Iterated crossed product of cyclic groups

Tam metin

(1)Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508 https://doi.org/10.1007/s41980-018-0103-0 ORIGINAL PAPER. Iterated Crossed Product of Cyclic Groups E. K. Çetinalp1 · E. G. Karpuz1 Received: 4 February 2017 / Accepted: 6 January 2018 / Published online: 21 June 2018 © Iranian Mathematical Society 2018. Abstract In Panaite [Iterated crossed products, J. Algebra Appl. 13(7), 14580036 (2014)], Panaite studied iterated crossed product construction from the point of algebraic structures. In this paper, we study iterated crossed product from the point of Combinatorial Group Theory and define a new version of the crossed product of groups. First, we give some conditions for this new product to be a group, then we obtain a presentation for iterated crossed product of cyclic groups. Additionally, using this presentation, we find a complete rewriting system and thus we obtain normal form structure of elements of this new group construction. This gives us the solvability of the word problem for this product. Keywords Crossed product · Iterated crossed product · Rewriting system · Normal form Mathematics Subject Classification Primary 16S15; Secondary 20E22 · 20M05. 1 Introduction and Preliminaries In Group Theory, it is important to classify the groups. For instance, in [7,10], the authors have identified new versions of crossed product of groups. However, some other part of the classification is based on the usage of automorphism groups (see, for example, [4,11]) and this gives an advantage to obtain some new groups in the meaning of products of groups. As a consequence of that, constructions such as direct and semidirect products of groups are current in mathematics. They are used when. Communicated by Hamid Mousavi.. B. E. G. Karpuz [email protected] E. K. Çetinalp [email protected]. 1. Department of Mathematics, Kamil Özda˘g Science Faculty, Karamano˘glu Mehmetbey University, Yunus Emre Campus, 70100 Karaman, Turkey. 123.

(2) 1494. Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. new groups are constructed that inherit some properties of initial groups and they are also used for some complex groups. In this paper, we follow this idea to get a new classification. As known crossed product construction appears in different areas of algebra such as Lie algebras, C ∗ -algebras and group theory. This product has also many applications in other fields of mathematics like group representation theory and topology. Here, by considering crossed product construction from view of Combinatorial Group Theory, we define an iterative version of this product. We call this product as iterated crossed product of groups. This product is more important than known group constructions since it contains direct, semidirect [8,9,14], twisted [16], knit [3] and crossed products of groups. One can also study this new product in many applications of Hopf algebra and C ∗ -algebra. In this paper, we study iterated crossed product construction in terms of Combinatorial Group Theory. Our beginning motivation is iterated semidirect product of free groups [6] and two-sided crossed product of cyclic groups [7,12]. By considering these papers, we want to have a more general group construction which includes a lot of group examples in algebra. Since crossed product construction contains semidirect product it is worth to study iterated crossed product contruction in terms of various properties. For example, in [15], Panaite studied iterated crossed product from the point of algebraic structures. In that paper, the author’s motivation was the so-called quasi-Hopf two-sided smash product on algebras. Let H and G be two groups. A crossed system of these groups is a quadruple (H , G, α, f ), where α : G → Aut(H ) and f : G × G → H are two maps (for the map α we use the notation α(g)(h) = g α h) such that the following compatibility conditions hold: g1 α (g2 α h) = f (g1 , g2 )((g1 g2 ) α h) f (g1 , g2 )−1 , f (g1 , g2 ) f (g1 g2 , g3 ) = (g1 α f (g2 , g3 )) f (g1 , g2 g3 ),. (1.1) (1.2). for all g1 , g2 , g3 ∈ G and h ∈ H . The crossed system (H , G, α, f ) is called normalized if f (1, 1) = 1. If (H , G, α, f ) is normalized crossed system then f (1, g) = f (g, 1) = 1 and 1 α h = h, for any g ∈ G and h ∈ H . While α(g) ∈ Aut(H ), we have that g α 1 = 1 and g α (h 1 h 2 ) = (g α h 1 )(g α h 2 ). f. Now let H #α G := H × G be a set with a binary operation defined by the formula: (h 1 , g1 ).(h 2 , g2 ) = (h 1 (g1 α h 2 ) f (g1 , g2 ), g1 g2 ), f. for all h 1 , h 2 ∈ H and g1 , g2 ∈ G. Then (H #α G, .) is a group with the unit 1 H # f G = α (1, 1) if and only if (H , G, α, f ) is a normalized crossed system [1]. In this case f the group H #α G is called the crossed product of H and G associated to the crossed system (H , G, α, f ).. 123.

(3) Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. 1495. The following result is one of the main applications of the crossed product construction. The proof of this result can be found in [1]. Proposition 1.1 Let E be a group, H be a normal subgroup of E and G be the quotient of E by H . Then there exist maps α : G → Aut(H ) and f : G × G → H such that f (H , G, α, f ) is normalized crossed system and E ∼ = (H #α G, ·). In this paper, since we will use complete rewriting system method to obtain normal form structure of elements of iterated crossed product, we give some information about complete rewriting system as in the following paragraphs. Let X be a set and let X ∗ be the free monoid that consists of all words obtained by the elements of X . A (string) rewriting system on X ∗ is a subset R ⊆ X ∗ × X ∗ and an element (u, v) ∈ R, also can be written as u → v, is called a rule of R. The idea for a rewriting system is an algorithm for substituting the right-hand side of a rule whenever the left-hand side appears in a word. In general, for a given rewriting system R, we write x → y for x, y ∈ X ∗ if x = uv1 w, y = uv2 w and (v1 , v2 ) ∈ R. Also, we write x →∗ y if x = y or x → x1 → x2 → · · · → y for some finite chain of reductions and ↔∗ is the reflexive, symmetric, and transitive closure of →. Furthermore, an element x ∈ X ∗ is called irreducible with respect to R if there is no possible rewriting (or reduction) x → y; otherwise x is called reducible. The rewriting system R is called • Noetherian if there is no infinite chain of rewritings x → x1 → x2 → · · · for any word x ∈ X ∗ , • Confluent if whenever x →∗ y1 and x →∗ y2 , there is a z ∈ X ∗ such that y1 →∗ z and y2 →∗ z, • Complete if R is both Noetherian and confluent. A critical pair of a rewriting system R is a pair of overlapping rules if one of the forms is satisfied. (i) (r1r2 , s), (r2 r3 , t)∈ R with r2 = 1 or (ii) (r1r2 r3 , s) (r2 , t)∈ R. A critical pair is resolved in R if there is a word z such that sr3 →∗ z and r1 t →∗ z in the first case or s →∗ z and r1 tr3 →∗ z in the second. A Noetherian rewriting system is complete if and only if every critical pair is resolved [17]. Knuth and Bendix have developed an algorithm for creating a complete rewriting system R which is equivalent to R, so that any word over X has an (unique) irreducible form with respect to R . By considering overlaps of left-hand sides of rules, this algorithm basically proceeds forming new rules when two reductions of an overlap word result in two distinct reduced forms. We finally note that the reader is referred to [5,13] and [17] for a detailed survey on (complete) rewriting systems. The organization of this paper is as follows: In the main section (Sect. 2), first, we define iterated crossed product of finite cyclic groups, then we present group structure of this new product. After that, we obtain a presentation and then we find a complete rewriting system for this product. As a result of this, by obtaining normal forms of elements of this group construction we get the solvability of the word problem. Throughout this paper, we order words in given alphabet in the deg-lex way by comparing two words first with their degrees (lengths), and then lexicographically when the lengths are equal. Additionally, the notations (i) ∩ ( j) and (i) ∪ ( j) denote the intersection and inclusion overlapping words of left-hand side of relations (i) and ( j), respectively.. 123.

