PLEASE SCROLL DOWN FOR ARTICLE

Transcription

1 This article was downloaded by: [Sun Yat-Sen University] On: 1 October 2008 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Communications in Algebra Publication details, including instructions for authors and subscription information: Gröbner-Shirshov Basis for HNN Extensions of Groups and for the Alternating Group Yuqun Chen a ; Chanyan Zhong a a School of Mathematical Sciences, South China Normal University, Guangzhou, China Online Publication Date: 01 January 2008 To cite this Article Chen, Yuqun and Zhong, Chanyan(2008)'Gröbner-Shirshov Basis for HNN Extensions of Groups and for the Alternating Group',Communications in Algebra,36:1, To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Communications in Algebra, 36: , 2008 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / GRÖBNER SHIRSHOV BASIS FOR HNN EXTENSIONS OF GROUPS AND FOR THE ALTERNATING GROUP Yuqun Chen and Chanyan Zhong School of Mathematical Sciences, South China Normal University, Guangzhou, China Key Words: In this article, we generalize the Shirshov s Composition Lemma by replacing the monomial order for others. By using Gröbner Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained. Alternating group; Gröbner Shirshov basis; HNN extension; Normal form Mathematics Subject Classification: 20E06; 16S15; 13P PRELIMINARIES It is known that in the Gröbner Shirshov basis theory, the Shirshov s Composition Diamond Lemma (Shirshov, 1962/1999) plays an important role. In the Composition Diamond Lemma, the order is asked to be monomial. In this article, we generalize the Composition Diamond Lemma by replacing the monomial order for others. From this result, we show by direct calculations of compositions that the presentation of the HNN extension is a Gröbner Shirshov basis under an appropriate ordering of group words, in which the order is not monomial order. By the generalized composition lemma, we immediately obtain the Normal Form Theorem for HNN extensions. In fact, this is the first time to find a Gröbner Shirshov basis by using a nonmonomial order. HNN extensions of groups were first invented by Higman et al. (1949) and independently by Novikov in 1952 (see Novikov, 1952, 1954, 1955). The Normal Form Theorem for certain HNN extensions of groups was first established by Bokut (see Bokut, 1966/1967, 1967, 1968 and see also Kalorkoti, 1982). In general, the Normal Form Theorem for HNN extensions of groups was proved in the text of Lyndon and Schupp (1977). For the alternating group A n, a presentation was given in the monograph of Jacobson (1985, p. 71). However, we still do not know what is the normal form for A n. In this article, we give the normal form theorem of A n with respect to the above presentation. Received August 30, 2006; Revised December 31, Communicated by V. A. Artamonov. Address correspondence to Yuqun Chen, School of Mathematical Sciences, South China Normal University, Guangzhou , China; Fax: ; yqchen@scnu@edu.cn 94

3 GRÖBNER SHIRSHOV BASIS 95 We first cite some concepts and results from the literature. Let k be a field, k X the free associative algebra over k generated by X and X the free monoid generated by X, where the empty word is the identity which is denoted by 1. For a word w X, we denote the length of w by deg w. Let X be a well-ordered set. Let f = a X f a a k X with the leading word f, where f a k. We say that f is monic if f has coefficient 1. We denote supp f = a X f a 0. Definition 1.1 (Shirshov, 1962/1999, see also Bokut, 1972, 1976; Bokut and Shum 2005). Let f and g be two monic polynomials in k X. Then, there are two kinds of compositions: (1) If w is a word such that w = fb = aḡ for some a b X with deg f + deg ḡ > deg w, then the polynomial f g w = fb ag is called the intersection composition of f and g with respect to w. (2) If w = f = aḡb for some a b X, then the polynomial f g w = f agb is called the inclusion composition of f and g with respect to w. In the above case, the transformation f f g w = f agb is called the elimination of the leading word (ELW) of g in f. Definition 1.2 (Bokut, 1972, 1976, cf. Shirshov, 1962/1999). Let S k X and < a well order on X. Then the composition f g w is called trivial modulo S if f g w = i a i s i b i, where each i k, a i b i X and a i s i b i <w. If this is the case, then we write f g w 0 In general, for p q k X, we write p q mod S w mod S w which means that p q = i a i s i b i, where each i k a i b i X and a i s i b i <w. Definition 1.3 (Bokut, 1972, 1976, cf. Shirshov, 1962/1999). We call the set S with respect to the well order < a Gröbner Shirshov set (basis) in k X if any composition of polynomials in S is trivial modulo S. Remark. Usually, in the definition of the Gröbner Shirshov basis, the order is asked to be monomial. A well order < onx is monomial if it is compatible with the multiplication of words, that is, for u v X, we have u>v w 1 uw 2 >w 1 vw 2 for all w 1 w 2 X The following lemma was proved by Shirshov (1962/1999) for the free Lie algebras (with deg lex ordering) in 1962 (see also Bokut, 1972). Bokut (1976) specialized the approach of Shirshov to associative algebras (see also Bergman,

