[R] J. J. Rotman, An introduction to the theory of groups, 4th ed, Springer-Verlag, Grad Texts in Math 148, 1995 25  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on H, the Ljubljana graph is semisymmetric. Next recall that there is a natural embedding : N AutL, given by (a) v; b) v; a b) which maps N isomorphically onto the group of covering transformations. Clearly the extension N AutL 21 splits, by the Zassenhaus lemma [12]. For the the sake of completeness we give a self contained proof that Aut L N o# 21 , and give its explicit action on vertex set V (H) N of L. First, by x10 there exists a homomorphism # : 21 Aut N satisfying g L = L g for every g 2 21 . An explicit calculation involving the ....

J. J. Rotman, An Introduction to the Theory of Groups (4th ed.), Springer-Verlag, New York, 1995. 15

....monoid an element is completable iff it has an inverse. Hence, hLab(G) fm : m ffi m = e; m 2 Lab(G)gi is a finitely generated abelian group and G can be considered as a valence grammar over this group. By the well known fundamental theorem for finitely generated abelian groups, see e.g. [17], hLab(G)i is either finite or isomorphic to some (ZZ ; 0) k 1. It is easy to show that valences over finite abelian groups do not increase the power of the basic model. ut As mentioned in the definitions, regulation by valences is very similar to regulation by matrices and unordered ....

J. J. Rotman. An Introduction to the Theory of Groups. New York: Springer, 5th edition, 1995.

....the form vv Gamma1 or v Gamma1 v. It can be proven that the reduced string fl obtained in this way is independent of the order of such eliminations. In this way, a product on F (V ) is defined, and it is easily shown that F (V ) becomes a (non commutative) group, called the free group over V [15]. 2.3 Group computation We will say that an ordered pair GCS = V; R) is a group computation structure if: 1. V is a set, called the vocabulary, or the set of generators 2. R is a subset of F (V ) called the lexicon, or the set of relators. 1 1 For readers familiar with group theory, this ....

....origin; the origin chosen does not matter: any other origin leads to a conjugate relator, and this does not affect the notion of result. 6.5 Fundamental theorem of combinatorial group theory We are now able to state what J. Rotman calls the fundamental theorem of combinatorial group theory [15]. We give the theorem in a slightly extended form, adapted to the case of the normal sub monoid closure; the standard case of normal subgroup closure follows immediately by taking a set or relators containing r Gamma1 along with r. Theorem 1 Let GCS = V; R) be a group computation structure ....

Rotman, J. J.: 1994, An Introduction to the Theory of Groups. SpringerVerlag, fourth edition.

....variants of finite automata over groups are concerned we shall show their considerable lack of accepting power. 1 2 Preliminaries We assume the reader familiar with the basic concepts in automata and formal language theory and in the group theory. For further details, we refer to [4] and [8], respectively. For an alphabet Sigma, we denote by Sigma the free monoid generated by Sigma under the operation of concatenation; the empty string is denoted by and the semigroup Sigma Gamma f g is denoted by Sigma . The length of x 2 Sigma is denoted by jxj. Let K = M; ....

....2 fr; sg; is in L 1 , contradiction. A similar reasoning for the relation L 2 = 2 L(EFA(K) is left to the reader. 2 4 EFA over non abelian groups In this section, we restrict our investigation to the free groups, since for any (non abelian) group K there is a homomorphism from a free group to K [8]. In this way, we get a characterization of the context free languages class in terms of languages accepted by extended finite automata over the free group with just two generators [2] The free group with n generators is denoted by F n . Recall from [2] Theorem 8 The family of context free ....

J. J. Rotman, An Introduction to the Theory of Groups, Springer-Verlag,

....element xyx 1 y 1 . The group G is 2 step nilpotent if [x, y] is in the center of G for every pair of elements x, y # G . A group is 1 step nilpotent if and only if it is Abelian. For the more general definition of n step nilpotent groups, see any standard textbook on group theory, such as [38]. Let R = R , 0, 1) be a countably infinite ring with unit of characteristic p 2. The deg(R) computably presentable group GR is defined to be the set of all triples (a, b, c) a, b, c # R , with multiplication given by the formula (a, b, c) x, y, z) a x, b y, b x c z) ....

