on terraces for abelian groups

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Discrete Mathematics 305 (2005) 250 – 263 www.elsevier.com/locate/disc On terraces for abelian groups M.A. Ollis Marlboro College, South Road, Marlboro,Vermont, 05344, USA Received 7 June 2004; received in revised form 30 May 2005; accepted 1 July 2005 Abstract It is known that elementary abelian 2-groups of order at least 4 do not have terraces. Bailey’s Conjecture is that these are the only groups which do not. We show that abelian groups, except possibly those of order coprime to 3 whose Sylow 2-subgroup is elementary abelian of order an odd power of two, satisfy Bailey’s conjecture. A consequence of this is that all 2-nilpotent groups whose Sylow 2-subgroups are abelian, but not elementary abelian, have terraces. © 2005 Elsevier B.V.All rights reserved. Keywords: 2-Sequencing; Bailey’s Conjecture; Match-point; R-sequencing; Terrace 1. Introduction Let G be a group of order n with identity e. Let a be the linear arrangement (a 1 ,a 2 ,...,a n ) of the elements of G and define b = (b 1 ,b 2 ,...,b n1 ), where b i = a 1 i a i +1 . If b contains one occurrence of each involution of G and two occurrences from each set {g,g 1 : g 2 = e} then a is a terrace for G and b is a 2-sequencing of G; we call b the 2-sequencing associated with a. If a group has a terrace then it is terraced. A terrace is directed if its associated 2-sequencing contains exactly one occurrence of each non-identity element of G. In this case the 2-sequencing is called a sequencing. A group which has a directed terrace is sequenceable. Although terraces and directed terraces for arbitrary groups were not introduced until 1984 and 1961, respectively (see below), similar ideas for cyclic groups were implicitly used by Lucas in 1892 [14] and Williams in 1949 [19]. The following example gives a E-mail address: [email protected]. 0012-365X/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2005.07.007

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Page 1: On terraces for abelian groups

Discrete Mathematics 305 (2005) 250–263www.elsevier.com/locate/disc

On terraces for abelian groups

M.A. OllisMarlboro College, South Road, Marlboro, Vermont, 05344, USA

Received 7 June 2004; received in revised form 30 May 2005; accepted 1 July 2005

Abstract

It is known that elementary abelian 2-groups of order at least 4 do not have terraces. Bailey’sConjecture is that these are the only groups which do not. We show that abelian groups, exceptpossibly those of order coprime to 3 whose Sylow 2-subgroup is elementary abelian of order an oddpower of two, satisfy Bailey’s conjecture. A consequence of this is that all 2-nilpotent groups whoseSylow 2-subgroups are abelian, but not elementary abelian, have terraces.© 2005 Elsevier B.V. All rights reserved.

Keywords: 2-Sequencing; Bailey’s Conjecture; Match-point; R-sequencing; Terrace

1. Introduction

Let G be a group of order n with identity e. Let a be the linear arrangement (a1, a2, . . . , an)

of the elements of G and define b = (b1, b2, . . . , bn−1), where bi = a−1i ai+1. If b contains

one occurrence of each involution of G and two occurrences from each set {g, g−1 : g2 �= e}then a is a terrace for G and b is a 2-sequencing of G; we call b the 2-sequencing associatedwith a. If a group has a terrace then it is terraced.

A terrace is directed if its associated 2-sequencing contains exactly one occurrence ofeach non-identity element of G. In this case the 2-sequencing is called a sequencing. Agroup which has a directed terrace is sequenceable.

Although terraces and directed terraces for arbitrary groups were not introduced until1984 and 1961, respectively (see below), similar ideas for cyclic groups were implicitlyused by Lucas in 1892 [14] and Williams in 1949 [19]. The following example gives a

E-mail address: [email protected].

0012-365X/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.disc.2005.07.007

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M.A. Ollis / Discrete Mathematics 305 (2005) 250–263 251

terrace for each cyclic group Zn. When n is even, it is directed and was implicitly used byLucas (who gave credit to Walecki) to construct solutions to the Children’s Round-DanceProblem. When n is odd, it is not directed and was used by Williams [19] to constructquasi-complete latin squares of odd order for use in experimental designs.

Example 1. The linear arrangement (0, 1, n − 1, 2, n − 2, . . .) of the elements of Zn is aterrace. It is directed if and only if n is even. The associated 2-sequencing is (1, n − 2, 3,

n − 4, 5, . . .).

Directed terraces for arbitrary groups were introduced by Gordon in [9], where it wasshown that the existence of a directed terrace in a group of order n is sufficient to constructa complete latin square of order n (see [6] for more details about complete latin squares ingeneral and this result in particular).