4 96 CHEN AND ZHONG 1978). For commutative polynomials, this lemma is known as the Buchberger s Theorem (see Buchberger, 1965), published in Buchberger (1970). Lemma 1.4 (Composition Diamond Lemma). Let A = k X S and < a monomial order on X. Then the following statements are equivalent: (i) S is a Gröbner Shirshov basis; (ii) For any f k X, 0 f Ideal S f = a sb for some s S, a b X ; (iii) The set is a linear basis of the algebra A. Red S = u X u a sb s S a b X If a subset S of k X is not a Gröbner Shirshov basis, then we can add to S all nontrivial compositions of polynomials of S, and by continuing this process (maybe infinitely) many times, we eventually obtain a Gröbner Shirshov basis S comp. Such a process is called the Shirshov algorithm. If S is a set of semigroup relations (that is, the polynomials of the form u v, where u v X ), then any nontrivial composition will have the same form. As a result, the set S comp also consists of semigroup relations. Let A = sgp X S be a semigroup presentation. Then S is a subset of k X and hence one can find a Gröbner Shirshov basis S comp. The last set does not depend on k, and as mentioned before, it consists of semigroup relations. We will call S comp a Gröbner Shirshov basis of A. This is the same as a Gröbner Shirshov basis of the semigroup algebra ka = k X S. 2. GENERALIZED COMPOSITION DIAMOND LEMMA In this section, we generalize the Composition Diamond Lemma which is useful in the sequel, by replacing the monomial order for others. The proof of the following lemma is essentially the same as in Bokut et al. (2000). For the sake of convenience, we give the details. Lemma 2.1 (Generalized Composition Diamond Lemma). Let S k X, A = k X S and < a well order on X such that: (A) asb = a sb for any a b X, s S; (B) for each composition s 1 s 2 w in S, there exists a presentation s 1 s 2 w = i i a i t i b i a i t i b i <w where t i S a i b i X i k such that for any c d X, we have ca i t i b i d < cwd (1)