(4) 1496. Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. 2 Iterated Crossed Product of Cyclic Groups We begin by defining iterated crossed product of finite cyclic groups. Let C1 , C2 , . . . , Cn be finite cyclic groups of order x1 , x2 , . . . , xn , respectively. A crossed system of these groups is a quadruple f. f. f. i+2 n−1 (Ci , Ci+1 #αi+1 i+1 C i+2 #αi+2 . . . #αn−1 C n , αi , f i ) (1 ≤ i ≤ n − 1),. f. f. f. i+2 n−1 where αi : Ci+1 #αi+1 i+1 C i+2 #αi+2 · · · #αn−1 C n → Aut(C i ) and. f. f. f. f. n−1 i+1 n−1 f i : (Ci+1 #αi+1 i+1 C i+2 · · · #αn−1 C n ) × (C i+1 #αi+1 C i+2 · · · #αn−1 C n ) → C i. are maps such that Eqs. (1.1), (1.2) and the following compatibility conditions hold: ⎫ x1,2 α1 (x2,2 α1 (x1,3 x2,3 α2 (· · · αn−2 (x1,n x2,n αn−1 x3,1 ) · · · )) ⎬ f 1 (x2,2 , x3,2 )) = x1,2 α1 (x1,3 α2 (· · · αn−2 (x1,n αn−1 [x2,2 α1 ⎭ (· · · αn−2 (x2,n αn−1 x3,1 ) · · · ) f 1 (x2,2 , x3,2 )]) · · · )).. (2.1). For the other condition, we use the notation f i (xm,i+1 , xn,i+1 ) instead of f i ((xm,i+1 , 1Ci+2 , . . . , 1Cn ), (xn,i+1 , 1Ci+2 , . . . , 1Cn )) to have more understandable expressions in multiplications. So, f i (xm,i+1 , xn,i+1 ) = xm,i (2 ≤ i ≤ n − 1),. (2.2). where x j,i is the j th element of i th group. The iterated crossed product of cyclic groups C1 , C2 , . . . , Cn associated to the crossed system with respect to the actions given above is the set C1 × C2 × · · · × Cn with the multiplication (x1,1 , x1,2 , . . . , x1,n )(x2,1 , x2,2 , . . . , x2,n ) = (x1,1 (x1,2 α1 (x1,3 α2 (· · · αn−2 (x1,n αn−1 x2,1 ) · · · ))) f 1 ( f 2 (· · · ( f n−1 (x1,n , x2,n ), x2,n−1 ), · · · ), x2,2 ), x1,2 x2,2 , x1,3 x2,3 , · · · , x1,n x2,n ) (2.3) for all x j,i ∈ Ci (1 ≤ i ≤ n). f f f We denote this product by C1 #α11 C2 #α22 · · · #αn−1 n−1 C n . f. f. f. 2.1 Group Structure for C1 #˛1 C2 #˛2 · · · #˛n−1 Cn 1. 2. n−1. In this subsection, we give the first main result of this paper. Theorem 2.1 Let C1 , C2 , . . . , Cn be finite cyclic groups. For all x j,i ∈ Ci (1 ≤ i ≤ n), let us consider the actions given in Eqs. (2.1) and (2.2). Then, the iterated normalized f f f crossed product C1 #α11 C2 #α22 · · · #αn−1 n−1 C n defines a group.. 123.

(5) Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508 f. 1497 f. f. Proof We verify the group properties for C1 #α11 C2 #α22 . . . #αn−1 n−1 C n . First, we show the associative property. To do that, for any x j,i ∈ Ci (1 ≤ i ≤ n), f f let (x1,1 , x1,2 , . . . , x1,n ), (x2,1 , x2,2 , . . . , x2,n ), (x3,1 , x3,2 , . . . , x3,n ) ∈ C1 #α11 C2 #α22 f · · · #αn−1 n−1 C n . So, we have [(x1,1 , x1,2 , . . . , x1,n )(x2,1 , x2,2 , . . . , x2,n )](x3,1 , x3,2 , . . . , x3,n ) = (x1,1 (x1,2 α1 (· · · αn−2 (x1,n αn−1 x2,1 ) · · · )) f 1 ( f 2 (· · · ( f n−1 (x1,n , x2,n ), x2,n−1 ), · · · ), x2,2 )(x1,2 x2,2 α1 (· · · αn−2 (x1,n x2,n αn−1 x3,1 ) · · · )) f 1 ( f 2 (· · · ( f n−1 (x1,n x2,n , x3,n )), · · · ), x3,2 ), x1,2 x2,2 x3,2 , · · · , x1,n x2,n x3,n ) = (x1,1 (x1,2 α1 (· · · αn−2 (x1,n αn−1 x2,1 ) · · · )) f 1 (x1,2 , x2,2 )(x1,2 x2,2 α1 (· · · αn−2 (x1,n x2,n αn−1 x3,1 ) · · · ))) f 1 (x1,2 x2,2 , x3,2 ), x1,2 x2,2 x3,2 , . . . , x1,n x2,n x3,n ) and (x1,1 , x1,2 , . . . , x1,n )[(x2,1 , x2,2 , . . . , x2,n )(x3,1 , x3,2 , . . . , x3,n )] = (x1,1 (x1,2 α1 (· · · αn−2 (x1,n αn−1 [x2,1 (x2,2 α1 (· · · αn−2 (x2,n αn−1 x3,1 ) · · · )) f 1 ( f 2 (· · · ( f n−1 (x2,n , x3,n ), x3,n−1 ), . . .), x3,2 )]) · · · )) f 1 ( f 2 (. . . f n−1 (x1,n , x2,n x3,n ), . . .), x2,2 x3,2 ), x1,2 x2,2 x3,2 , . . . , x1,n x2,n x3,n ) = (x1,1 (x1,2 α1 (· · · αn−2 (x1,n αn−1 x2,1 ) · · · )) (x1,2 α1 (· · · αn−2 (x1,n αn−1 [x2,2 α1 (· · · αn−2 (x2,n αn−1 x3,1 ) . . .) f 1 (x2,2 , x3,2 )]) . . .) f 1 (x1,2 , x2,2 x3,2 ), x1,2 x2,2 x3,2 , . . . , x1,n x2,n x3,n . Hence, the multiplication given by Eq. (2.3) is associative if and only if f 1 (x1,2 , x2,2 )(x1,2 x2,2 α1 (· · · αn−2 (x1,n x2,n αn−1 x3,1 ) · · · )) f 1 (x1,2 x2,2 , x3,2 ) = (x1,2 α1 (· · · αn−2 (x1,n αn−1 [x2,2 α1 (· · · αn−2 (x2,n αn−1 x3,1 ) · · · ) f 1 (x2,2 , x3,2 )]) · · · )) f 1 (x1,2 , x2,2 x3,2 ). Now, let us show that this equality holds. f 1 (x1,2 , x2,2 )(x1,2 x2,2 α1 (x1,3 α2 (· · · αn−2 (x1,n x2,n αn−1 x3,1 ) · · · ))) (by (1.1)) f 1 (x1,2 x2,2 , x3,2 ) = (x1,2 α1 (x2,2 α1 [x1,3 x2,3 α2 (· · · αn−2 (x1,n x2,n αn−1 x3,1 ) · · · )])) f 1 (x1,2 , x2,2 ) f 1 (x1,2 x2,2 , x3,2 ) (by (1.2)) = (x1,2 α1 (x2,2 α1 [x1,3 x2,3 α2 (· · · αn−2 (x1,n x2,n αn−1 x3,1 ) · · · )])). 123.