Gordon also proved the following result.

Theorem 2 (Gordon [9]). An abelian group has a directed terrace if and only if it has asingle involution.

An abelian group having a single involution is equivalent to its Sylow 2-subgroup beingnon-trivial and cyclic. It is known [9] that the quaternion group of order 8 and the dihedralgroups of orders 8 and 10 are not sequenceable. Keedwell [13] conjectures that all othernon-abelian groups are sequenceable. Progress made towards proving this conjecture issurveyed in [15].

In 1984, Bailey [4] introduced terraces for arbitrary groups. In much the same way thatGordon showed that a directed terrace for a group of order n leads to a complete latin squareof order n, Bailey showed that a terrace for a group of order n is sufficient to construct aquasi-complete latin square of order n. Bailey also proved the following result.

Theorem 3 (Bailey [4]). All abelian groups of odd order are terraced.

Anderson and Ihrig [2] generalised this result as follows.

Theorem 4 (Anderson and Ihring [2]). Let G be a group with a normal subgroup N of oddorder. If G/N is terraced then G is terraced.

Corollary 5 (Anderson and Ihring [2]). All groups of odd order are terraced.

Proof. As all groups of odd order are soluble (by the Feit–Thompson Theorem [7]) and wehave a terrace for Zp whenever p is an odd prime (by Example 1) it follows from repeatedapplications of Theorem 4 that all groups of odd order are terraced. �

Bailey [4] also showed that elementary abelian 2-groups of order at least 4 are not terraced(a terrace for such a group must be directed and no directed terrace exists by Theorem 2).Bailey’s conjecture [4] states that all groups except the elementary abelian 2-groups of orderat least 4 are terraced. Progress made towards proving this conjecture is summarised in [15].

The only non-cyclic abelian groups of even order known to be terraced are those of theform Z2m×Z2, for m > 1 [3]; those of order at most 86, except possibly those of order 64 [1]

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252 M.A. Ollis / Discrete Mathematics 305 (2005) 250–263

and those which can be terraced using these two classes of groups and applying Theorem 4.In this paper we show that every abelian group whose Sylow 2-subgroup is not elementaryabelian of order an odd power (greater than 1) satisfies Bailey’s Conjecture. We also showthat every abelian group whose order is divisible by 3 satisfies Bailey’s Conjecture.

To construct terraces for these groups we use a generalisation of R-sequencings (to bedefined in the next section). This generalisation is interesting in its own right as it allows usto construct quasi-orthogonal latin squares (analogously to the construction of orthogonallatin squares from an R-sequencing). Quasi-orthogonal latin squares are described in [5].

We consider what consequences this has for Bailey’s conjecture in non-abelian groupsin Section 4. In particular, we show that a 2-nilpotent group whose Sylow 2-subgroup isabelian, but not elementary abelian, is terraced.

2. R-sequencings and R-terraces

In this section and the next we assume that all groups are abelian, and use additivenotation. Note that many of the concepts apply to more general groups, but the centralresult (Theorem 16) requires that the groups involved are abelian. We often consider listsof group elements [g1, g2, . . . , gm] which are to be considered as a cycle. In this case weshall write the list in square brackets and assume without comment that the subscripts areto be taken modulo the length of the list.

Let G be a group of order n and a be a cycle [a1, a2, . . . , an−1] containing each of thenon-identity elements of G. Define b = [b1, b2, . . . , bn−1], where bi = ai+1 − ai . If b alsoconsists of all of the non-identity elements of G then b is an R-sequencing of G. We calla a directed R-terrace for G. There is a unique R-sequencing associated with a directedR-terrace and also a unique directed R-terrace associated with an R-sequencing. A groupthat has an R-sequencing is called R-sequenceable.

This definition is the same as the one used by Headley [11]. The circular arrangement[b1, b2, . . . , bn−1] is an R-sequencing according to this definition if and only if the lin-ear arrangement (0, b1, b2, . . . , bn−1) is an R-sequencing according to the definition of[6, Chapter 2].

R-sequencings were introduced by Paige in [17], where it is shown that an R-sequencingfor a group of order n is sufficient to construct a pair of orthogonal latin squares of ordern. More details on R-sequencings and their connection to orthogonal latin squares may befound in [6, Chapters 2 and 3].

Friedlander et al. [8] give the following result.

Theorem 6 (Friedlander et al. [8]). The cyclic group of odd order, Z2r+1, has a directedR-terrace.