5 GRÖBNER SHIRSHOV BASIS 97 Then, the following statements hold. (i) S is a Gröbner Shirshov basis; (ii) For any f k X, 0 f Ideal S f = a sb for some s S, a b X ; (iii) The set is a linear basis of the algebra A. Red S = u X u a sb s S a b X Proof. (i) is clear. Now, we prove (ii). Let n f = i a i s i b i i k s i S a i b i X i=1 Assume that w i = a i s i b i w 1 = w 2 = =w l >w l+1 We will use the induction on l and w 1 to prove that f = a sb, for some s S and a b X. If l = 1, then by (A), f = a 1 s 1 b 1 and hence the result holds. Assume that l 2. Then f = a 1 s 1 b a 2 s 2 b 2 a 1 s 1 b 1 + For w 1 = w 2, there are three cases to consider. Case 1. Assume that b 1 = bs 2 b 2 and a 2 = a 1 s 1 b. Then we have a 2 s 2 b 2 a 1 s 1 b 1 = a 1 s 1 b s 2 s 2 b 2 a 1 s 1 s 1 bs 2 b 2 For any t supp s 2 s 2, by (A), a 1 s 1 btb 2 = a 1 s 1 btb 2 <a 1 s 1 bs 2 b 2 = w 1 and similarly, we have a 1 t 1 bs 2 b 2 <w 1, for any t 1 supp s 1 s 1. Case 2. Assume that b 1 = bb 2, a 2 = a 1 a, s 1 b = as 2 and deg s 1 + deg s 2 > deg as 1. Then a 2 s 2 b 2 a 1 s 1 b 1 = a 1 as 2 s 1 b b 2 By (B), there exist j k, u j v j X, t j S such that u j t j v j <w= s 1 b, s 1 b as 2 = j j u j t j v j and a 1 u j t j v j b 2 <a 1 s 1 bb 2. Now, by (A), for any j, we have a 1 u j t j v j b 2 = a 1 u j t j v j b 2 <a 1 s 1 bb 2 = w 1. Case 3. Assume that b 2 = bb 1, a 2 = a 1 a, and s 1 = as 2 b. Then a 2 s 2 b 2 a 1 s 1 b 1 = a 1 as 2 b s 1 b 1 by (A) and (B), there exist j k, u j v j X, t j S such that u j t j v j <w= s 1, s 1 as 2 b = j j u j t j v j and for any j, we have a 1 u j t j v j b 1 = a 1 u j t j v j b 1 <a 1 s 1 b 1 = w 1. (iii) follows from (ii).

6 98 CHEN AND ZHONG 3. NORMAL FORM FOR HNN EXTENSION OF GROUP In this section, by using the Gröbner Shirshov basis, we provide a new proof of the normal form theorem of HNN extension of a group. Definition 3.1 (Higman et al., 1949; Novikov, 1952, 1954, 1955). Let G be a group and let A and B be subgroups of G with A B an isomorphism. Then the HNN extension of G relative to A B and is the group = gp G t t 1 at = a a A Let G = G 1 1, where G 1 = G\ 1 = g, and let G/A = g i A i I G/B = h j B j J where g i i I and h j j J are the coset representatives of A and B in G, respectively. We assume that all sets, I, J are well ordered and so are the sets g, g i i I, h j j J. Then we get a new presentation of the group as a semigroup: = sgp G 1 t t 1 gg = gg gt = g A t a g gt 1 = g B t 1 1 b g t t = 1 g g G 1 =±1 where gg G; g = g A a g, g A g i i I, g g A, a g A; g = g B b g, g B h j j J, g g B, b g B. Now, we order the set G in three different ways: (1) Let 1 <g <g < < be a well order of G. Then we denote this order by G > and call it an absolute order; (2) For any g g G, suppose that g = g A a g g = g A a g. Then g> A g if and only if g A a g > g A a g is ordered lexicographically (elements g A g A by I, elements a g a g by (1)). We denote this order by G > A and call it the A-order. In particular, if g g A, then g> A g A, for g A a g > g A 1, a g 1; (3) For any g g G, suppose that g = g B b g g = g B b g. Then g> B g if and only if g B b g > g B b g is ordered lexicographically (elements g B g B by J, elements b g b g by (1)). We denote this order by G > B and call it the B-order. Then we order the set G 1 in three different ways too: (1) The absolute order G 1 is deg lex order, to compare words g 1 g n n 0 first by length and then lexicographically using absolute order of G 1 ; (2) The A-order G 1 A is deg-lex A order, firstly to compare words g 1 g n n 0 by length, secondly for n 1, g 1 g n 1 lexicographically by absolute order, and finally the last elements g n by A-order; (3) The B-order G 1 B is similar to (2) replacing > A by > B. Each element in G 1 t t 1 has a unique form u = u 1 t 1 u2 t 2 uk t k uk+1, where each u i G 1, i =±1, k 0. Suppose that v = v 1 t 1 v2 v l t l vl+1 G 1 t t 1.

