scholarly journals Finitely presented algebras defined by permutation relations of dihedral type

2016 ◽  
Vol 26 (01) ◽  
pp. 171-202 ◽  
Author(s):  
Ferran Cedó ◽  
Eric Jespers ◽  
Georg Klein
Keyword(s):  
Normal Subgroup ◽  
Normal Form ◽  
Symmetric Group ◽  
Cyclic Group ◽  
Index Formula ◽  
Universal Group ◽  
Product Group ◽  
Unique Product ◽  
Dihedral Type ◽  
Finitely Presented

The class of finitely presented algebras over a field [Formula: see text] with a set of generators [Formula: see text] and defined by homogeneous relations of the form [Formula: see text], where [Formula: see text] runs through a subset [Formula: see text] of the symmetric group [Formula: see text] of degree [Formula: see text], is investigated. Groups [Formula: see text] in which the cyclic group [Formula: see text] is a normal subgroup of index [Formula: see text] are considered. Certain representations by permutations of the dihedral and semidihedral groups belong to this class of groups. A normal form for the elements of the underlying monoid [Formula: see text] with the same presentation as the algebra is obtained. Properties of the algebra are derived, it follows that it is an automaton algebra in the sense of Ufnarovskij. The universal group [Formula: see text] of [Formula: see text] is a unique product group, and it is the central localization of a cancellative subsemigroup of [Formula: see text]. This, together with previously obtained results on such semigroups and algebras, is used to show that the algebra [Formula: see text] is semiprimitive.

Unitary Representations of Generalized Symmetric Groups

1969 ◽  
Vol 21 ◽  
pp. 28-38 ◽  
Author(s):  
B. M. Puttaswamaiah
Keyword(s):  
Normal Subgroup ◽  
Symmetric Group ◽  
Permutation Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Symmetric Groups ◽  
Unitary Representations ◽  
Complex Field ◽  
Permutation Matrices ◽  
Image Position

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute withObviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) andthen {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).

VERIFYING THAT A GROUP IS VIRTUALLY FREE

1991 ◽  
Vol 01 (03) ◽  
pp. 339-351
Author(s):  
ROBERT H. GILMAN
Keyword(s):  
Finite Index ◽  
Free Subgroup ◽  
Universal Group ◽  
Finitely Presented Groups ◽  
Finite Presentation ◽  
Finitely Presented

This paper is concerned with computation in finitely presented groups. We discuss a procedure for showing that a finite presentation presents a group with a free subgroup of finite index, and we give methods for solving various problems in such groups. Our procedure works by constructing a particular kind of partial groupoid whose universal group is isomorphic to the group presented. When the procedure succeeds, the partial groupoid can be used as an aid to computation in the group.

scholarly journals Automorphisms of Regular Wreath Product -Groups

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Cyclic Group ◽  
Product Group ◽  
Finite Cyclic Group ◽  
Arbitrary Finite Group

We present a useful new characterization of the automorphisms of the regular wreath product group of a finite cyclic -group by a finite cyclic -group, for any prime , and we discuss an application. We also present a short new proof, based on representation theory, for determining the order of the automorphism group Aut(), where is the regular wreath product of a finite cyclic -group by an arbitrary finite -group.

scholarly journals On characters in the principal 2-block

1978 ◽  
Vol 25 (3) ◽  
pp. 264-268 ◽  
Author(s):  
Thomas R. Berger ◽  
Marcel Herzog
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Cyclic Group ◽  
Complex Number ◽  
Irreducible Character ◽  
Odd Order ◽  
A Finite Group

AbstractLet k be a complex number and let u be an element of a finite group G. Suppose that u does not belong to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(1) – X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G if and only if G/O(G) is isomorphic either to C2, a cyclic group of order 2, or to PSL (2, 2n), n ≧ 2.

scholarly journals Chain conditions and semigroup graded rings

1988 ◽  
Vol 45 (3) ◽  
pp. 372-380 ◽  
Author(s):  
E. Jespers
Keyword(s):  
Inverse Semigroup ◽  
Sufficient Conditions ◽  
Cancellative Semigroup ◽  
Graded Rings ◽  
Semigroup Rings ◽  
Graded Ring ◽  
Product Group ◽  
Chain Conditions ◽  
Unique Product

AbstractThe following questions are studied: When is a semigroup graded ring left Noetherian, respectively semiprime left Goldie? Necessary sufficient conditions are proved for cancellative semigroup-graded subrings of rings weakly or strongly graded by a polycyclic-by-finite (unique product) group. For semigroup rings R[S] we also give a solution to the problem in case S is an inverse semigroup.

EVANS' NORMAL FORM THEOREM REVISITED

2007 ◽  
Vol 17 (08) ◽  
pp. 1577-1592 ◽  
Author(s):  
JONATHAN D. H. SMITH
Keyword(s):  
Normal Form ◽  
Symmetric Group ◽  
Word Problem ◽  
Latin Squares ◽  
Form Theorem ◽  
Left Regular ◽  
Normal Form Theorem ◽  
Partial Latin Squares

Evans defined quasigroups equationally, and proved a Normal Form Theorem solving the word problem for free extensions of partial Latin squares. In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols. Triality leads to considerable simplifications in the proof of Evans' Normal Form Theorem, and makes it directly applicable to each of the six major varieties of quasigroups defined by subgroups of the symmetric group. Normal form theorems for the corresponding varieties of idempotent quasigroups are obtained as immediate corollaries.

ON THE AUTOMORPHISM GROUPS OF A FAMILY OF BINARY QUANTUM ERROR-CORRECTING CODES

2006 ◽  
Vol 04 (06) ◽  
pp. 1013-1022
Author(s):  
TAILIN LIU ◽  
FENGTONG WEN ◽  
QIAOYAN WEN
Keyword(s):  
Semidirect Product ◽  
Automorphism Groups ◽  
Error Correcting Code ◽  
Error Correcting Codes ◽  
Index Formula ◽  
Product Group ◽  
Simplex Code ◽  
Quantum Error ◽  
The Relationship ◽  
Free Element

Based on the classical binary simplex code [Formula: see text] and any fixed-point-free element f of [Formula: see text], Calderbank et al. constructed a binary quantum error-correcting code [Formula: see text]. They proved that [Formula: see text] has a normal subgroup H, which is a semidirect product group of the centralizer Z(f) of f in GLm(2) with [Formula: see text], and the index [Formula: see text] is the number of elements of Ff = {f, 1 - f, 1/f, 1 - 1/f, 1/(1 - f), f/(1 - f)} that are conjugate to f. In this paper, a theorem to describe the relationship between the quotient group [Formula: see text] and the set Ff is presented, and a way to find the elements of Ff that are conjugate to f is proposed. Then we prove that [Formula: see text] is isomorphic to S3 and H is a semidirect product group of [Formula: see text] with [Formula: see text] in the linear case. Finally, we generalize a result due to Calderbank et al.

scholarly journals Fitting quotients of finitely presented abelian-by-nilpotent groups

2014 ◽  
Vol 17 (1) ◽  
pp. 1-12
Author(s):  
J. R. J. Groves ◽  
Ralph Strebel
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Finitely Presented

Abstract.We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

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