Proof. For 1� i�r define

ai ={

(i + 1)/2 if i is odd,

r + 1 − i/2 if i is even.

Then [a1, a2, . . . , ar , ar + r, ar−1 + r, . . . , a1 + r] is a directed R-terrace for Z2r+1.

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M.A. Ollis / Discrete Mathematics 305 (2005) 250–263 253

The associated R-sequencing [b1, b2, . . . , b2r ] is as follows. For 1� i�r − 1 take

bi ={

r − i if i is odd,

i − r if i is even

and br+i = −br−i . Take br = r and b2r = −r . �

The following lemma motivates our generalisation of R-sequencings. Let b be an R-sequencing for a group, with the notation above. If bi = −ai+1 then we say that i is aright match-point of the R-sequencing. Note that the idea of right match-points of an R-sequencing (and below, right match-points of R-2-sequencings) is similar to the idea ofright match-points of sequencings and 2-sequencings defined in [16].

Lemma 7. Let G be a group of order n. Let [a1, a2, . . . , an−1] be a directed R-terrace ofG with associated R-sequencing b = [b1, b2, . . . , bn−1]. If b has a right match-point thenG is terraced.

Proof. Let i be the right match-point of b. The sequence

(e, ai+1, ai+2, . . . , an−1, a1, a2, . . . , ai)

is a terrace for G.It has 2-sequencing (−bi, bi+1, bi+2, . . . , bn−1, b1, b2, . . . , bi−1). �

We now generalise the idea of an R-sequencing. Let G be a group of order n. Let abe a circular arrangement [a1, a2, . . . , an−1] of the non-identity elements of G and letb=[b1, b2, . . . , bn−1], where bi =ai+1 −ai . If b contains one occurrence of each involutionof G and two occurrences from each set {g, −g : 2g �= 0}, then we call b an R-2-sequencingof G. We call a an R-terrace for G. We say that a group which has an R-terrace is R-terraced.As with R-sequencings and directed R-terraces, an R-2-sequencing uniquely determines anR-terrace and vice versa.

The same argument for acquiring a pair of orthogonal latin squares from an R-sequencingmay be applied to an R-2-sequencing to obtain a pair of quasi-orthogonal latin squares. Thisuse of R-terraces, or any other, does not seem to appear in the literature. For more detailsabout quasi-orthogonal latin squares see [5].

Let b be an R-2-sequencing for a group, with the notation above. Similarly to the R-sequencing case, if bi=−ai+1 then we say that i is a right match-point of the R-2-sequencing.We can now give a more general result along the lines of Lemma 7.

Lemma 8. Let G be a group of order n. Let [a1, a2, . . . , an−1] be an R-terrace of G withassociated R-2-sequencing b = [b1, b2, . . . , bn−1]. If b has a right match-point then G isterraced.

Proof. Let i be the right match-point of b. As in Lemma 7, the sequence

(e, ai+1, ai+2, . . . , an−1, a1, a2, . . . , ai)

is a terrace for G.

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254 M.A. Ollis / Discrete Mathematics 305 (2005) 250–263

It has 2-sequencing (−bi, bi+1, bi+2, . . . , bn−1, b1, b2, . . . , bi−1). �

Example 9. Consider the group Z6 × Z2, with the element (x, y) abbreviated to xy. Thesequence [41, 51, 01, 20, 21, 10, 30, 11, 10, 31, 20] is an R-2-sequencing for Z6 × Z2 withassociated R-terrace [41, 20, 11, 10, 30, 51, 01, 31, 40, 50, 21].

This R-2-sequencing has right match-point 2 (and also 9) and so we can use Lemma8 to obtain the terrace (00, 11, 10, 30, 51, 01, 31, 40, 50, 21, 41, 20) for Z6 × Z2. Its 2-sequencing is (11, 01, 20, 21, 10, 30, 11, 10, 31, 20, 41).

The following result gives a necessary condition an abelian group must satisfy if it is tobe R-sequenceable.

Theorem 10 (Frielander et al. [8]). Abelian groups which have a single involution do nothave R-sequencings.

Proof. The argument is contained in the proof of the next theorem. �

The general result for R-2-sequencings is the following.

Theorem 11. An abelian group whose Sylow 2-subgroup is isomorphic to Z2 does not havean R-2-sequencing.

Proof. Let G be an abelian group of order n. Suppose that a is the R-terrace[a1, a2, . . . , an−1] for G, with R-2-sequencing b = [b1, b2, . . . , bn−1]. We have that

a1 + b1 + b2 + · · · + bn−1 = a1.