7 GRÖBNER SHIRSHOV BASIS 99 Then wt u = k t 1 t k u 1 u k u k+1 wt v = l t 1 t l v 1 v l v l+1 We define u v if wt u > wt v lexicographically, using the order of natural numbers and the following orders: (a) t>t 1 ; (b) u i > A v i if i = 1, 1 i k; (c) u i > B v i if i = 1, 1 i k; (d) u k+1 >v l+1 k = l, the absolute order of G 1. Now, we can easily verify the following lemma. Lemma 3.2. Let the order on G 1 t t 1 be defined as above. Then the order is a well order but not monomial, for example, g g does not necessarily imply that gt g t. Equipping with the above notation, we have the following lemma. Lemma 3.3. Let X = G 1 t t 1. Suppose that the order on X is defined as above and S = gg gg, gt g A t a g, gt 1 g B t 1 1 b g, t t 1 g g G 1, =±1 is as above too. Then S satisfies conditions (A) (B) in Lemma 2.1. Proof. For any c d X, suppose that c = c 1 t 1 cn t n cn+1, d = d 1 t 1 dm t m dm+1, c i d j G 1, i j =±1. We firstly check (A). Then there are four cases to consider. For example, the second case is for polynomials gt g A t a g, g g A. We need to prove that cgtd cg A t a g d for any c d. Since wt cgtd = n + m + 1 t 1 c 1 c n c n+1 g d 1 d m d m+1 wt cg A t a g d = n + m + 1 t 1 c 1 c n c n+1 g A a g d 1 d m d m+1 and c n+1 g> A c n+1 g A (since g> A g A ), we have cgtd cg A t a g d. Secondly, we check that (B) holds in Lemma 2.1. By noting that there is no inclusion compositions in S, we need only to consider the cases of intersection compositions. For any a b X, s 1 s 2 S, suppose that as 1 = s 2 b with deg s 1 + deg s 2 > deg as 1. Then, we consider the following cases: w = gg g gg t g g A gg t 1 g g B gt t t t t =±1 For example, the second case is as follows. Let s 1 = gg, s 2 = g t, w = gg t. Then, by noting that gg = gg A a gg = gg A Aa gg A a g implies that gg A = gg A A and a gg = a gg A a g, we know that

8 100 CHEN AND ZHONG s 2 s 1 w = gg t gg At a g = gg t gg A t a gg gg A gg A t a g gg A t a g gg A At a gg A a g. We denote s 1 = gg t gg A t a gg, s 2 = gg A gg A, b 2 = t a g, s 3 = gg A t a g gg A At a gg A a g. Clearly, s 1, s 2, s 3 S or 0 (if gg = gg A ). Since wt cs 1 d = n + m + 1 t 1 c 1 c n c n+1 gg d 1 d m d m+1 wt cs 2 b 2d = n + m + 1 t 1 c 1 c n c n+1 gg A a g d 1 d m d m+1 wt cs 3 d = n + m + 1 t 1 c 1 c n c n+1 gg A a g d 1 d m d m+1 wt cwd = n + m + 1 t 1 c 1 c n c n+1 gg d 1 d m d m+1 and c n+1 gg > A c n+1 gg, c n+1 gg A, c n+1 gg A, we have cwd cs 1 d, c s 2 b 2d, cs 3 d. All other cases are treated the same. The proof is finished. Now, by Lemmas 2.1 and 3.3, we obtain the following theorem. Theorem 3.4. A Gröbner Shirshov basis of HNN extension = gp G t t 1 at = a a A consists of the relations (3.1) gg = gg, (3.2) gt = g A t a g, where g = g A a g, g A g i i I, a g A, (3.3) gt 1 = g B t 1 1 b g, where g = g B b g, g B h j j J, b g B, (3.4) tt 1 = 1, t 1 t = 1, where g i i I and h j j J are the coset representatives of A and B = A in G, respectively. Thus S is a Gröbner Shirshov basis of the algebra k G 1 t t 1 with Red S = u G 1 t t 1 u a sb s S a b G 1 t t 1 which is the normal form of the HNN extension. From this result, the normal form theorem for HNN Extension of the Group G easily follows. Theorem 3.5 (The Normal Form Theorem for HNN Extension, Lyndon and Schupp, 1977, Theorem 4.2.1). Let = gp G t t 1 at = a a A be an HNN extension of group G. If g i i I, and h j j J are the sets of representatives of the left cosets of A and B = A in G, respectively, then every element w of has a unique representation w = g 1 t 1 gn t n gn+1 n 0, l =±1, where, for 1 l n, the following conditions are satisfied: (1) If l = 1, then g l g i i I ; (2) If l = 1, then g l h j j J ;

