective reduction homomorphism from a complete discrete valuation ring of characteristic 0 to its residue field of characteristic p. We prove that this reduction map always has a section which is a group homomorphism.

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22

KASSEL, CHRISTIAN. "ON AN ACTION OF THE BRAID GROUP B2g+2 ON THE FREE GROUP F2g." International Journal of Algebra and Computation 23, no. 04 (June 2013): 819–31. http://dx.doi.org/10.1142/s0218196713400110.

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We construct an action of the braid group B2g+2 on the free group F2g extending an action of B4 on F2 introduced earlier by Reutenauer and the author. Our action induces a homomorphism from B2g+2 into the symplectic modular group Sp 2g(ℤ). In the special case g = 2 we show that the latter homomorphism is surjective and determine its kernel, thus obtaining a braid-like presentation of Sp4(ℤ).

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23

Alkan, Mustafa, and Yilmaz Simsek. "The actions on the generating functions for the family of the generalized Bernoulli polynomials." Filomat 31, no. 1 (2017): 35–44. http://dx.doi.org/10.2298/fil1701035a.

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In this paper, we study the generalization Bernoulli numbers and polynomials attached to a periodic group homomorphism from a finite cyclic group to the set of complex numbers and derive new periodic group homomorphism by using a fixed periodic group homomorphism. Hence, we obtain not only multiplication formulas, but also some new identities for the generalized Bernoulli polynomials attached to a periodic group homomorphism.

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24

Mehmood, Faisal, Fu-Gui Shi, Khizar Hayat, and Xiao-Peng Yang. "The Homomorphism Theorems of M-Hazy Rings and Their Induced Fuzzifying Convexities." Mathematics 8, no. 3 (March 13, 2020): 411. http://dx.doi.org/10.3390/math8030411.

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In traditional ring theory, homomorphisms play a vital role in studying the relation between two algebraic structures. Homomorphism is essential for group theory and ring theory, just as continuous functions are important for topology and rigid movements in geometry. In this article, we propose fundamental theorems of homomorphisms of M-hazy rings. We also discuss the relation between M-hazy rings and M-hazy ideals. Some important results of M-hazy ring homomorphisms are studied. In recent years, convexity theory has become a helpful mathematical tool for studying extremum problems. Finally, M-fuzzifying convex spaces are induced by M-hazy rings.

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25

Zelenyuk, Yevhen. "Local Homomorphisms of Topological Groups." Journal of the Australian Mathematical Society 83, no. 1 (August 2007): 135–48. http://dx.doi.org/10.1017/s1446788700036430.

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AbstractA mapping f : G → s from a left topological group G into a semigroup S is a local homomorphism if for every x є G \ {e}, there is a neighborhood Ux of e such that f (xy) = f (x)f (y) for all y є Ux \ {e}. A local homomorphism f : G → S is onto if for every neighborhood U of e, f(U \ {e}) = S. We show that(1) every countable regular left topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto N,(2) it is consistent that every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto any countable semigroup,(3) it is consistent that every countable nondiscrete maximally almost periodic topological group admits a local homomorphism onto the countably infinite right zero semigroup.

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26

Johnson, F. E. A. "Homotopy classification and the generalized Swan homomorphism." Journal of K-theory 4, no. 3 (January 7, 2009): 491–536. http://dx.doi.org/10.1017/is008012013jkt072.

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AbstractIn his fundamental paper on group cohomology [20] R.G. Swan defined a homomorphism for any finite group G which, in this restricted context, has since been used extensively both in the classification of projective modules and the algebraic homotopy theory of finite complexes ([3], [18], [21]). We extend the definition so that, for suitable modules J over reasonably general rings Λ, it takes the form here is the quotient of the category of Λ-homomorphisms obtained by setting ‘projective = 0’. We then employ it to give an exact classification of homotopy classes of extensions 0 → J → Fn → … → F0 → F0 → M → 0 where each Fr is finitely generated free.

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27

FERNÓS, TALIA, and POOJA SINGLA. "ON IMAGES OF REAL REPRESENTATIONS OF SPECIAL LINEAR GROUPS OVER COMPLETE DISCRETE VALUATION RINGS." Glasgow Mathematical Journal 58, no. 1 (July 21, 2015): 263–72. http://dx.doi.org/10.1017/s001708951500018x.

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AbstractIn this paper, we investigate the abstract homomorphisms of the special linear group SLn($\mathfrak{O}$) over complete discrete valuation rings with finite residue field into the general linear group GLm($\mathbb{R}$) over the field of real numbers. We show that for m < 2n, every such homomorphism factors through a finite index subgroup of SLn($\mathfrak{O}$). For $\mathfrak{O}$ with positive characteristic, this result holds for all m ∈ ${\mathbb N}$.