Therefore, the sum of the elements in the R-2-sequencing is 0.In the proof of Theorem 2, Gordon [9] uses results of Paige [17] and Hall and Paige [10]

to show that the sum of all of the non-zero elements of an abelian group G is 0 if and onlyif the Sylow 2-subgroup of G is not cyclic. This proves Theorem 10.

In abelian groups there is a well-defined concept of parity. An element g of an abeliangroup is even if g = 2h for some h ∈ G; otherwise g is odd. The sum of two elements ofthe same parity is even; the sum of two elements of different parity is odd.

Suppose now that G is an abelian group with cyclic Sylow 2-subgroup. We can obtainthe multiset {b1, b2, . . . , bn−1} from the set of all non-zero elements of G by negating someof them. However, this does not change the parity of the total as replacing g by −g altersthe total by 2g. As 0 is even, a necessary condition for G to have an R-terrace is that thesum of the non-zero elements is even. This occurs if and only if the Sylow 2-subgroup hasorder at least 4, and the result follows. �

Despite the superficial similarity of sequencings and R-sequencings, this result, taken withTheorem 2, shows that no abelian group can have both a sequencing and an R-sequencing.

The following example gives an R-2-sequencing for a group which fails the necessarycondition to have an R-sequencing.

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M.A. Ollis / Discrete Mathematics 305 (2005) 250–263 255

Example 12. The sequence [3, 10, 9, 4, 6, 2, 1, 7, 11, 8, 5] is an R-terrace for Z12. It hasR-2-sequencing [7, 11, 7, 2, 8, 11, 6, 4, 9, 9, 10].

We need the following group-theoretic result, taken from [12, pp. 488–489].

Theorem 13. Suppose that G is abelian with subgroup S. Then G has a subgroup N withN�G/S and G/N�S.

Now, if G is an abelian group with a terraced subgroup S of odd index we may applyTheorems 13, 3 and 4 to obtain a terrace for G. This is the strategy we will use, with S beingthe Sylow 2-subgroup of G if this is not elementary abelian and S being a slightly largergroup otherwise.

3. Constructing R-2-sequencings

Our aim in this chapter is to construct R-2-sequencings that have match-points for a widerange of groups which might be used as “S” in the strategy outlined at the end of the previoussection. We then apply the strategy to find terraces for many abelian groups. (Recall that allgroups in this section are assumed to be abelian.)

Our main tool is a generalisation of a theorem of Headley [11] which constructs an R-sequencing for a group by combining an R-sequencing of a subgroup and an R-sequencingof its quotient group. Before we can state Headley’s result we need another definition.

Let G be a group of order n. Let a be the R-terrace [a1, a2, . . . , an−1] for G. If ai =ai−1 + ai+1 for some i, then a is an R∗-terrace. The associated R-2-sequencing is anR∗-2-sequencing. Directed R∗-terraces and R∗-sequencings are defined similarly. The ter-minology “R∗-sequencing” was introduced in [8]. As we will be interested in the relativepositions of right match-points and the element of the R∗-terrace that is the sum of itsneighbours, we define an R∗-terrace to be standard if i = 1 in the above definition. Wealso refer to the R∗-2-sequencing associated with a standard R∗-terrace as standard. AnyR∗-terrace can be made standard by relabeling. The R-terraces in Examples 9 and 12 arestandard R∗-terraces.

Theorem 14 (Frielander et al. [8]). Every cyclic group of odd order has a directed R∗-terrace.

Proof. Consider Z2r+1. If r ≡ 0 (mod 3) or r ≡ 1 (mod 3) then the directed R-terrace fromTheorem 6 is an R∗-terrace. If r ≡ 0 (mod 3), we have ai = ai−1 + ai+1 for i = 2r/3. Ifr ≡ 1 (mod 3), we have ai = ai−1 + ai+1 for i = (4r + 2)/3.

When r ≡ 2 (mod 3) we need to define an alternative directed R∗-terrace. For 1� i�r

define

ai =

⎧⎪⎨⎪⎩

(i + 3)/2 if i ≡ 1 (mod 4),

r − i/2 if i ≡ 2 (mod 4),

(i − 1)/2 if i ≡ 3 (mod 4),

r + 2 − i/2 if i ≡ 0 (mod 4),

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256 M.A. Ollis / Discrete Mathematics 305 (2005) 250–263

except that

ar−1 = (r + 1)/2 if r ≡ 5 (mod 12),

ar−3 = (r + 5)/2 if r ≡ 5 (mod 12),

ar−1 = (r + 3)/2 if r ≡ 11 (mod 12).