(6) 1498. Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. (x1,2 α1 f 1 (x2,2 , x3,2 )) f 1 (x1,2 , x2,2 x3,2 ) = (x1,2 α1 [x2,2 α1 (x1,3 x2,3 α2 (· · · αn−2 (x1,n x2,n αn−1 x3,1 ) · · · )) (by (2.1)) f 1 (x2,2 , x3,2 )]) f 1 (x1,2 , x2,2 x3,2 ) (by (2.1)) = x1,2 α1 (· · · αn−2 (x1,n αn−1 [x2,2 α1 (· · · αn−2 (x2,n αn−1 x3,1 ) · · · ) f 1 (x2,2 , x3,2 )]) · · · ) f 1 (x1,2 , x2,2 x3,2 ). Let 1C1 , 1C2 , . . . , 1Cn be the identity elements of groups C1 , C2 , . . . , Cn , respectively. We have (x1,1 , x1,2 , . . . , x1,n )(1C1 , 1C2 , . . . , 1Cn ) = (x1,1 (x1,2 α1 (· · · αn−2 (x1,n αn−1 1C1 ) · · · )) f 1 ( f 2 (· · · ( f n−1 (x1,n , 1Cn ), 1Cn−1 ), · · · ), 1C2 ), x1,2 , . . . , x1,n ) = (x1,1 1C1 , x1,2 , . . . , x1,n ) = (x1,1 , x1,2 , . . . , x1,n ) and (1C1 , 1C2 , . . . , 1Cn )(x1,1 , x1,2 , . . . , x1,n ) = (1C1 (1C2 α1 (· · · αn−2 (1Cn αn−1 x1,1 ) · · · )) f 1 ( f 2 (· · · ( f n−1 (1Cn , x1,n ), x1,n−1 ), . . .), x1,2 ), x1,2 , . . . , x1,n ) = (1C1 x1,1 , x1,2 , . . . , x1,n ) = (x1,1 , x1,2 , . . . , x1,n ). Finally, let us find inverse element of (x1,1 , x1,2 , . . . , x1,n ) f f ∈ C1 #α11 C2 · · · #αn−1 n−1 C n .. (x1,1 , x1,2 , . . . , x1,n )(x1,1 , x1,2 , . . . , x1,n ) = (1C1 , 1C2 , . . . , 1Cn ). ⇒ (x1,1 (x1,2 α1 (x1,3 α2 (· · · (x1,n αn−1 x1,1 ) · · · ))) f 1 ( f 2 (· · ·. ( f n−1 (x1,n , x1,n ), x1,n−1 ), . . .), x1,2 ), x1,2 x1,2 , . . . , x1,n x1,n ) = (1C1 , 1C2 , . . . , 1Cn ). ⇒ (x1,1 (x1,2 α1 (x1,3 α2 (· · · (x1,n αn−1 x1,1 ) · · · ))) f 1 (x1,2 , x1,2 ),. x1,2 x1,2 , . . . , x1,n x1,n ) = (1C1 , 1C2 , . . . , 1Cn ).. −1 −1 −1 −1 −1 Hence, we obtain x1,1 = f 1 (x1,2 , x1,2 )−1 (x1,2 α1 (x1,3 α2 (· · · (x1,n αn−1 x1,1 ) · · · ))). −1 and x1,i = x1,i (2 ≤ i ≤ n).. 123.