9 GRÖBNER SHIRSHOV BASIS 101 (3) There does not exist subwords tt 1 and t 1 t; (4) g n+1 is an arbitrary element of G. Remark. In Theorem of Lyndon and Schupp (1977), the right cosets were considered. We notice that the above Theorem 3.5 is essentially the same as Theorem in Lyndon and Schupp (1977). 4. NORMAL FORM FOR ALTERNATING GROUP In this section, we first find a Gröner Shirshov basis for the alternating group A n and then we give the normal form theorem of A n. Let S n be the group of the permutations of 1 2 n. Then the subset A n of all even permutations in S n is a normal subgroup of S n. We call A n the alternating group of degree n. The following presentation of A n was given in the monograph of Jacobson (1985, p. 71): A n = gp x i 1 i n 2 x 3 1 = 1 x i 1x i 3 = x 2 i = 1 2 i n 2 x i x j 2 = 1 1 i<j 1 j n 2 where x i = 12 i + 1 i + 2, i = 1 2 n 2, ij the transposition. Thus, we give a presentation of the group A n as a semigroup: A n = sgp x 1 1 x i 1 i n 2 x 1 x 1 1 = x 1 1 x 1 = x 3 1 = 1 x i 1x i 3 = x 2 i = 1 2 i n 2 x i x j 2 = 1 1 i<j 1 j n 2 We now order the generators in the following way: x 1 1 <x 1 <x 2 < <x n 2 Let X = x 1 1 x 1 x 2 x n 2. Then, with the above notations, we can order the words of X by the deg lex order, i.e., compare two words first by their degrees, then order them lexicographically when the degrees are equal. Clearly, this order is a monomial order. Now we define the words x ji = x j x j 1 x i where j>i>1 and x j1 = x j x j 1 x 1 =±1. The proof of the following lemma is straightforward. We omit the detail. Lemma 4.1. For =±1, the following relations hold in the alternating group A n : (4.1) x1 2 = x1 ; (4.2) xi 2 = 1 i>1 ; (4.3) x j x i = x i x j j 1 >i 2 ; (4.4) x j x1 = x 1 x j j>2 ; (4.5) x ji x j = x j 1 x ji j>i 2 ; (4.6) x j1 x j = x j 1 x j1 j>2 ;