J. J. Rotman, An Introduction to the Theory of Groups, vol. 148 of Graduate Texts in Mathematics, 4th ed. (Springer-Verlag, New York, 1995). 52

....element xyx 1 y 1 . The group G is 2 step nilpotent if [x; y] is in the center of G for every pair of elements x; y 2 jGj. A group is 1 step nilpotent if and only if it is Abelian. For the more general de nition of n step nilpotent groups, see any standard textbook on group theory, such as [38]. Let R = jRj; 0; 1) be a countably in nite ring with unit of characteristic p 2. The deg(R) computably presentable group GR is de ned to be the set of all triples (a; b; c) a; b; c 2 jRj, with multiplication given by the formula (a; b; c) x; y; z) a x; b y; b x c z) It ....

J. J. Rotman, An Introduction to the Theory of Groups, vol. 148 of Graduate Texts in Mathematics, 4th ed. (Springer-Verlag, New York, 1995). 52

....N , m 2 M , v 2 Z k , i (A; 0) w; m; v) holds in G. 2 Before proving the main result of this section, we give some auxiliary results from the theory of commutative monoids. The rst lemma is known as the Fundamental Theorem for nitely generated Abelian (i.e. commutative) groups [20]. Sequential Grammars and Automata with Valences 15 Lemma 4.16 Any nitely generated commutative group is isomorphic to some group M Z k , where k 0 and M is a nite commutative group. Theorem 4.17 Let M be a commutative monoid and X 2 fREG;CF ; CFg. Then, the class L(Val; X; M) equals ....

J. J. Rotman. An Introduction to the Theory of Groups. New York: Springer, 5th edition, 1995.

....of SR symmetric groups for convenience. Assume that G is a symmetric group on R. We de ne a relation on R by the rule for r 1 ; r 2 2 R; r 1 r 2 i there exists 2 G such that (r 1 ) r 2 : is an equivalence on R. Its equivalence classes are called orbits of G. We refer the reader to [21] for more information. Now suppose that O 1 ; Om constitute a partition of R. O i is said to be proper if it contains more than one element. For any 1 2 SO1 ; m 2 SOm , let be the following permutation on R: r) i (r) if r 2 O i : By G = SO1 SOm we denote ....

Rotman, J. J., An Introduction to the Theory of Groups, Springer-Verlag, New York, 1994.

....this procedure, we finally obtain a valence grammar G 0 , equivalent to G, such that all elements of Lab(G 0 ) are completable in hLab(G 0 )i. If M is Abelian, then hLab(G 0 )i is an Abelian group. By the well known fundamental theorem for finitely generated Abelian groups, see e.g. [19], hLab(G 0 )i is isomorphic to some M Theta Z k , where M is a finite Abelian group, and k 0. For any valence grammar G over M Theta Z k , one can find by a triple construction an equivalent valence grammar over Z k . 2 As mentioned in the definitions, regulation by valences is very ....

J. J. Rotman. An Introduction to the Theory of Groups. New York: Springer, 5th edition, 1995.

....paper can be viewed as some representative examples. 3 2 Basic knowledge: symmetric groups and logic programs In this section, we describe some basic notions and results on which our following discussions are directly based. For those we do not de ne but use in the paper, the reader may consult [23] and [16] 2.1 Symmetric groups Let R be a non empty set. A permutation on R is a bijection from R to R. Let SR be the set of all permutations on R. SR forms a group under the operation of function composition. To be convenient, in the following we call any a subgroup of SR a symmetric group (on ....

Rotman, J. J., An Introduction to the Theory of Groups, Springer-Verlag, New York, 1994.

....effective sequence (A s ) s2 of finite sets of natural numbers, such that A is the pointwise limit of the A s . Assume further that for every s, A s is contained in [0; s] and 0 2 A s . For the following we need two notions from group theory, the definition of which we include (see, e.g. Rotman [Ro95]) Definition. 1. If A and B are groups and : B Aut(A) is a group automorphism (that is, B acts on A) we define the semidirect product of A and B (which also depends on ) to be the set A Theta B equipped with the group operation (a 1 ; b 1 ) Delta (a 2 ; b 2 ) a 1 Delta (a 2 ) b ....

Rotman, J., An introduction to the theory of groups, 4th edition, Graduate Texts in Mathematics No. 148, Springer-Verlag, New York, 1995.