(7) Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508 f. f. 1499. f. 2.2 A Presentation for C1 #˛1 C2 #˛2 · · · #˛n−1 Cn 1. 2. n−1. In this subsection, we give the second main result of this paper which gives a presentation of iterated crossed product of cyclic groups. To do that, let Ci (1 ≤ i ≤ n) be finite cyclic groups presented by ai ; aixi = 1. f. f. Theorem 2.2 A group E is isomorphic to iterated crossed product C1 #α11 C2 #α22 · · · f #αn−1 n−1 C n if and only if E is a group generated by generators ai (1 ≤ i ≤ n) and relations ⎫ a1x1 = 1, ⎪ ⎬ xp a p = Wa1 Wa2 · · · Wa p−1 (2 ≤ p ≤ n), (2.4) ⎪ ⎭ −1 a j ai a j = Wa1 Wa2 . . . Wai (1 ≤ i ≤ n − 1, 2 ≤ j ≤ n, j − i ≥ 1), where Wak (1 ≤ k ≤ n) are positive words obtained by the powers of generators ak . Proof We make the proof by induction process. When n = 2, it follows readily that f the group C1 #α11 C2 has a presentation. a1 , a2 ; a1x1 = 1, a2x2 = a1i1 , a2−1 a1 a2 = a1i2 for all 0 ≤ i 1 , i 2 ≤ x1 − 1. Thus, relations (2.4) are hold for n = 2. Now we take n = 3 and suppose that the group E is isomorphic to iterated crossed f f product C1 #α11 C2 #α22 C3 . So, there exists a normal subgroup C1 of E such that E/C1 ∼ = f f C2 #α22 C3 . Let us consider crossed product C2 #α22 C3 with a presentation. a2 , a3 ; a2x2 = 1, a3x3 = a2k1 , a3−1 a2 a3 = a2k2 , ∼ C2 #αf2 C3 and C1 E, we obtain that where 0 ≤ k1 , k2 ≤ x2 − 1. Since E/C1 = 2 x2 t1 x3 −k1 t2 −k a2 = a1 , a3 a2 = a1 , a3−1 a2 a3 a2 2 = a1t3 and a2−1 a1 a2 = a1l1 , a3−1 a1 a3 = a1l2 (1 ≤ t1 , t2 , t3 , l1 , l2 ≤ x1 − 1), respectively. x Thus, a1x1 = 1, a pp = Wa1 Wa2 . . . Wa p−1 (2 ≤ p ≤ 3) and a −1 j ai a j = Wa1 Wa2 . . . Wai (1 ≤ i ≤ 2, 2 ≤ j ≤ 3, j − i ≥ 1). So, relations (2.4) are hold for n = 3 as well. Now we assume that the statement holds when n = k − 1 and iterated crossed f f f product C1 #α11 C2 #α22 . . . #αk−2 k−2 C k−1 has a presentation x. ai (1 ≤ i ≤ k − 1); a1x1 = 1, a pp = Wa1 Wa2 . . . Wa p−1 , a −1 j ai a j = Wa1 Wa2 . . . Wai ,. (2.5). where 1 ≤ i ≤ k − 2, 2 ≤ j ≤ k − 1, j − i ≥ 1. We prove that the statement holds when n = k using the assumption above. To do that, let us suppose that the group E is isomorphic to iterated crossed product. 123.

(8) 1500. Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. f f f ∼ C1 #α11 C2 #α22 . . . #αk−1 k−1 C k . So, there exists a normal subgroup C 1 of E such that E/C 1 = f k−1 f f2 f3 f f C2 #α2 C3 #α3 . . . #αk−1 Ck . Since E/C1 ∼ = C2 #α22 C3 #α33 . . . #αk−1 k−1 C k , we obtain that x. . . . Wa−1 Wa−1 = Wa1 (2 ≤ p ≤ k), a pp Wa−1 p−1 3 2.

(9). −1 −1 −1 a −1 j ai a j Wai . . . Wa3 Wa2 = Wa1 (2 ≤ i ≤ k − 1, 3 ≤ j ≤ k, j − i ≥ 1).. (2.6) In a similar way, from C1 E, we obtain l1 a −1 j a1 a j = a1 (2 ≤ j ≤ n).. (2.7). From Eqs. (2.6) and (2.7), we obtain the relations given in Eq. (2.4). Conversely, let us suppose that the relations in Eq. (2.4) are hold. We aim to show f f n−1 f −1 a l g ∈ C that Ct Ct #αtt Ct+1 #αt+1 t t+1 · · · #αn−1 C n = E t (1 ≤ t ≤ n − 1), that is g t (0 ≤ l ≤ xt − 1), for every g ∈ E t . Since g ∈ E t, we can take g = g1 g2 . . . gk −1 (k ∈ N) and gs ∈ at , at−1 , at+1 , at+1 , · · · , an , an−1 (0 ≤ s ≤ k). This gives that g −1 atl g = gk−1 . . . g2−1 g1−1 atl g1 g2 · · · gk . It is easy to see by a direct computation that g1−1 atl g1 ∈ Ct for every . −1 , . . . , an , an−1 and so, by induction it follows that g −1 atl g ∈ g1 ∈ at , at−1 , at+1 , at+1 Ct . Hence Ct E t . By a similar way, it can be showed that every element of group E t can be written as p pt+1 p . . . an n for pt ∈ Z (1 ≤ t ≤ n). Hence |E t | = xt xt+1 . . . xn and so |E/Ct | = at t at+1 xt+1 . . . xn for 1 ≤ t ≤ n − 1. So, E/Ct = Wat+1 Ct , Wat+2 Ct , . . . , Wan Ct , that is, the group E t has normal subgroup Ct . Therefore, by [2, Theorem 1.3], there exists f f t+2 f n−1 crossed system (Ct , Ct+1 #αt+1 t+1 C t+2 #αt+2 . . . #αn−1 C n , αt , f t ) such that f f n−1 Et ∼ = Ct #αftt Ct+1 #αt+1 t+1 . . . #αn−1 C n (1 ≤ t ≤ n).. . Hence the result. f. f. f. 2.3 A Complete Rewriting System for C1 #˛1 C2 #˛2 . . . #˛n−1 Cn 1. 2. f. n−1. f. f. To obtain a complete rewriting system for C1 #α11 C2 #α22 . . . #αn−1 n−1 C n , we should write the words Wak (1 ≤ k ≤ n) given in Eq. (2.4) more clearly. So, we use the notation aks (0 ≤ s ≤ xk − 1) instead of Wak (1 ≤ k ≤ n). Then, it is not hard to see that the f f f presentation for iterated crossed product C1 #α11 C2 #α22 . . . #αn−1 n−1 C n is as follows. k. x. p−1 ai (1 ≤ i ≤ n); a1x1 = 1, a1k1 a2k2 . . . a p−1 = a pp (2 ≤ p ≤ n),. a j a1l1 a2l2 . . . aili = ai a j (1 ≤ i ≤ n − 1, 2 ≤ j ≤ n, j − i ≥ 1), where 0 ≤ ki , li < xi (1 ≤ i ≤ n).. 123. (2.8).