Then [a1, a2, . . . , ar , ar + r, ar−1 + r, . . . , a1 + r] is a directed R∗-terrace for Z2r+1, unlessr = 5.

The R∗-2-sequencing is [b1, b2, . . . , b2r ], where, for 1� i�r − 1,

bi ={

r − i − 2 if i ≡ 1 (mod 4),

r − i + 2 if i ≡ 3 (mod 4),

i − r if i is even,

except that

br−1 = 1 if r ≡ 5 (mod 12),

br−2 = 2 if r ≡ 5 (mod 12),

br−3 = −4 if r ≡ 5 (mod 12),

br−4 = 3 if r ≡ 5 (mod 12),

br−1 = −2 if r ≡ 11 (mod 12),

br−2 = 1 if r ≡ 11 (mod 12)

and set br+i = −br−i . Also set br = r and b2r = −r .When r ≡ 2 (mod 6), we have ai =ai−1 +ai+1 for i = (4r + 4)/3. When r ≡ 5 (mod 6),

but r �= 5, we have ai = ai−1 + ai+1 for i = (2r + 2)/3.For Z11, the sequence [5, 6, 9, 3, 7, 4, 2, 1, 8, 10] is a standard R∗-terrace with [1, 3, 5, 4,

8, 9, 10, 7, 2, 6] as its associated R∗-2-sequencing. �

Headley’s theorem is the following.

Theorem 15 (Headley [11]). If an abelian group G is an extension of Z2 × Z2 by anR∗-sequenceable group then G is R∗-sequenceable.

This theorem was used in [11], along with other results concerning R-sequencings, toshow that an abelian group whose Sylow 2-subgroup is non-cyclic and not of order 8 isR∗-sequenceable.

The construction given in the following result is exactly Headley’s construction, but ap-plied to R∗-2-sequencings and an arbitrary subgroup of order 4, in place of R∗-sequencingsand Z2 × Z2, respectively.

Theorem 16. Let G be an abelian group with a subgroup H of order 4. If G/H is R∗-terraced then G is R∗-terraced.

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M.A. Ollis / Discrete Mathematics 305 (2005) 250–263 257

Proof. Let |G/H |=m. Let [K1, K2, . . . , Km−1] be a standard R∗-terrace for G/H . Chooseki , for 1� i�m − 1, such that ki ∈ Ki and km−1 + k2 = k1. Each element of G can beuniquely expressed in the form h + k, where h ∈ H and k ∈ {0, k1, k2, . . . , km−1}.

Let [h1, h2, h3] be an R-terrace for H.We have that such an R-terrace exists; if H=Z2×Z2use [01, 11, 10], if H = Z4 use [1, 3, 2]. There are no other groups of order 4.

We now list the elements of an R-terrace a for G, in the form h + k. We list the hcomponents and k components separately. The h component list varies as m varies modulo3. In all cases the list is given by the successive rows of the appropriate 4 × m matrix. Thenotation (h1, h2, h3)t−2 denotes t − 2 repetitions of the sequence (h1, h2, h3).

Case 1: m = 3t , for t > 1. Then⎡⎢⎢⎢⎣

0 0 . . . 0 h2 h1

h3 (h1, h2, h3)t−2 h1 h3 h3 h2 h1

h3 h2 (h3, h1, h2)t−2 h3 h2 h1 h3

h2 h1 (h2, h3, h1)t−2 h2 h1 0

⎤⎥⎥⎥⎦

is the h component matrix.Case 2: m = 3t + 1, for t �1. Then⎡

⎢⎢⎢⎣0 0 . . . h2 h1

h3 (h2, h1, h3)t−1 h3 h2 h1

h3 h2 (h1, h3, h2)t−1 h1 h3

h2 h1 (h3, h2, h1)t−1 0

⎤⎥⎥⎥⎦

is the h component matrix.Case 3: m = 3t + 2, for t �1. Then⎡

⎢⎢⎢⎣0 0 . . . 0 h2 h1

h3 (h2, h1, h3)t−1 h2 h3 h2 h1

h3 h2 (h1, h3, h2)t−1 h1 h1 h3

h2 h1 (h3, h2, h1)t−1 h3 0

⎤⎥⎥⎥⎦

is the h component matrix.In all cases⎡

⎢⎢⎢⎣k1 k2 . . . km−1 0

k2 k3 . . . km−1 k1 k1

k1 k2 . . . km−1 0

0 k2 k3 . . . km−1

⎤⎥⎥⎥⎦

is the k component matrix.When m = 3 we have |G| = 12. So in this case we have that G is either Z6 × Z2 or Z12.