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28

Snaith, V. P. "Hecke Algebras and Class-Group Invariants." Canadian Journal of Mathematics 49, no. 6 (December 1, 1997): 1265–80. http://dx.doi.org/10.4153/cjm-1997-062-x.

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AbstractLet G be a finite group. To a set of subgroups of order two we associate a mod 2 Hecke algebra and construct a homomorphism, ψ, from its units to the class-group of Z[G]. We show that this homomorphism takes values in the subgroup, D(Z[G]). Alternative constructions of Chinburg invariants arising fromthe Galois module structure of higher-dimensional algebraic K-groups of rings of algebraic integers often differ by elements in the image of ψ. As an application we show that two such constructions coincide.

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29

Reid, Colin D., and Phillip R. Wesolek. "Homomorphisms into totally disconnected, locally compact groups with dense image." Forum Mathematicum 31, no. 3 (May 1, 2019): 685–701. http://dx.doi.org/10.1515/forum-2018-0017.

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Abstract Let {\phi:G\rightarrow H} be a group homomorphism such that H is a totally disconnected locally compact (t.d.l.c.) group and the image of ϕ is dense. We show that all such homomorphisms arise as completions of G with respect to uniformities of a particular kind. Moreover, H is determined up to a compact normal subgroup by the pair {(G,\phi^{-1}(L))} , where L is a compact open subgroup of H. These results generalize the well-known properties of profinite completions to the locally compact setting.

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30

Satoh, Takao. "On the Universal SL2-Representation Rings of Free Groups." Proceedings of the Edinburgh Mathematical Society 60, no. 4 (January 30, 2017): 973–1001. http://dx.doi.org/10.1017/s0013091516000456.

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AbstractIn this paper, we give an explicit realization of the universal SL2-representation rings of free groups by using ‘the ring of component functions’ of SL(2, ℂ)-representations of free groups. We introduce a descending filtration of the ring, and determine the structure of its graded quotients. Then we study the natural action of the automorphism group of a free group on the graded quotients, and introduce a generalized Johnson homomorphism. In the latter part of this paper, we investigate some properties of these homomorphisms from a viewpoint of twisted cohomologies of the automorphism group of a free group.

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31

SATOH, TAKAO. "The abelianization of the congruence IA-automorphism group of a free group." Mathematical Proceedings of the Cambridge Philosophical Society 143, no. 1 (July 2007): 255–56. http://dx.doi.org/10.1017/s0305004107000382.

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Çelik, Mehmet, Moges Shalla, and Necati Olgun. "Fundamental Homomorphism Theorems for Neutrosophic Extended Triplet Groups." Symmetry 10, no. 8 (August 3, 2018): 321. http://dx.doi.org/10.3390/sym10080321.

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In classical group theory, homomorphism and isomorphism are significant to study the relation between two algebraic systems. Through this article, we propose neutro-homomorphism and neutro-isomorphism for the neutrosophic extended triplet group (NETG) which plays a significant role in the theory of neutrosophic triplet algebraic structures. Then, we define neutro-monomorphism, neutro-epimorphism, and neutro-automorphism. We give and prove some theorems related to these structures. Furthermore, the Fundamental homomorphism theorem for the NETG is given and some special cases are discussed. First and second neutro-isomorphism theorems are stated. Finally, by applying homomorphism theorems to neutrosophic extended triplet algebraic structures, we have examined how closely different systems are related.

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33

Barlak, Selçuk, and Gábor Szabó. "Sequentially split ∗-homomorphisms between C*-algebras." International Journal of Mathematics 27, no. 13 (December 2016): 1650105. http://dx.doi.org/10.1142/s0129167x16501056.

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We define and examine sequentially split ∗-homomorphisms between C*-algebras and C*-dynamical systems. For a ∗-homomorphism, the property of being sequentially split can be regarded as an approximate weakening of being a split-injective inclusion of C*-algebras. We show for a sequentially split ∗-homomorphism that a multitude of C*-algebraic approximation properties pass from the target algebra to the domain algebra, including virtually all important approximation properties currently used in the classification theory of C*-algebras. We also discuss various settings in which sequentially split ∗-homomorphisms arise naturally from context. One particular class of examples arises from compact group actions with the Rokhlin property. This allows us to recover and extend the presently known permanence properties of Rokhlin actions with a unified conceptual approach and a simple proof. Moreover, this perspective allows us to obtain new results about such actions, such as a generalization of Izumi’s original [Formula: see text]-theory formula for the fixed point algebra, or duality between the Rokhlin property and approximate representability.

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34

Yaqoob, Naveed, and Shamsul Haq. "Generalized Rough Γ-Hyperideals in Γ-Semihypergroups." Journal of Applied Mathematics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/658252.