(10) Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. 1501. Let us order the generators as an > an−1 > · · · > a2 > a1 . Now, we have the another main result of this paper as follows. Theorem 2.3 A complete rewriting system for the presentation given in Eq. (2.8) consists of the following relations: Case 1 Let x p ≥ k j (1 ≤ j ≤ p − 1). We have k. x. (1) a1x1 → 1,. p−1 (2) a pp → a1k1 a2k2 . . . a p−1 (2 ≤ p ≤ n),. (3) a j a1l1 a2l2 . . . aili → ai a j (1 ≤ i ≤ n − 1, 2 ≤ j ≤ n, j − i ≥ 1), k. k. p−1 p−1 (4) a p a1k1 a2k2 . . . a p−1 → a1k1 a2k2 . . . a p−1 a p (2 ≤ p ≤ n).. Case 2 Let x p <. . k j (1 ≤ j ≤ p − 1). We have k. (1) a1x1 → 1,. x. p−1 (2)a1k1 a2k2 . . . a p−1 → a pp (2 ≤ p ≤ n),. (3) a j a1l1 a2l2 . . . aili → ai a j (1 ≤ i ≤ n − 1, 2 ≤ j ≤ n, j − i ≥ 1). • For li + ki + ki+1 + · · · + k p−1 + 1 > l1 + k1 + x p , we have p−1 i+1 · · · a p−1 → a j a1l1 −k1 a pp (4) ai a j aiki −li ai+1 (2 ≤ i ≤ n − 1, 3 ≤ j ≤ n, 3 ≤ p ≤ n, i ≤ p − 1, j > i),. k. k. −k. x. p−1 p−1 (5) a j a1l1 −k1 a pp a p−1 a pp · · · aili → ai a j ( p < i < j, 2 ≤ p ≤ n, 3 ≤ i ≤ n, 4 ≤ j ≤ n).. l. x. l. • For li + ki + ki+1 + · · · + k p−1 + 1 ≤ l1 + k1 + x p , we have p−1 i+1 · · · a p−1 (4) a j a1l1 −k1 a pp → ai a j aiki −li ai+1. x. k. k. (2 ≤ i ≤ n − 1, 3 ≤ j ≤ n, 3 ≤ p ≤ n, i ≤ p − 1, j > i), −k p−1 l p ap. p−1 (5) a j a1l1 −k1 a pp a p−1. l. x. · · · aili → ai a j. ( p < i < j, 2 ≤ p ≤ n, 3 ≤ i ≤ n, 4 ≤ j ≤ n), p−1 i+1 lm (6) ai a j aiki −li ai+1 · · · a p−1 a pp · · · am → am a j (i < p < m < j, 2 ≤ i ≤ n − 1).. k. l. l. Proof Since we have the ordering an > an−1 > · · · > a2 > a1 , there are no infinite reduction steps for all overlapping words. Hence the rewriting system is Noetherian for both cases given in theorem. Now, to catch up the aim, we need to show that the confluent property holds for each cases separately. • In Case 1; we have the following overlapping words and corresponding critical pairs, respectively.. 123.

(11) 1502. Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. (1) ∩ (1) : a1x1 +1 , (a1 , a1 ), x +1. (2) ∩ (2) : a pp. k. k. p−1 p−1 (2 ≤ p ≤ n), (a1k1 a2k2 · · · a p−1 a p , a p a1k1 a2k2 · · · a p−1 ),. x. (2) ∩ (3) : a j j a1l1 a2l2 · · · aili (1 ≤ i ≤ n − 1, 2 ≤ j ≤ n, j − i ≥ 1), x −1. k. j−1 l1 l2 (a1k1 a2k2 · · · a j−1 a1 a2 · · · aili , a j j. (2) ∩ (4) (3) ∩ (1) (3) ∩ (2) (3) ∩ (4). ai a j ), k p−1 x p k1 k2 : a p a1 a2 · · · a p−1 (2 ≤ p ≤ n), k p−1 k1 k2 k p−1 k p−1 x −1 a1 a2 · · · a p−1 , a pp a1k1 a2k2 · · · a p−1 a p ), (a1k1 a2k2 · · · a p−1 x1 x1 −l1 : a j a1 (2 ≤ j ≤ n), (a1 a j a1 , a j ), li−1 xi l1 l2 : a j a1 a2 · · · ai−1 ai (2 ≤ i ≤ n − 1, 3 ≤ j ≤ n, j − i ≥ 1), li−1 k1 k2 ki−1 (ai a j aixi −li , a j a1l1 a2l2 · · · ai−1 a1 a2 · · · ai−1 ), k p−1 l p k1 k2 l1 l2 : a j a1 a2 · · · a p a1 a2 · · · a p−1 (3 ≤ j ≤ n, 2 ≤ p ≤ n − 1, j > p), l −1 k1 k2 a1 a2. k. p−1 , a j a1l1 a2l2 · · · a pp (a p a j a1k1 a2k2 · · · a p−1. (4) ∩ (1) : a p a1x1 (2 ≤ p ≤ n), (a1k1 a p a1x1 −k1 , a p ), k. k. p−1 · · · a p−1 ap). x. p−2 p−1 a p−1 (2 ≤ p ≤ n), (4) ∩ (2) : a p a1k1 a2k2 · · · a p−2. k. x. p−1 p−1 a p a p−1 (a1k1 a2k2 · · · a p−1. −k p−1. k. k. p−2 k1 k2 p−2 , a p a1k1 a2k2 · · · a p−2 a1 a2 · · · a p−2 ),. k. p−1 l1 l2 a1 a2 · · · aili (4) ∩ (3) : a p a1k1 a2k2 · · · a p−1 (3 ≤ p ≤ n, 1 ≤ i ≤ n − 1, p − 1 > i),. k. k. p−1 p−1 a p a1l1 a2l2 · · · aili , a p a1k1 a2k2 · · · a p−1 (a1k1 a2k2 · · · a p−1. −1. ai a p−1 ).. In fact, all these above critical pairs are resolved by reduction steps. We show one of them as an example. x. k. p−1 (2) ∩ (4) : a pp a1k1 a2k2 · · · a p−1 ,. k. k. x −1. k. p−1 k1 k2 p−1 p−1 (a1k1 a2k2 · · · a p−1 a1 a2 · · · a p−1 , a pp a1k1 a2k2 · · · a p−1 a p ), ⎧ k p−1 k1 k2 k p−1 ⎪ a1k1 a2k2 · · · a p−1 a1 a2 · · · a p−1 ⎪ ⎪ ⎪ ⎪ k p−1 k p−1 x −1 ⎪ ⎪ a pp a1k1 a2k2 · · · a p−1 a p → a1k1 a2k2 · · · a p−1 ⎪ ⎪ ⎨ k p−1 k p−1 x −1 k1 k2 a pp a1k1 a2k2 · · · a p−1 −→ a p a1 a2 · · · a p−1 a p ⎪ ⎪ ⎪ ⎪ k p−1 k1 k2 ⎪ ⎪ a1 a2 ···a p−1 ⎪ ⎪ ⎪ ⎩ → a k1 a k2 · · · a k p−1 a k1 a k2 · · · a k p−1 1 2 p−1 1 2 p−1. • For li + ki + ki+1 + · · · + k p−1 + 1 > l1 + k1 + x p in Case 2; we have the following overlapping words and corresponding critical pairs, respectively.. 123.