As we noted at the start of this section, Examples 9 and 12 give standard R∗-terraces forZ6 × Z2 and Z12, respectively.

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258 M.A. Ollis / Discrete Mathematics 305 (2005) 250–263

It is straightforward to verify that a is an R-terrace using the following two points:(i) The k component differences are given by the successive rows of the matrix below, as

we have 0 − km−1 = k2 − k1 and k2 − 0 = k1 − km−1.⎡⎢⎢⎢⎣

k2−k1 k3−k2 . . . km−1−km−2 k2−k1 k1−km−1

k3−k2 k4−k3 . . . km−1−km−2 k1−km−1 0 0

k2 − k1 k3−k2 . . . km−1−km−2 k2−k1 0

k1−km−1 k3−k2 k4−k3 . . . km−1−km−2 k1−km−1

⎤⎥⎥⎥⎦.

(ii) In the h component matrices, each of the repeated 3-element sequences is an R-terrace. Within a matrix, each of them starts with a different element, but the correspondingpositions in the k component matrix have the same element. For example, the first elementof the first occurrence of each 3-element sequence is always paired with k3.

As (0 + km−1) + (0 + k2) = (0 + k1) the R-terrace we have constructed is a standardR∗-terrace. �

For our purposes we need R∗-2-sequencings that have right match-points. The followingresult shows that, in most cases, if we feed an R∗-2-sequencing with a right match-pointinto the above construction then the result is an R∗-2-sequencing with a right match-point.

Theorem 17. Let G be an abelian group with a subgroup H of order 4. Let |G/H | = m. IfG/H has a standard R∗-2-sequencing with i as a right match-point, for some 1� i�m−3,then G has a standard R∗-2-sequencing with i as a right match-point.

Proof. Apply Theorem 16, choosing values for the sequence (k1, k2, . . . , km−1) to give(ki+1−ki)=−ki+1. The resulting R∗-terrace a is standard. The associated R∗-2-sequencinghas i as a match-point as the h component of both ai and ai+1 is 0. �

We now have the tools to construct R∗-2-sequencings with right match-points and henceterraces. Our first target is 2-groups. Elementary abelian 2-groups do not have terraces andso cannot have R∗-2-sequencings with right match-points. We also omit cyclic 2-groupsfrom consideration—these are terraced by Theorem 2.

Theorem 18. Let G be a non-cyclic abelian 2-group of order n which is not elementaryabelian. Then G has an R∗-2-sequencing with right match-point i, for some 1� i�n − 3.

Proof. The conditions of the theorem imply that G has order at least 8. We use inductionon n, with Theorem 17 providing the inductive step. Suppose that n�32. Then G has asubgroup H of order 4 with G/H a 2-group which is neither cyclic nor elementary abelian.So it suffices to find suitable R∗-2-sequencings for all non-cyclic groups of orders 8 and 16which are not elementary abelian.

Below, we omit the brackets and commas from the notation of elements in direct productgroups.

The group Z4 × Z2 has [21, 30, 01, 20, 10, 11, 31] as a standard R∗-terrace with R∗-2-sequencing [11, 11, 21, 30, 01, 20, 30], which has right match-point 5.

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M.A. Ollis / Discrete Mathematics 305 (2005) 250–263 259

The group Z8 × Z2 has

[31, 70, 71, 51, 11, 40, 21, 10, 20, 50, 61, 01, 30, 60, 41]as a standard R∗-terrace with R∗-2-sequencing

[41, 01, 60, 40, 31, 61, 71, 10, 30, 11, 20, 31, 30, 61, 70],which has right match-point 6.

The group Z4 × Z4 has

[13, 32, 33, 31, 22, 02, 23, 01, 30, 03, 10, 20, 12, 11, 21]as a standard R∗-terrace with R∗-2-sequencing

[23, 01, 02, 31, 20, 21, 22, 31, 13, 11, 10, 32, 03, 10, 32],which has right match-point 6.

The group Z4 × Z2 × Z2 has

[211, 311, 101, 310, 110, 011, 001, 010, 100, 301, 201, 200, 111, 210, 300]as a standard R∗-terrace with R∗-2-sequencing

[100, 210, 211, 200, 301, 010, 011, 110, 201, 300, 001, 311, 101, 110, 311],which has right match-point 12. �

Corollary 19. Every abelian group whose Sylow 2-subgroup is not elementary abelian isterraced.

Proof. Let G be an abelian group whose Sylow 2-subgroup is not elementary abelian. Thenwe have G�A × B, where A is a 2-group which is not elementary abelian and |B| is odd.