....are equal. The converse is obvious. Let be a permutation that maps QN (v N ; q) to QN (w N ; s) i.e. QN (w N ; s) QN (v N ; q) where (QN (v N ; q) denotes the quorum set obtained by replacing i by (i) in the quorum set QN (v N ; q) From the theory of permutations (see for example [Rot84] we know that a permutation can be factorized into disjoint cycles and this factorization is unique except for the order in which the cycles are written. Also, each r cycle (i 1 i 2 . i r ) can be written as a product of r Gamma 1 transpositions as follows: i r i 1 ) i r Gamma1 i 1 ) ....

Joseph J. Rotman. An Introduction to the Theory of Groups. Allyn and Bacon, Inc., 1984.

....0 is a maximal pure independent subset consisting of homogenous elements, then fgA jA 2 S 0 g is a maximal pure independent subset. In order to show that Phi A2S0 hg A i is a basic subgroup of G, it is sufficient to proof that G= Phi A2S0 hg A i is divisible, which follows from Lemma 10.31 in [15]. Indeed, Lemma 10.31 is formulated for p groups but is valid for arbitrary abelian torsion groups. For this one has to derive Lemma 10.29 in [15] for abelian torsion groups and then the proof of the general case is entirely the same as the proof for p groups. 2) Since H is a direct sum of cyclic ....

....that Phi A2S0 hg A i is a basic subgroup of G, it is sufficient to proof that G= Phi A2S0 hg A i is divisible, which follows from Lemma 10.31 in [15] Indeed, Lemma 10.31 is formulated for p groups but is valid for arbitrary abelian torsion groups. For this one has to derive Lemma 10.29 in [15] for abelian torsion groups and then the proof of the general case is entirely the same as the proof for p groups. 2) Since H is a direct sum of cyclic groups, there exists a basis H 0 ae H such that H = Phi g2H0 hgi. Then S 0 = fg ord (g) jg 2 H 0 g ae B(G) is independent of type (ord (g) ....

Rotman J., An introduction to the theory of groups, Allyn and Bacon, 1984.

....homeomorphism used in topology are not equivalent, cf. 21] 3 Complexes and their covering In this section we recall quickly for the reader s convenience basic properties of coverings of simplicial complexes. All the facts exposed in this section are standard and can be found for example in [21, 20] (and in numerous textbooks on algebraic topology) A complex K (or more precisely an abstract simplicial complex) is a family of nonempty finite subset, called simplexes, of a set V = V (K) of vertices, such that for each vertex v of V , the singleton set fvg is a simplex of K, if A is a ....

.... for us) then the fundamental group does not depend on v, thus when v does not matter then we shall note the fundamental group as (K) Let us recall that a finitely presentable group (or fp group for short) is a group that can be represented by a finite number of generators and relations (see [20, 13]) Theorem 6. If K is a finite connected complex then the fundamental group (K) is finitely presentable, moreover its finite presentation can be deduced effectively from K. Conversely, if G is a finitely presentable group then there exists a finite connected complex K, that can be constructed ....

Joseph J. Rotman. An Introduction to the Theory of Groups. Springer Verlag, 1995.

....are equal. The converse is obvious. Let be a permutation that maps QN (r; V N ) to QN (s; W N ) i.e. QN (s; W N ) QN (r; V N ) where (QN (r; V N ) denotes the read coterie obtained by replacing i by (i) in the read coterie QN (r; V N ) From the theory of permutations (see for example [19]) we know that a permutation can be factorized into disjoint cycles and this factorization is unique except for the order in which the cycles are written. Also, each r cycle (i 1 i 2 : i r ) can be written as a product of r Gamma 1 transpositions as follows: i r i 1 ) i r Gamma1 i 1 ) ....

J. J. Rotman, An Introduction to the Theory of Groups. Allyn and Bacon, Inc., 1984.

....= CA(fiX n X) Proof. Define Upsilon : C (X) C (fiX n X) by Upsilon(f ) f fi j fiXnX ; Upsilon is the homomorphism that induces the isomorphism in Lemma 5. We define a function Upsilon # : PSA(X) CA(fiX n X) by A 7 f Upsilon(f ) f 2 Ag. Now, from the correspondence theorem ([RO], p. 26) from elementary algebra, Upsilon # is an order preserving bijection between the set of subalgebras of C (X) which contain ker( Upsilon) C 0 (X) and the subalgebras of C (fiX n X) However, for a locally compact space, an algebra A 2 CA(X) is in PSA(X) if and only if C 0 (X) ae ....