(12) Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. 1503. (1) ∩ (1) : a1x1 +1 , (a1 , a1 ), p−1 p−1 (2 ≤ p ≤ n), (a2k2 · · · a p−1 , a1x1 −k1 a pp ), (1) ∩ (2) : a1x1 a2k2 a3k3 · · · a p−1. k. k. x. k. (2) ∩ (3) : a1k1 a2k2 · · · a j j a1l1 a2l2 · · · aili (1 ≤ i ≤ n − 1, 2 ≤ j ≤ n, k −1. x. j+1 l1 l2 j − i ≥ 1), (a j+1 a1 a2 · · · aili , a1k1 a2k2 · · · a j j. a j a1k1 a2k2. k p−1 l p · · · a p−1 ap. ai a j ),. · · · aili. (2) ∪ (3) : (l1 ≥ k1 , lt = kt (2 ≤ t ≤ p − 2), l p−1 ≥ k p−1 , p − 1 ≤ i), −k p−1 l p a p · · · aili ), k p−1 ki+1 · · · a p−1 ( j > i, i ≤ p − 1), : a1k1 a2k2 · · · aiki a j aiki −li ai+1 k p−1 x xi+1 ki −li ki+1 k1 k2 ai+1 · · · a p−1 , a1 a2 · · · aiki −1 a j a1l1 −k1 a pp ), (ai+1 a j ai k x l p−1 −k p−1 l p : a1k1 a2k2 · · · a j j a1l1 −k1 a pp a p−1 a p · · · aili ( p < i < j), x j+1 l1 −k1 x p l p−1 −k p−1 l p k −1 (a j+1 a1 a p a p−1 a p · · · aili , a1k1 a2k2 · · · a j j ai a j ), : a j a1x1 (2 ≤ j ≤ n), (a1 a j a1x1 −l1 , a j ), k p−1 (l1 ≥ k1 , lt = kt (2 ≤ t ≤ i − 1), ki ≥ li , : a j a1l1 a2k2 · · · a p−1 k p−1 x ki −li · · · a p−1 , a j a1l1 −k1 a pp ), i ≤ p − 1), (ai a j ai k p−1 ki+1 · · · a p−1 (i < s, j, p − 1 ≥ i), : as a1l1 a2l2 · · · aili a j aiki −li ai+1 k p−1 x ki −li ki+1 ai+1 · · · a p−1 , as a1l1 a2l2 · · · aili −1 a j a1l1 −k1 a pp ), (ai as a j ai l x l p−1 −k p−1 l p a p · · · aili ( p < i < j < s), : as a1l1 a2l2 · · · a jj a1l1 −k1 a pp a p−1 l −1 x l p−1 −k p−1 l p (a j as a1l1 −k1 a pp a p−1 a p · · · aili , as a1l1 a2l2 · · · a jj ai a j ), k p−1 l1 l2 ki+1 · · · a p−1 a1 a2 · · · atlt : ai a j aiki −li ai+1 p−1 (ai a j , a j a1l1 −k1 a pp a p−1. x. (2) ∩ (4) (2) ∩ (5) (3) ∩ (1) (3) ∩ (2) (3) ∩ (4) (3) ∩ (5) (4) ∩ (3). l. ( j > i, p − 1 > t, i ≤ p − 1), p−1 (a j a1l1 −k1 a pp a1l1 a2l2 · · · atlt , ai a j aiki −li · · · a p−1. k. x. ki+1 ai a j aiki −li ai+1. −1. k p−1 l1 −k1 xt lt−1 −kt−1 lt · · · a p−1 a1 at at−1 at. (4) ∩ (5) : ( j > i, t < m < p − 1, i ≤ p − 1), t−1 (a j a1l1 −k1 a pp a1l1 −k1 atxt at−1. −kt−1 lt at. p−1 i+1 · · · a p−1 ai a j aiki −li ai+1. am a p−1 ),. x. k. l. k. −1. x l p−1 −k p−1 l p ap a j a1l1 −k1 a pp a p−1. (5) ∩ (3) : (i > t, p < i < j). at a p−1 ),. lm · · · am. lm · · · am ,. · · · aili a1l1 a2l2 · · · atlt ,. −k p−1 l p a p · · · aili −1 at ai ), x l p−1 −k p−1 l p ki+1 km−1 a p · · · aili at aiki −li ai+1 · · · am−1 : a j a1l1 −k1 a pp a p−1 ki+1 km−1 (i < t, i ≤ m − 1, p < i < j), (ai a j at aiki −li ai+1 · · · am−1 , li −1 l1 −k1 x p l p−1 −k p−1 l p l1 −k1 xm a p a p−1 a p · · · ai at a1 am ). a j a1 p−1 (ai a j a1l1 a2l2 · · · atlt , a j a1l1 −k1 a pp a p−1. x. (5) ∩ (4). l. 123.