If A is cyclic then we can terrace G using Theorems 2 or 3. If A is not cyclic then we canterrace A using Theorem 18 and Lemma 8 and then terrace G using Theorem 4. �

We want to use a similar approach when the Sylow 2-subgroup of G is elementary abelian,but clearly this will not work immediately as elementary abelian 2-groups are not terraced.To get around this, we take a subgroup which contains the Sylow 2-subgroup and terracethis. Define an augmented Sylow 2-subgroup of a group G to be a subgroup of G isomorphicto S×Zp, where S is the Sylow 2-subgroup of G and p is the smallest odd prime dividing |G|.All abelian groups which are not 2-groups have at least one augmented Sylow 2-subgroup.

We first consider the case when the Sylow 2-subgroup has order an even power of 2.

Theorem 20. Let p be an odd prime and k be an even number. The group Zk2 × Zp has a

standard R∗-2-sequencing with right match-point i, for some 1� i�2kp − 3, unless k = 0and p ∈ {3, 5}.

Proof. The argument is similar to that used in the proof of Theorem 18. If k�2 then it issufficient to find a suitable R∗-2-sequencing for Zk−2

2 ×Zp and then apply Theorem 16 with

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260 M.A. Ollis / Discrete Mathematics 305 (2005) 250–263

H = Z2 × Z2. Therefore, finding standard R∗-2-sequencings for Zp, for all odd primes p,which have a right match-point i, for some 1� i�p − 3, would prove the result. However,Z3 and Z5 do not have such an R∗-2-sequencing and we must do something slightly differentwhen p is 3 or 5. We first deal with the cases p > 5.

Let p = 2r + 1. After standardising, the R∗-sequencing for Z2r+1 of Theorem 14 hasright match-point i, where

i =

⎧⎪⎨⎪⎩

(2r + 3)/3 when r ≡ 0 (mod 3),

(4r − 1)/3 when r ≡ 1 (mod 3),

(4r − 5)/3 when r ≡ 2 (mod 6),

(2r − 1)/3 when r ≡ 5 (mod 6) and r > 5.

The standard R∗-sequencing of Z11 given in Theorem 14 has right match-point 7.Now suppose that p ∈ {3, 5}. If k�4 then it is sufficient to find a suitable R∗-2-sequencing

for Zk−22 × Zp and then apply Theorem 16 with H = Z2 × Z2. So we need standard R∗-

2-sequencings for Z2 × Z2 × Z3 and Z2 × Z2 × Z5 which have a right match-point i, forsome 1� i�p − 3. Note that Z2 × Z2 × Z3�Z2 × Z6 and Z2 × Z2 × Z5�Z2 × Z10. Weomit the brackets and commas from the notation for elements of direct product groups.

The sequence [14, 02, 11, 01, 03, 15, 10, 13, 04, 05, 12] is a standard R∗-terrace for Z2×Z6 with R∗-2-sequencing [14, 15, 10, 02, 12, 01, 03, 11, 01, 13, 02], which has right match-point 2.

The sequence

[01, 11, 16, 15, 02, 18, 12, 08, 06, 03, 04, 17, 05, 14, 09, 07, 19, 13, 10]is a standard R∗-terrace for Z2 × Z10 with R∗-2-sequencing

[10, 05, 09, 17, 16, 04, 16, 08, 07, 01, 13, 18, 19, 15, 08, 12, 04, 07, 11],which has right match-point 9 (and also 11). �

Corollary 21. Every abelian group whose Sylow 2-subgroup is elementary abelian of orderan even power of 2 satisfies Bailey’s conjecture.

Proof. Let G be an abelian group whose Sylow 2-subgroup is elementary abelian of orderan even power of 2 (but G not a 2-group). Then, by Theorem 13, G has a subgroup B of oddorder with G/B�S, where S is an augmented Sylow 2-subgroup of G.

If S is isomorphic to either of Z3 or Z5 then we can terrace G using Theorem 3 as |G| isodd. Otherwise, we can terrace S using Theorem 20 and Lemma 8 and then terrace G usingTheorem 4. �

So far we have shown that Bailey’s conjecture holds for all abelian groups, except possiblythose whose Sylow 2-subgroup is isomorphic to an elementary abelian 2-group of orderan odd power (greater than 1) of 2. When we apply a similar argument to that of Theorem18 or 20 in this case we find, in the first instance, that it is sufficient to find suitable R∗-2-sequencings for Z2 × Zp for all odd primes p. However, Theorem 11 says that such groupsdo not have any R-2-sequencings.