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Davvaz (2008) introduced the concept of set-valued homomorphism andT-rough sets in a group. In this paper, we consider the set-valued homomorphismTonΓ-semihypergroupHto interpret the lower and upper approximations. We study the roughness of(m,n)bi-Γ-hyperideals and(m,n)quasi-Γ-hyperideals in terms of set-valued homomorphisms, which are extended notions of(m,n)bi-Γ-hyperideals and(m,n)quasi-Γ-hyperideals ofΓ-semihypergroups.

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35

Selvakumar, R. "Unconventional construction of DNA codes: Group Homomorphism." Journal of Discrete Mathematical Sciences and Cryptography 17, no. 3 (May 4, 2014): 227–37. http://dx.doi.org/10.1080/09720529.2013.858476.

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36

Ismagilova, A. S. "A homomorphism of the group GL2(R)." Journal of Mathematical Sciences 144, no. 2 (July 2007): 3938–48. http://dx.doi.org/10.1007/s10958-007-0246-7.

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37

BHUTANI, KIRAN R., and JOHN N. MORDESON. "VAGUE GROUPS AND Ω-VAGUE GROUPS." New Mathematics and Natural Computation 01, no. 02 (July 2005): 229–42. http://dx.doi.org/10.1142/s1793005705000135.

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Given a group G, we show how one can define a vague group structure on G via a chain of subgroups of G. We discuss how a group homomorphism f from a vague group X onto a group Y induces a vague group structure on Y with f satisfying the vague homomorphism property. The notion of Ω-vague groups is introduced, where Ω is a fuzzy subset. The direct product G1 × G2 of two vague groups and the internal vague direct product of subgroups of a vague group is introduced.

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38

Zhu, Jun. "On Singular Braids." Journal of Knot Theory and Its Ramifications 06, no. 03 (June 1997): 427–40. http://dx.doi.org/10.1142/s0218216597000285.

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In Vassiliev theory, there is a natural monoid homomorphism from n-strand singular braids to the group algebra of n-strand braid group. J. Birman conjectured that this monoid homomorphism is injective. We show that the monoid homomorphism is injective on braids with up to three singularities and that Birman's conjecture is equivalent to that singular braids are distinguishable by Vassiliev braid invariants.

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39

Massuyeau, Gwénaël, and Takuya Sakasai. "Morita’s trace maps on the group of homology cobordisms." Journal of Topology and Analysis 12, no. 03 (October 10, 2018): 775–818. http://dx.doi.org/10.1142/s179352531950064x.

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Morita introduced in 2008 a [Formula: see text]-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His [Formula: see text]-cocycle contains all the “traces” of Johnson homomorphisms which he introduced 15 years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita’s [Formula: see text]-cocycle based on a simple and explicit construction. Our [Formula: see text]-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization of the group of homology cobordisms is non-trivial. Besides, we apply some of our algebraic methods to compare two natural filtrations on the automorphism group of a finitely-generated free group.

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40

Van Oystaeyen, Fred, and Yinhuo Zhang. "Embedding the Hopf Automorphism Group into the Brauer Group." Canadian Mathematical Bulletin 41, no. 3 (September 1, 1998): 359–67. http://dx.doi.org/10.4153/cmb-1998-048-8.

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AbstractLet H be a faithfully projective Hopf algebra over a commutative ring k. In [8, 9] we defined the Brauer group BQ(k, H) of H and an homomorphism π from Hopf automorphism group AutHopf(H) to BQ(k,H). In this paper, we show that the morphism π can be embedded into an exact sequence.

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41

Labruere, C. "Generalized Braid Groups and Mapping Class Gropus." Journal of Knot Theory and Its Ramifications 06, no. 05 (October 1997): 715–26. http://dx.doi.org/10.1142/s021821659700039x.

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Given a chord system of D2, we associate a generalized braid group, a surface and a homomorphism from this braid group to the mapping class group of the surface. We disprove a conjecture stated in an article by Perron and Vannier by showing that generally this homomorphism is not injective.

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42

Massuyeau, Gwénaël, and Jean-Baptiste Meilhan. "Characterization of Y2-Equivalence for Homology Cylinders." Journal of Knot Theory and Its Ramifications 12, no. 04 (June 2003): 493–522. http://dx.doi.org/10.1142/s0218216503002585.

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For Σ a compact connected oriented surface, we consider homology cylinders over Σ: these are homology cobordisms with an extra homological triviality condition. When considered up to Y2-equivalence, which is a surgery equivalence relation arising from the Goussarov-Habiro theory, homology cylinders form an Abelian group. In this paper, when Σ has one or zero boundary component, we define a surgery map from a certain space of graphs to this group. This map is shown to be an isomorphism, with inverse given by some extensions of the first Johnson homomorphism and Birman-Craggs homomorphisms.