(13) 1504. Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. All these above critical pairs are resolved by reduction steps. We show one of them as follows. p−1 i+1 · · · a p−1 , (3) ∩ (4) : as a1l1 a2l2 · · · aili a j aiki −li ai+1. k. k. p−1 i+1 (ai as a j aiki −li ai+1 · · · a p−1 , as a1l1 a2l2 · · · aili −1 a j a1l1 −k1 a pp ), ⎧ k p−1 ki+1 ⎪ ai as a j aiki −li ai+1 · · · a p−1 ⎪ ⎪ ⎪ x ⎨ → ai as ai−1 a j a1l1 −k1 a pp k p−1 li ki −li ki+1 l1 l2 as a1 a2 · · · ai a j ai ai+1 · · · a p−1 −→ x ⎪ ⎪ as a1l1 a2l2 · · · aili −1 a j a1l1 −k1 a pp ⎪ ⎪ ⎩ l1 −k1 x p −1 → ai as ai a j a1 ap. k. k. x. (We note that since we have the relation k p−1 x ki+1 ai a j aiki −li ai+1 · · · a p−1 = a j a1l1 −k1 a pp , according to the deg-lex way in this equation, it is written by k p−1 x ki+1 · · · a p−1 ←→ ai−1 a j a1l1 −k1 a pp as well.) a j aiki −li ai+1 • For li + ki + ki+1 + · · · + k p−1 + 1 ≤ l1 + k1 + x p in Case 2, we have the following overlapping words and corresponding critical pairs, respectively. (1) ∩ (1) : a1x1 +1 , (a1 , a1 ), p−1 p−1 (2 ≤ p ≤ n), (a2k2 a3k3 · · · a p−1 , a1x1 −k1 a pp ), (1) ∩ (2) : a1x1 a2k2 a3k3 · · · a p−1. k. k. x. k. (2) ∩ (3) : a1k1 a2k2 · · · a j j a1l1 a2l2 · · · aili (1 ≤ i ≤ n − 1, 2 ≤ j ≤ n, k −1. x. j+1 l1 l2 j − i ≥ 1), (a j+1 a1 a2 · · · aili , a1k1 a2k2 · · · a j j. (2) ∪ (3) :. a j a1k1 a2k2. k p−1 l p · · · a p−1 ap. · · · aili. ai a j ),. (l1 ≥ k1 , lt = kt (2 ≤ t ≤ p − 2), −k p−1 l p ap. p−1 l p−1 ≥ k p−1 , p − 1 ≤ i), (ai a j , a j a1l1 −k1 a pp a p−1. l. x. · · · aili ),. (2) ∩ (4) : a1k1 a2k2 · · · a j j a1l1 −k1 a pp (i < j, i ≤ p − 1), k. x. k −1. j+1 l1 −k1 (a j+1 a1 a pp , a1k1 a2k2 · · · a j j. x. x. k. (3) ∩ (1) (3) ∩ (2) (3) ∩ (4). 123. k. k. −k p−1 l p a p · · · aili ( p < i < j), x j+1 l1 −k1 x p l p−1 −k p−1 l p k −1 a1 a p a p−1 a p · · · aili , a1k1 a2k2 · · · a j j ai a j ), (a j+1 l ki+1 lm · · · a pp · · · am (2 ≤ i ≤ n − 1, : a1k1 a2k2 · · · aiki a j aiki −li ai+1 xi+1 ki −li ki+1 lm i < p < m < j), (ai+1 a j ai ai+1 · · · am , a1k1 a2k2 · · · aiki −1 am a j ), : a j a1x1 (2 ≤ j ≤ n), (a1 a j a1x1 −l1 , a j ), k p−1 (l1 ≥ k1 , lt = kt (2 ≤ t ≤ i − 1), ki ≥ li , : a j a1l1 a2k2 · · · a p−1 k p−1 x , a j a1l1 −k1 a pp ), i ≤ p − 1), (ai a j aiki −li · · · a p−1 l x : ak a1l1 a2l2 · · · a jj a1l1 −k1 a pp (i < j < k, i ≤ p − 1), l −1 k p−1 x ki+1 (a j ak a1l1 −k1 a pp , ak a1l1 a2l2 · · · a jj ai a j aiki −li ai+1 · · · a p−1 ),. p−1 (2) ∩ (5) : a1k1 a2k2 · · · a j j a1l1 −k1 a pp a p−1. (2) ∩ (6). p−1 i+1 ai a j aiki −li ai+1 · · · a p−1 ),. x. l.

(14) Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. 1505. −k p−1 l p a p · · · aili ( p < i < j < k), l −1 x l p−1 −k p−1 l p (a j ak a1l1 −k1 a pp a p−1 a p · · · aili , ak a1l1 a2l2 · · · a jj ai a j ), l ki+1 lm · · · a pp · · · am : ak a1l1 a2l2 · · · aili a j aiki −li ai+1 l ki+1 lm (i < k, i < p < m < j), (ai ak a j aiki −li ai+1 · · · a pp · · · am , li −1 l1 l2 ak a1 a2 · · · ai am a j ), x lm : a j a1l1 −k1 a pp a1l1 a2l2 · · · am ( j > i, p > m, i ≤ p − 1), k p−1 l1 l2 x −1 ki −li ki+1 lm ai+1 · · · a p−1 a1 a2 · · · am , a j a1l1 −k1 a pp am a p ), (ai a j ai x l p−1 −k p−1 l p lm a p · · · am (i < p < m < j) : a j a1l1 −k1 a pp a p−1 k p−1 l p−1 −k p−1 l p ki −li ki+1 lm ai+1 · · · a p−1 a p−1 a p · · · am , am a j ), (ai a j ai k p −l p k p+1 lm ls l1 −k1 x p a p at a p a p+1 · · · am · · · as : a j a1. p−1 (3) ∩ (5) : ak a1l1 a2l2 · · · a jj a1l1 −k1 a pp a p−1. l. (3) ∩ (6). (4) ∩ (3) (4) ∪ (5) (4) ∩ (6). x. l. ( p < m < s < t, i < p, j), k −l p k p+1 a p+1. p−1 i+1 · · · a p−1 at a pp (ai a j aiki −li ai+1. k. k. x −1. lm · · · am · · · asls ,. a j a1l1 −k1 a pp. as at ), l1 −k1 x p l p−1 −k p−1 l p a p a p−1 ap a j a1. (5) ∩ (3) : (i > m, p < i < j),. lm · · · aili a1l1 a2l2 · · · am. −k p−1 l p a p · · · aili −1 am ai ), x l p−1 −k p−1 l p a p · · · aklk a1l1 −k1 amxm : a j a1l1 −k1 a pp a p−1 (k > i, p < k < j, i ≤ m − 1), (ak a j a1l1 −k1 amxm , x l p−1 −k p−1 l p ki+1 km−1 a p · · · aklk −1 ai ak aiki −li ai+1 · · · am−1 ), a j a1l1 −k1 a pp a p−1 l −k x l k p−1 p−1 i+1 a pp · · · aili am aiki −li ai+1 · · · asls · · · atlt : a j a1l1 −k1 a pp a p−1 ki+1 (i < s < j < m), (ai a j am aiki −li ai+1 · · · asls · · · atlt , x l p−1 −k p−1 l p a p · · · aili −1 at am ), a j a1l1 −k1 a pp a p−1 l −1 l ki+1 lm l1 l2 · · · a pp a pp · · · am a1 a2 · · · atlt (m > t, : ai a j aiki −li ai+1 i < p < m < j), (am a j a1l1 a2l2 · · · atlt , l −1 l ki+1 lm −1 · · · a pp a pp · · · am at am ), ai a j aiki −li ai+1 lp ki −li ki+1 lm l1 −k1 xt ai+1 · · · a p · · · am a1 at : ai a j ai l1 −k1 xt (m > s, i < p < m < j), (am a j a1 at , lp kt−1 ki −li ki+1 lm −1 ks −ls ks+1 ai+1 · · · a p · · · am as am as as+1 · · · at−1 ), ai a j ai l p −1 l p ki −li ki+1 lm l1 −k1 xt lt−1 −kt−1 lt ai+1 · · · a p a p · · · am a1 at at−1 at · · · asls : ai a j ai lm p−1 , a j a1l1 −k1 a pp a p−1 (ai a j a1l1 a2l2 · · · am x. (5) ∩ (4). (5) ∩ (6). (6) ∩ (3). (6) ∩ (4). (6) ∩ (5). l. (t < s < m, i < p < m < j), −kt−1 lt at l −1 l k i+1 · · · a pp a pp ai a j aiki −li ai+1 t−1 (am a j a1l1 −k1 atxt at−1. l. · · · asls , lm −1 · · · am as am ).. 123.