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M.A. Ollis / Discrete Mathematics 305 (2005) 250–263 261

However, as in the p ∈ {3, 5} case of Theorem 20, if we start the induction at the nextstage then we find that it is sufficient to find suitable R∗-2-sequencings for Z3

2 × Zp for allodd primes p. We do not have a general construction in this case, but the following exampledeals with the case p = 3.

Example 22. Consider Z6 × Z2 × Z2. The sequence

[011, 300, 500, 210, 201, 301, 510, 100, 101, 010, 111, 001, 200, 401, 511, 400,

410, 310, 211, 411, 110, 501, 311]is a standard R∗-terrace for this group. It has R∗-2-sequencing

[311, 200, 310, 011, 100, 211, 210, 001, 511, 101, 510, 201, 201, 110, 511, 010,

500, 501, 200, 301, 411, 410, 300]which has right match-point 13. Observe that Z3

2 × Z3�Z6 × Z2 × Z2.

Corollary 23. Let G be an abelian group whose Sylow 2-subgroup is elementary abelianof order an odd power of 2. If |G| is divisible by 3 then G is terraced.

Proof. By Theorem 13, G has a subgroup B of odd order with G/B�S, where S is anaugmented Sylow 2-subgroup of G. We can terrace S using Theorem 17, Example 22 andLemma 8 and then terrace G using Theorem 4. �

Collecting together Corollaries 19, 21 and 23 gives

Corollary 24. All abelian groups satisfy Bailey’s conjecture, except possibly those of ordercoprime to 3 with Sylow 2-subgroup an elementary abelian 2-group of order an odd power(greater than 1) of 2.

4. Non-abelian groups

In this section we consider how we may use the terraces we have constructed along withTheorem 4 and Corollary 5 to construct terraces for some classes of non-abelian groups.

A group is called 2-nilpotent if it has normal subgroup of odd order and index a power of 2.Nilpotent groups are 2-nilpotent, and it follows from [18, Theorem 5.4.8] that supersolublegroups are 2-nilpotent. Thus 2-nilpotence lies between solubility and supersolubility. Moreinformation about the structure of 2-nilpotent groups may be found in [18].

Theorem 25. All 2-nilpotent groups whose Sylow 2-subgroups are abelian, but not ele-mentary abelian of order at least 4, are terraced.

Proof. Let G be a group of the form in the theorem. Then G has a normal subgroup B ofodd order with index a power of 2. G/B is isomorphic to a Sylow 2-subgroup of G andhence is terraced by Corollary 19. Corollary 5 and Theorem 4 now give a terrace for G. �

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262 M.A. Ollis / Discrete Mathematics 305 (2005) 250–263

If the Sylow 2-subgroup of a 2-nilpotent group G is elementary abelian of order 2k thenit can be shown that G has a normal subgroup N of index 2kp, where p is the smallest oddprime dividing |G|. There are two options for G/N ; it is either isomorphic to Zk

2 × Zp

or isomorphic to Zk−12 × D2p, where D2p is the dihedral group of order 2p. We saw in

Corollaries 21 and 23 that many groups of type Zk2 × Zp are terraced.

Note that the discussion of groups in this section, taken along with the remarks beforeExample 22, show that to prove that all 2-nilpotent groups which have abelian Sylow 2-subgroups satisfy Bailey’s Conjecture it suffices to prove that

(1) groups of the form Z32 × Zp have standard R∗-2-sequencings with a right match-point

in one of the first 23p − 3 positions for all odd primes p,(2) groups of the form Zk

2 × D2p are terraced for all odd primes p and positive integers k.

More generally, to show that all 2-nilpotent groups satisfy Bailey’s Conjecture it sufficesto show, in addition to the above, that all non-abelian 2-groups are terraced.

In pursuing this last goal, it is worth noting some generalisations of results from Section2 to non-abelian groups. It follows from [8, Section 6] that if G is a group whose Sylow2-subgroup is cyclic and non-trivial then the product of all the elements of G cannot bethe identity. Hence such groups are not R-sequenceable, generalising Theorem 10. Usingsimilar methods, it is also possible to generalise Theorem 11 to show that any group oforder twice an odd number has no R-2-sequencing.

Acknowledgements

Most of this work was completed during a Ph.D. studentship at Queen Mary, Universityof London, funded by the EPSRC. I am grateful to my supervisor, Prof. R. A. Bailey, foruseful comments about this work and to P. Spiga, of Queen Mary, University of London,for discussions about the structure of soluble groups. I am also grateful to two refereesfor their suggestions, particularly concerning the generalisations of Theorems 10 and 11 tonon-abelian groups.

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