10 102 CHEN AND ZHONG (4.7) x 2 x 1 x 2 = x1 x 2x1 ; (4.8) x1 x 1 = 1. Now, we can state the normal form theorem for the group A n. Theorem 4.2. Let A n = gp x i 1 i n 2 x 3 1 = 1 x i 1x i 3 = x 2 i = 1 2 i n 2 x i x j 2 = 1 1 i<j 1 j n 2 be the alternating group of degree n. Let S = x1 2 x 1 x2 i 1 i > 1 x j x i x i x j j 1 >i 2 x j x1 x 1 x j j > 2 x ji x j x j 1 x ji j>i 2 x j1 x j x j 1 x j1 j > 2 x 2 x 1 x 2 x1 x 2x1 x 1 x 1 1 =±1, where x ji = x j x j 1 x i, j>i>1, x i1 = x i x i 1 x 1, =±1. Then: (i) S is a Gröbner Shirshov basis of the alternating group A n ; (ii) Every element w of A n has a unique representation w = x 1j1 x 2j2 x n 2jn 2, where x tt = x t t > 1, x ii+1 = 1, x i1 = x i x i 1 x 1, 1 j i i + 1, 1 i n 2, = ±1 (here we use x i1 instead of x i1 ). Proof. By Lemma 4.1, it is easy to see that every element w of A n has a representation w = x 1j1 x 2j2 x n 2jn 2 j i i + 1. Here x 1j1 may have 3 possibilities 1 x 1 x 1 1 x 2j 2 4 possibilities, and generally, x iji i + 2 possibilities, 1 i n 2. So there are n!/2 words. From this it follows that each representation is unique since A n =n!/2. On the other hand, it is clear that Red S consists of the same words w. Hence, by Composition Diamond Lemma, S is a Gröbner Shirshov basis of the alternating group A n. Remark. According to Bokut and Shiao (2001), normal form in S n 1 is x 1j1 x 2j2 x n 2jn 2, but here x i1 = x i x i 1 x 1. ACKNOWLEDGMENT The authors would like to express their deepest gratitude to Professor L. A. Bokut for his kind guidance, useful discussions, and enthusiastic encouragement during his visit to the South China Normal University. The research is supported by the National Natural Science Foundation of China (Grant No ) and the Natural Science Foundation of Guangdong Province (Grant No ; ). REFERENCES Bergman, G. M. (1978). The diamond lemma for ring theory. Adv. in Math. 29: Bokut, L. A. (1966/1967). On one property of the Boone group. Algebra i Logika 5(5):5 23; II, 6(1): Bokut, L. A. (1967). On the Novikov groups. Algebra i Logika 6(1): Bokut, L. A. (1968). Degrees of unsolvability of the conjugacy problems for finitely presented groups, I, II. Algebra i Logika 7(5):4 70; 7(6):4 52.

11 GRÖBNER SHIRSHOV BASIS 103 Bokut, L. A. (1972). Unsolvability of the word problem, and subalgebras of finitely presented Lie algebras. Izv. Akad. Nauk. SSSR Ser. Mat. 36: Bokut, L. A. (1976). Imbeddings into simple associative algebras. Algebra i Logika 15: Bokut, L. A., Fong, Y., Ke, W.-F. (2000). Gröbner Shirshov bases and composition lemma for associative conformal algebras: An example. Contemporary Mathematics Vol. 264, AMS, pp Bokut, L. A., Shiao, L.-S. (2001). Gröbner Shirshov bases for coxeter groups. Communications in Algebra 29: Bokut, L. A., Shum, K. P. (2005). Gröbner and Gröbner Shirshov bases in algebra: an elementary approach. SEA Bull. Math. 29: Buchberger, B. (1965). An Algorithm for Finding a Basis for the Residue Class Ring of a Zero-Dimensional Polynomial Ideal [in German]. Ph.D. thesis, University of Innsbruck, Austria. Buchberger, B. (1970). An algorithmical criteria for the solvability of algebraic systems of equations [in German]. Aequationes Math. 4: Higman, G., Neumann, B. H., Neumann, H. (1949). Embedding theorems for groups. J. London Math. Soc. 24: Jacobson, N. (1985). Basic Algebra I. W. H. Freeman and Company. Kalorkoti, K. (1982). Decision problems in group theory. Proc. Lond. Math. Soc., III. Ser. 44: Lyndon, R. C., Schupp, P. E. (1977). Combinatorial Group Theory. Springer-Verlag. Novikov, P. S. (1952). On algorithmic unsolvability of the problem of identity. Dokl. Akad. Nauk SSSR 86: Novikov, P. S. (1954). Unsolvability of the conjugacy problem in the theory of groups. Izv. Akad. Nauk SSSR Ser. Mat. 18: Novikov, P. S. (1955). On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov 44:143. Shirshov, A. I. (1962/1999). Some algorithmic problem for Lie algebras. Sibirsk. Mat. Z. 3: (in Russian); English translation in SIGSAM Bull. 33(2):3 6.