Groups with a Cyclic Group as Lattice-Homomorph

1948 ◽  
Vol 49 (2) ◽  
pp. 347
Author(s):  
Philip M. Whitman
Keyword(s):  
Cyclic Group

scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

On the Uniqueness of the Cyclic Group of Order n

1992 ◽  
Vol 99 (6) ◽  
pp. 545-547 ◽  
Author(s):  
Dieter Jungnickel
Keyword(s):  
Cyclic Group

scholarly journals A note on frame starters

1996 ◽  
Vol 53 (2) ◽  
pp. 293-297
Author(s):  
Cheng-De Wang
Keyword(s):  
Cyclic Group ◽  
Unique Element ◽  
Left Frame

We construct frame starters in Z2n − {0, n}, for n ≡ 0, 1 mod 4, where Z2n denotes the cyclic group of order 2n. We also construct left frame starters in Q2n − {e, αn}, where Q2n is the dicyclic group of order 4n and αn is the unique element of order 2 in Q2n.

scholarly journals Semigroups with few endomorphisms

1969 ◽  
Vol 10 (1-2) ◽  
pp. 162-168 ◽  
Author(s):  
Vlastimil Dlab ◽  
B. H. Neumann
Keyword(s):  
Cyclic Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Infinite Groups ◽  
Infinite Cyclic Group

Large finite groups have large automorphism groups [4]; infinite groups may, like the infinite cyclic group, have finite automorphism groups, but their endomorphism semigroups are infinite (see Baer [1, p. 530] or [2, p. 68]). We show in this paper that the corresponding propositions for semigroups are false.

scholarly journals The solution of length three equations over groups

1983 ◽  
Vol 26 (1) ◽  
pp. 89-96 ◽  
Author(s):  
James Howie
Keyword(s):  
Free Product ◽  
Cyclic Group ◽  
Normal Closure ◽  
Infinite Cyclic Group ◽  
Canonical Map ◽  
Equations Over Groups

Let G be a group, and let r = r(t) be an element of the free product G * 〈G〉 of G with the infinite cyclic group generated by t. We say that the equation r(t) = 1 has a solution in G if the identity map on G extends to a homomorphism from G * 〈G〉 to G with r in its kernel. We say that r(t) = 1 has a solution over G if G can be embedded in a group H such that r(t) = 1 has a solution in H. This property is equivalent to the canonical map from G to 〈G, t|r〉 (the quotient of G * 〈G〉 by the normal closure of r) being injective.

scholarly journals PSL(2, q) as an image of the extended modular group with applications to group actions on surfaces

1987 ◽  
Vol 30 (1) ◽  
pp. 143-151 ◽  
Author(s):  
David Singerman
Keyword(s):  
Free Product ◽  
Cyclic Group ◽  
Modular Group ◽  
Prime Power ◽  
Group Actions ◽  
Index 2 ◽  
Image Position

The modular group PSL(2, ℤ), which is isomorphic to a free product of a cyclicgroupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed that PSL(2, q) is an image of the modular group if q ≠ 9. (Here, as usual, q is a prime power.) The extended modular group PGL(2, ℤ) contains PSL{2, ℤ) with index 2. It has a presentationthe subgroup PSL(2, ℤ) being generated by UV and VW.

On multiplicative systems defined by generators and relations

1953 ◽  
Vol 49 (4) ◽  
pp. 579-589 ◽  
Author(s):  
Trevor Evans
Keyword(s):  
Automorphism Group ◽  
Finite Number ◽  
Cyclic Group ◽  
Complete Information ◽  
Isomorphism Problem ◽  
Infinite Cyclic Group ◽  
Generators And Relations ◽  
Decision Method ◽  
Multiplicative Systems ◽  
Finitely Related

The techniques developed in (9) are used here to study the properties of multiplicative systems generated by one element (monogenie systems). The results are of two kinds. First, we obtain fairly complete information about the automorphisms and endo-morphisms of free and finitely related loops. The automorphism group of the free monogenie loop is the infinite cyclic group, each automorphism being obtained by mapping the generator on one of its repeated inverses. A monogenie loop with a finite, non-empty set of relations has only a finite number of endomorphisms. These are obtained by mapping the generator on some of the components, or their repeated inverses, occurring in the relations. We use the same methods to solve the isomorphism problem for monogenie loops, i.e. we give a method for determining whether two finitely related monogenie loops are isomorphic. The decision method consists essentially of constructing all homomorphisms between two given finitely related monogenie loops.

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