(15) 1506. Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. All these above critical pairs are resolved by reduction steps. We show one of them as an example. i+1 lm l1 l2 (6) ∩ (3) : ai a j aiki −li ai+1 · · · a pp · · · am a1 a2 · · · atlt ,. l. k. i+1 lm −1 (am a j a1l1 a2l2 · · · atlt , ai a j aiki −li ai+1 · · · a pp · · · am at am ), ⎧ l1 l2 ⎪ am a j a1 a2 · · · atlt ⎪ ⎪ ⎪ −1 a a ⎪ ⎪ → am a j am t m ⎪ ⎨ l ki+1 k −l ki+1 lm l1 l2 ai a j aiki −li ai+1 · · · a pp · · · am a1 a2 · · · atlt −→ ai a j ai i i ai+1 ⎪ ⎪ ⎪ · · · a l p · · · a lm −1 a a ⎪ ⎪ t m m p ⎪ ⎪ ⎩ → a a a −1 a a. l. k. m j m. t m. (We note that since we have the relation am a1l1 a2l2 · · · atlt = at am , according to the −1 a a as well.) deg-lex way in this equation, it is written by a1l1 a2l2 · · · atlt ←→ am t m After all the above processes, we see that all critical pairs can be resolved. So, the rewriting system is complete. Hence the result. By Theorem 2.3, we have the following result. f. f. f. Corollary 2.4 Let us consider the word u ∈ C1 #α11 C2 #α22 · · · #αn−1 n−1 C n . Thus, the normal form C(u) of the word u is. C(u) = Wa1 Wa2 · · · Wan Wa1 Wa2 · · · Wan · · · Wa1 Wa2 · · · Wan ,. where Wai , Wai and Wai are reduced words generated by ai (1 ≤ i ≤ n). By Theorem 2.3 and Corollary 2.4, we have the following result. f. f. f. Corollary 2.5 The word problem for the group C1 #α11 C2 #α22 · · · #αn−1 n−1 C n is solvable. 2.4 Example Part We consider three cyclic groups and give an application of Theorems 2.2 and 2.3. Let C1 = a1 ; a17 = 1, C2 = a2 ; a25 = 1 and C3 = a3 ; a38 = 1. By Theorem f f 2.2, a presentation for iterated crossed product C1 #α11 C2 #α22 C3 can be given as a1 , a2 , a3 ;. a17 = 1, a25 = a13 , a38 = a12 a23 , a2 a14 = a1 a2 , a3 a15 = a1 a3 , a3 a16 a24 = a2 a3 .. Let us order the generators as a3 > a2 > a1 . By considering deg-lex ordering, we f f have the following complete rewriting system for C1 #α11 C2 #α22 C3 .. 123.

(16) Bulletin of the Iranian Mathematical Society (2018) 44:1493–1508. 1507. (1) a17 → 1, (2) a25 → a13 , (3) a38 → a12 a23 , (4) a2 a14 → a1 a2 , (5) a3 a15 → a1 a3 , (6) a3 a16 a24 → a2 a3 , (7) a2 a13 → a13 a2 , (8) a3 a12 a23 → a12 a23 a3 . Since we have the ordering a3 > a2 > a1 , there are no infinite reduction steps for all overlapping words. Hence the rewriting system is Noetherian. To show that the confluent property holds, we give the following overlapping words and corresponding critical pairs, respectively. (1) ∩ (1) : a18 , (a1 , a1 ),. (2) ∩ (2) : a26 , (a13 a2 , a2 a13 ),. (2) ∩ (4) : a25 a14 , (a13 a14 , a24 a1 a2 ),. (2) ∩ (7) : a25 a13 , (a13 a13 , a24 a13 a2 ,. (3) ∩ (3) : a39 , (a12 a23 a3 , a3 a12 a23 ), (3) ∩ (5) : a38 a15 , (a12 a23 a15 , a37 a1 a3 ), (3) ∩ (6) : a38 a16 a24 , (a12 a23 a16 a24 , a37 a2 a3 ), (3) ∩ (8) : a38 a12 a23 , (a12 a23 a12 a23 , a37 a12 a23 a3 ), (4) ∩ (1) : a2 a17 , (a1 a2 a13 , a2 ),. (5) ∩ (1) : a3 a17 , (a1 a3 a12 , a3 ),. (5) ∪ (6). : a3 a16 a24 , (a1 a3 a1 a24 , a2 a3 ),. (6) ∩ (2). : a3 a16 a25 , (a2 a3 a2 , a3 a16 a13 ),. (6) ∩ (4). : a3 a16 a24 a14 , (a2 a3 a14 , a3 a16 a23 a1 a2 ),. (6) ∩ (7). : a3 a16 a24 a13 , (a2 a3 a13 , a3 a16 a23 a13 a2 ),. (7) ∩ (1). : a2 a17 , (a13 a2 a14 , a2 ),. (7) ∪ (4). : a2 a14 , (a13 a2 a1 , a1 a2 ),. (8) ∩ (2). : a3 a12 a25 , (a12 a23 a3 a22 , a3 a12 a13 ),. (8) ∩ (4). : a3 a12 a23 a14 , (a12 a23 a3 a14 , a3 a12 a22 a1 a2 ),. (8) ∩ (7). : a3 a12 a23 a13 , (a12 a23 a3 a13 , a3 a12 a22 a13 a2 ).. All these above critical pairs are reduced by reduction steps. Here we note that new relations (7) and (8) are of the form (4) in Case 1 of Theorem 2.3. f f Now we take a word u ∈ C1 #α11 C2 #α22 C3 . C(u) is of the form. Wa1 Wa2 Wa3 Wa1 Wa2 Wa3 · · · Wa1 Wa2 Wa3 ,. where Wai , Wai and Wai are reduced words generated by ai (1 ≤ i ≤ 3). Acknowledgements The authors would like to thank Professor A. S. Çevik for his enthusiastic encouragement in writing up this paper. This work is supported by the Scientific Research Fund of Karamano˘glu Mehmetbey University Project No: 17-M-16.. References 1. Agore, A.L., Militaru, G.: Crossed product of groups, applications. Arab. J. Sci. Eng. 33, 1–17 (2008) 2. Agore, A.L., Fratila, D.: Crossed product of cyclic groups. Czechoslov. Math. J. 60, 889–901 (2010). 123.

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