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43

Karyati, Karyati, and Rifki Chandra Utama. "FUZZY RINGS AND ITS PROPERTIES." Jurnal Sains Dasar 5, no. 1 (January 20, 2017): 32. http://dx.doi.org/10.21831/jsd.v5i1.12662.

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Abstract One of algebraic structure that involves a binary operation is a group that is defined an un empty set (classical) with an associative binary operation, it has identity elements and each element has an inverse. In the structure of the group known as the term subgroup, normal subgroup, subgroup and factor group homomorphism and its properties. Classical algebraic structure is developed to algebraic structure fuzzy by the researchers as an example semi group fuzzy and fuzzy group after fuzzy sets is introduced by L. A. Zadeh at 1965. It is inspired of writing about semi group fuzzy and group of fuzzy, a research on the algebraic structure of the ring is held with reviewing ring fuzzy, ideal ring fuzzy, homomorphism ring fuzzy and quotient ring fuzzy with its properties. The results of this study are obtained fuzzy properties of the ring, ring ideal properties fuzzy, properties of fuzzy ring homomorphism and properties of fuzzy quotient ring by utilizing a subset of a subset level and strong level as well as image and pre-image homomorphism fuzzy ring. Keywords: fuzzy ring, subset level, homomorphism fuzzy ring, fuzzy quotient ring

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44

SATOH, TAKAO. "On the SL(2, C)-representation rings of free abelian groups." Mathematical Proceedings of the Cambridge Philosophical Society 167, no. 02 (May 28, 2018): 229–47. http://dx.doi.org/10.1017/s0305004118000300.

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AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

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45

Leiderman, Arkady, and Mikhail Tkachenko. "Separable Quotients of Free Topological Groups." Canadian Mathematical Bulletin 63, no. 3 (November 29, 2019): 610–23. http://dx.doi.org/10.4153/s0008439519000699.

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AbstractWe study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$-compact spaces, the class of connected locally connected spaces, and some others.We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.

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46

Rosendal, Christian. "Universally measurable subgroups of countable index." Journal of Symbolic Logic 75, no. 3 (September 2010): 1081–86. http://dx.doi.org/10.2178/jsl/1278682216.

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AbstractIt is proved that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group S∞ is continuous. It is also shown that a universally measurable homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous.

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47

Zhang, Xiaohong, Xiaoyan Mao, Florentin Smarandache, and Choonkil Park. "On Homomorphism Theorem for Perfect Neutrosophic Extended Triplet Groups." Information 9, no. 9 (September 18, 2018): 237. http://dx.doi.org/10.3390/info9090237.

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Some homomorphism theorems of neutrosophic extended triplet group (NETG) are proved in the paper [Fundamental homomorphism theorems for neutrosophic extended triplet groups, Symmetry 2018, 10(8), 321; doi:10.3390/sym10080321]. These results are revised in this paper. First, several counterexamples are given to show that some results in the above paper are not true. Second, two new notions of normal NT-subgroup and complete normal NT-subgroup in neutrosophic extended triplet groups are introduced, and their properties are investigated. Third, a new concept of perfect neutrosophic extended triplet group is proposed, and the basic homomorphism theorem of perfect neutrosophic extended triplet groups is established.

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48

Knop, Friedrich. "A Harish-Chandra Homomorphism for Reductive Group Actions." Annals of Mathematics 140, no. 2 (September 1994): 253. http://dx.doi.org/10.2307/2118600.

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Kim, S., and V. O. Manturov. "Artin’s braids, braids for three space, and groups Γn4 and Gnk." Journal of Knot Theory and Its Ramifications 28, no. 10 (September 2019): 1950063. http://dx.doi.org/10.1142/s0218216519500639.

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We construct a group [Formula: see text] corresponding to the motion of points in [Formula: see text] from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on [Formula: see text] strands to the product of copies of [Formula: see text]. We will also study the group of pure braids in [Formula: see text], which is described by a fundamental group of the restricted configuration space of [Formula: see text], and define the group homomorphism from the group of pure braids in [Formula: see text] to [Formula: see text]. At the end of this paper, we give some comments about relations between the restricted configuration space of [Formula: see text] and triangulations of the 3-dimensional ball and Pachner moves.

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50

Klug, Michael. "Functoriality of group trisections." Proceedings of the National Academy of Sciences 115, no. 43 (October 22, 2018): 10875–79. http://dx.doi.org/10.1073/pnas.1717167115.

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Building on work by Stallings, Jaco, and Hempel in three dimensions and a more recent four-dimensional analog by Abrams, Kirby, and Gay, we show how the splitting homomorphism and group trisection constructions can be extended to functors between appropriate categories. This further enhances the bridge between smooth four-dimensional topology and the group theory of free and surface groups.

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