Joseph Rotman, An Introduction to the Theory of Groups, Wm. C. Brown Publishers, Dubuque, Iowa, 1988.

....an element is completable iff it has an inverse. Hence, hLab(G) fm 0 : m ffi m 0 = e; m 2 Lab(G)gi is a finitely generated abelian group and G can be considered as a valence grammar over this group. By the well known fundamental theorem for finitely generated abelian groups, see e.g. [17], hLab(G)i is either finite or isomorphic to some (ZZ k ; 0) k 1. It is easy to show that valences over finite abelian groups do not increase the power of the basic model. ut As mentioned in the definitions, regulation by valences is very similar to regulation by matrices and unordered ....

J. J. Rotman. An Introduction to the Theory of Groups. New York: Springer, 5th edition, 1995.

....with a permutation matrix (each row and column of the matrix has exactly one 1) as Q. A linear complement permutation with matrix P = Q j k) Q an n Theta n boolean matrix and k some n bit vector, has the same properties as the linear permutation corresponding to Q has. In the literature [2, 6, 14], the linear permutations are termed as the non singular linear transformations of the n dimensional vector space over the field GF (2) the field consisting of two elements: 0, and 1. There are exactly 2 n elements in this vector space, and each element corresponds to a processor index in ....

....all tags assigned to processor with address 0, under the linear transform T . From the property of linear transforms, T (0) 0; so, j ker T j 1. Also, T can be treated as a homomorphism from the group underlying the vector space of processor indices to itself. From the first isomorphism theorem [14], we get that each processor with at least one tag will have the same number of tags that processor 0 has under the linear transform T . Therefore, we have the following lemma. Lemma 2 If the tags are distributed among the processors such that some processors have one tag, some other processors ....

J. J. Rotman. An introduction to the theory of groups. Wm. C. Brown Publishers, third edition, 1988.

....Cl p (H) described here, can be found also in [4] with all the necessary justifications. It is adapted from a first algorithm due to Ribes and Zalesskii [5] The main property of p groups used here is the well known fact that in a finite p group, every maximal proper subgroup has index exactly p [6]. This is used to prove the following lemma. Lemma 3.1 Let H be a p open subgroup of F (A) Then there exists a morphism : F (A) Gamma Z=pZsuch that H ker . This in turn is used to characterize the p dense subgroups of F (A) that is, the subgroups which are not contained in a p open proper ....

....a subgroup By pro nil, we mean pro G nil , where G nil is the pseudovariety of finite nilpotent groups. We write Cl nil (H) for the pro nil closure of H . The main property of finite nilpotent groups used here is the well known fact that any finite nilpotent group is a direct product of p groups [6]. This is used to prove the following result. Proposition 3.4 Let H be a finitely generated subgroup of the free group. Then Cl nil (H) T p prime Cl p (H) Moreover, Cl nil (H) has finite rank. The second part of the statement follows from an earlier remark: since the p closures of H are ....

J. Rotman. An introduction to the theory of groups (4th edition), Springer, New York, 1995.

....automorphism of G which cyclically permutes the generators in S We give a partial answer to this problem in Proposition 4.13. Notice that it is a classical result of group theory that if G = S n with n 6= 2; 6, then the only group automorphisms of G are the inner automorphisms (see for example [23]) But this result is not sufficient since, for example, the hypercube H(d) is a Cayley graph on a proper subgroup of S d (see Appendix A) 2.4 Rotational graphs We say for short that a graph Gamma is rotational if there exist a group G and a set of generators S such that Gamma = Cay(G; S) and ....

J. J. Rotman. Introduction to the theory of groups, third edition. Allyn and Bacon (Boston), 1984.

.... a word of a group (as a product of its generators) is equal to the identity [Sti93, page 46] Novikov [Nov55] showed that the word problem is undecidable: there exists a group G such that no algorithm can decide whether a word of this group is equal to the identity (for a textbook proof see [Rot95, chapter 12]) Notice that the group G need not be part of the input, although for our purposes the weaker version of the result when the group is part of the input will suffice. We will make use of a stronger version of Proposition 2. We first observe that the contractibility problem is undecidable for ....

