Projection of a free product and group isomorphisms

Arshinov, M N

doi:10.1007/bf00970229

Cited by 4 publications

(12 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Now/s =7*=/since S is ^-unitary, so/5 G <j>. But since e < xx" 1 and <p E is an isomorphism,/ < g = &$"', so that / = f(ss~l) = (fs)s l G <i>. Therefore e G <x> and by the properties above e is a zero for <x>.…”

Section: If S Is E-unitary Andmentioning

confidence: 93%

“…(If one group is trivial then G gp H is isomorphic with the other.) In fact Holmes [10] and Arsinov [1] proved that each lattice isomorphism of G gp H upon a group K is induced by a unique isomorphism of G gp H upon K.…”

Section: Free Products Of Groupsmentioning

confidence: 99%

“…(I) a cyclic group; (II) the free elementary inverse semigroup I 3 r 9 1 ; (III) defined by the relations x n x~" = x""x", x n + m = x", n > 2, m > 0, whence S is completely semisimple, with a group kernel generated by x n + 1 x~" (as an inverse subsemigroup); P.R.Jones [4] (IV) bicyclic; or (V) defined by the relation (x"x~n)(x'"x") = x"x", n > 2, (or its dual) whence S has a bicyclic kernel, generated by x" + i x~" (or, dually, by x~i n+i) x").…”

Section: 10) That // S Is E-unitary and £(S) = £{T) Then T Is Also mentioning

confidence: 99%

“…Again denoting gO G by g, (g E G) we show first that g^1 = (g~!),. Clearly we may assume g ¥= e. Let gj~' = u, for some u G G. (So (uefu' l )<p = gf'Ag,; we must show u = g" 1 …”

mentioning

confidence: 99%

See 3 more Smart Citations

Inverse semigroups determined by their lattices of inverse subsemigroups

Jones

1981

J Aust Math Soc A

View full text Add to dashboard Cite

In a previous paper ([14]) the author showed that a free inverse semigroup W9 X is determined by its lattice t(^i x ) of inverse subsemigroups, in the sense that for any inverse semigroup T, tCSi x ) =* £(7") implies 9$ x =s T. (In fact, the lattice isomorphism is induced by an isomorphism of < 3$ x upon T.) In this paper the results leading up to that theorem are generalized (from completely semisimple to arbitrary inverse semigroups) and applied to various classes, including simple, fundamental and £-unitary inverse semigroups. In particular it is shown that the free product of two groups in the category of inverse semigroups is determined by its lattice of inverse subsemigroups.

show abstract

Section: If S Is E-unitary Andmentioning

confidence: 93%

Section: Free Products Of Groupsmentioning

confidence: 99%

Section: 10) That // S Is E-unitary and £(S) = £{T) Then T Is Also mentioning

confidence: 99%

“…Again denoting gO G by g, (g E G) we show first that g^1 = (g~!),. Clearly we may assume g ¥= e. Let gj~' = u, for some u G G. (So (uefu' l )<p = gf'Ag,; we must show u = g" 1 …”

mentioning

confidence: 99%

See 2 more Smart Citations

Inverse semigroups determined by their lattices of inverse subsemigroups

Jones

1981

J Aust Math Soc A

View full text Add to dashboard Cite

show abstract

“…It was proved by Sadovskii [31] that G gp Я is determined by its lattice of sub groups when G and Я are non-trivial. (If one group is trivial then G gp Я is isomorphic with the other|./In fact Holmes [10] and Arsinov [1] proved that each lattice iso morphism of G gp Я upon a group К is induced by a unique isomorphism of G gp Я upon K,…”

Section: Free Products Of Groupsmentioning

confidence: 99%

Inverse semigroups determined by their lattices of inverse subsemigroups

Jones¹

1981

Czech. Math. J.

View full text Add to dashboard Cite

show abstract

Lattice definability of matrix groups and certain other groups

Yakovlev¹

1986

Algebra and Logic

View full text Add to dashboard Cite

Let Z~G~ denote the lattice of the subgroups of the group~.The groups ~ and Gr are said to be lattice isomorphic if there exists an isomorphism ~ of the lattice ~G~ onto ~C~I~ If /~ is a subgroup of ~, then H f denotes the image of H under the lattice isomorphism ~ ; in particular, ~= ~IWe say that a group ~ is determined by the lattice of its subgroups or, briefly, it is lattice-determined, if ~ is isomorphic to ~I whenever ~(~ is isomorphic to ~I~r). The question of the definability of a group by the lattice of its subgroups is one of the fundamental problems in the theory of lattice isomorphisms of groups. Information on results obtained in this direction can be found in [I-4]. From more recent investigations we mention the results obtained by R. Schmidt; in his investigations [5,6] it is proved that the following groups are lattice-determined: firstly, the triply transitive and certain doubly transitive permutation groups of degree ~ , generated by involutions; secondly, the groups ~(~ over a skew field ~, where ~ ~ (for ~=~ I~I~3 ), and the groups ~(~ over a field ~ for I~I~ ; thirdly, the projective images of symplectic, orthogonal, unitary groups over a field with several additional restrictions; and, finally, the finite simple Suzuki groups, Ree groups, the Higman-Sims group, the Conway group, as well as some groups from other classes.In this paper we prove the lattice definability of groups of certain classes. If we oversimplify it somewhat, then these are the groups possessing a system V of generators such that each lattice isomorphism of the group ~1~f "~3 ~ is induced by a group isomorphism for any mutually distinct elements ~7,~z,~jE ~ , and also some groups of matrices of order ~# over an associative ring with identity, generated by transvections and, moreover, such that the additive group of the ring is either nonperiodic or is generated by elements of prime order; finally, these are some groups generated by involutions.More refined results are formulated in the theorems of Secs. 6 and 8. The proof of the theorem is carried out by a method in which the starting point consists of Theorem i.i and its corollary (Sec. i) and a fundamental role is played by Theorem 2.1 (Sec. 2). For applications Theorem 3.1 (Sec.3) is fundamental. In Secs. 4 and 5 we prove local and approximation theorems, which also play an important role in applications. -Basically, we make use of the standard notations. The meaning of the nonstandard notations is explained in the text at the appropriate place.

show abstract

Projection of a free product and group isomorphisms

Cited by 4 publications

References 0 publications

Inverse semigroups determined by their lattices of inverse subsemigroups

Inverse semigroups determined by their lattices of inverse subsemigroups

Inverse semigroups determined by their lattices of inverse subsemigroups

Lattice definability of matrix groups and certain other groups

Contact Info

Product

Resources

About