Joseph J. Rotman. An Introduction to the Theory of Groups. Graduate texts in mathematics. Springer-Verlag, New York, 4th edition, 1995.

....group H = hX j Si, we must effectively produce (a presentation of) a finitely presented group G such that G is torsion free iff H is. We note that the Higman Embedding Theorem embeds a recursively presented group H into a finitely presented group G. A careful analysis of the proof (e.g. in [Ro95]) shows that G is obtained from H by a finite sequence of HNN extensions. But, by Britton s Lemma, HNN extensions preserve torsion freeness, i.e. G is torsion free iff H is, as desired. Xi Proof of Proposition 2. Given (a presentation of) an infinitely recursively presented group H = hX j Si, ....

Rotman, J. J., An introduction to the theory of groups, Springer-Verlag, New York, 1995.

....membership problem for pseudovarieties of monoids of the form J fl m V, where J is the class of J trivial monoids [22] 1 The pro V topology on a group Here we present general results on profinite topologies on groups. For a general reference on the theory of groups, the reader is referred to [19]. For basic results on profinite groups, see [20, 16, 5, 6] A more general approach, involving profinite monoids, can be found in [1, 2] Profinite topologies on groups were introduced by M. Hall [9] 1.1 Definitions A pseudovariety of groups is a class of finite groups closed under taking ....

J. Rotman. An introduction to the theory of groups (4th edition), Springer, New York, 1995.

....a fixed prime, the intersection of the p Sylows of a group G constitutes its greatest normal subgroup in G p . On the other hand, the pseudovariety G n of all nilpotent groups is not closed under extension. Yet each finite group has a greatest normal nilpotent subgroup, called the Fitting subgroup [16]. Theorem 3.5. If H is a pseudovariety of groups such that each group admits a greatest congruence over H, then each semigroup admits a greatest congruence over N H, K H, D H and LH. Before we prove Theorem 3.5, let us recall a few facts about completely simple semigroups (see [7] Every ....

Rotman, J., "An introduction to the theory of groups", Wm. C. Brown Publishers, Dubuque, 1988.

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[R] J. J. Rotman, An introduction to the theory of groups, 4th ed, Springer-Verlag, Grad Texts in Math 148, 1995 25

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[R] J. J. Rotman, An introduction to the theory of groups, 4th ed, Springer-Verlag, Grad Texts in Math 148, 1995 25

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J.Rotman (1984) An introduction to the theory of groups, Allyn and Bacon.

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J. J. Rotman. An Introduction to the Theory of Groups, 4th ed. Springer-Verlag, New York NY, 1995.

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Rotman, J.J. (1995). An Introduction to the Theory of Groups. Berlin: SpringerVerlag.

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Rotman, "Introduction to the Theory of Groups", fourth edition, Graduate Texts in Mathematics 148, , Springer-Verlag, New York, 1995.

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Rotman, \Introduction to the Theory of Groups", fourth edition, Graduate Texts in Mathematics 148, , Springer-Verlag, New York, 1995.

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J.J. Rotman, An Introduction to the Theory of Groups, SpringerVerlag, 1995.

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J. J. Rotman, An Introduction to the Theory of Groups, vol. 148 of Graduate Texts in Mathematics, 4th ed. (Springer-Verlag, New York, 1995). 52

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Rotman, J. 1984. An Introduction to the Theory of Groups. Allyn and Bacon, Inc., Newton Massachusetts.

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J. J. Rotman, \An introduction to the theory of groups (fourth ed.)", Springer Verlag (1995), New York. 12

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Rotman, J. J. (1995). An Introduction to the Theory of Groups, fourth edition, Springer-Verlag.

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Joseph J. Rotman, An Introduction to the Theory of Groups, SpringerVerlag, New York, 1995.

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J.J. Rotman, An Introduction to the Theory of Groups, 3rd Edition, Wm. C. Brown Publishers, Dubuque, 1988.

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Joseph J. Rotman. An Introduction to the Theory of Groups (Springer, New York, 1995).

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Joseph J. Rotman, An Introduction to the Theory of Groups, SpringerVerlag, New York, 1995.

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J. J. Rotman, An Introduction to the Theory of Groups, 3rd. ed., Wm. C. Brown Publishers, Dubuque, IA, 1988.

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