Constructing R-sequencings and terraces for groups of even order
The problem of finding R-sequencings for abelian groups of even orders has been reduced to that of finding R*-sequencings for abelian groups of odd orders except in the case when the Sylow 2-subgroup is a non-cyclic non-elementary-abelian group of order 8. We partially address this exception, includ...
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irk-123456789-1551452019-06-17T01:26:17Z Constructing R-sequencings and terraces for groups of even order Ollis, M. The problem of finding R-sequencings for abelian groups of even orders has been reduced to that of finding R*-sequencings for abelian groups of odd orders except in the case when the Sylow 2-subgroup is a non-cyclic non-elementary-abelian group of order 8. We partially address this exception, including all instances when the group has order 8t for t congruent to 1, 2, 3 or 4 (mod7). As much is known about which odd-order abelian groups are R*-sequenceable, we have constructions of R-sequencings for many new families of abelian groups. The construction is generalisable in several directions, leading to a wide array of new R-sequenceable and terraceable non-abelian groups of even order. 2015 Article Constructing R-sequencings and terraces for groups of even order / M. Ollis // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 297-316. — Бібліогр.: 21 назв. — англ. 1726-3255 2010 MSC:Primary 20D60; Secondary 05B99. http://dspace.nbuv.gov.ua/handle/123456789/155145 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The problem of finding R-sequencings for abelian groups of even orders has been reduced to that of finding R*-sequencings for abelian groups of odd orders except in the case when the Sylow 2-subgroup is a non-cyclic non-elementary-abelian group of order 8. We partially address this exception, including all instances when the group has order 8t for t congruent to 1, 2, 3 or 4 (mod7). As much is known about which odd-order abelian groups are R*-sequenceable, we have constructions of R-sequencings for many new families of abelian groups. The construction is generalisable in several directions, leading to a wide array of new R-sequenceable and terraceable non-abelian groups of even order. |
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Ollis, M. |
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Ollis, M. Constructing R-sequencings and terraces for groups of even order Algebra and Discrete Mathematics |
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Ollis, M. |
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Ollis, M. |
| title |
Constructing R-sequencings and terraces for groups of even order |
| title_short |
Constructing R-sequencings and terraces for groups of even order |
| title_full |
Constructing R-sequencings and terraces for groups of even order |
| title_fullStr |
Constructing R-sequencings and terraces for groups of even order |
| title_full_unstemmed |
Constructing R-sequencings and terraces for groups of even order |
| title_sort |
constructing r-sequencings and terraces for groups of even order |
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Інститут прикладної математики і механіки НАН України |
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2015 |
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http://dspace.nbuv.gov.ua/handle/123456789/155145 |
| citation_txt |
Constructing R-sequencings and terraces for groups of even order / M. Ollis // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 297-316. — Бібліогр.: 21 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT ollism constructingrsequencingsandterracesforgroupsofevenorder |
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2025-07-14T07:14:11Z |
| last_indexed |
2025-07-14T07:14:11Z |
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1837605586784485376 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 2, pp. 297–316
© Journal “Algebra and Discrete Mathematics”
Constructing R-sequencings and terraces
for groups of even order
M. A. Ollis
Communicated by R. I. Grigorchuk
Abstract. The problem of finding R-sequencings for abelian
groups of even orders has been reduced to that of finding R∗-
sequencings for abelian groups of odd orders except in the case
when the Sylow 2-subgroup is a non-cyclic non-elementary-abelian
group of order 8. We partially address this exception, including all
instances when the group has order 8t for t congruent to 1, 2, 3
or 4 (mod 7). As much is known about which odd-order abelian
groups are R∗-sequenceable, we have constructions of R-sequencings
for many new families of abelian groups. The construction is gen-
eralisable in several directions, leading to a wide array of new R-
sequenceable and terraceable non-abelian groups of even order.
1. Introduction
There are several problems, usually arising from methods to construct
combinatorial objects, that require elements of a finite group to be listed
in a way that satistfies various constraints. In this paper we consider
R-sequenceability and terraceability, the combinatorial consequences of
which include constructions of graph decompositions, quasi-complete Latin
squares and neighbor-balanced designs, among others.
First we look at R-sequenceability and R∗-sequenceability; secondly
we see how we can relax some of the constraints to give R∗-terraces, and
2010 MSC: Primary 20D60; Secondary 05B99.
Key words and phrases: 2-sequencing; Bailey’s Conjecture; R-sequencing; ter-
race.
298 R-sequencings and terraces
from there terraces. These results allow us to construct R-sequencings
and terraces for many groups that were not previously known to possess
them, edging closer to answering the longstanding questions of exactly
which groups are R-sequenceable or terraceable.
We often need to consider circular lists, where the first element is taken
to be to the right of the last element. As in [10], in such a case we add
a hooked arrow (a1, a2, . . . , an ←֓) and calculate subscripts modulo the
length of the list. Some groups we will refer to throughout the paper: Let Zr
be the additively written cyclic group on the symbols {0, 1, . . . , r − 1},
let D2r be the dihedral groups of order 2r defined by
D2r = 〈u, v : ur = e = v2, vu = u−1v〉,
let Q4r be the dicyclic group of order 4r defined by
Q4r = 〈u, v : u2r = e, v2 = ur, vu = u−1v〉,
and let A4 be the alternating group on 4 symbols.
Let G be a group of order n and let a = (a1, a2, . . . , an−1 ←֓) be
a circular arrangement of the non-identity elements of G. Define b =
(b1, b2, . . . , bn−1 ←֓) by bi = a−1
i ai+1 for each i. If the elements of b are
also all of the non-identity elements of G then b is a rotational sequencing
or R-sequencing of G and a is the corresponding directed rotational terrace
or directed R-terrace of G. If G has an R-sequencing then it is said to
be R-sequenceable. If, in addition, we have that a2an−1 = a1 = an−1a2
then a is a directed R∗-terrace, b is an R∗-sequencing and G is said to be
R∗-sequenceable.
Inspired by a map-colouring problem of Ringel, R-sequenceability was
introduced by Friedlander, Gordon and Miller in [5]. Various different
definitions, all equivalent to the above, are used in the literature; see, for
example, [1, 5, 9, 12,17].
Example 1. The following is a directed R∗-terrace for Z11:
(5, 6, 9, 3, 7, 4, 2, 1, 8, 10 ←֓).
Its R∗-sequencing is
(1, 3, 5, 4, 8, 9, 10, 7, 2, 6 ←֓).
Much is known about the R-sequenceability of abelian groups. Fried-
lander, Gordon and Miller [5] conjecture that the only abelian groups that
M. A. Ollis 299
are not R-sequenceable are those with exactly one involution (which they
prove cannot be R-sequenced). For even-order abelian groups the only
groups for which the conjecture is open are those with non-cyclic Sylow
2-subgroups of order 8. Some new infinite families of groups whose Sylow
2-subgroups are isomorphic to Z4×Z2 are shown to be R-sequenceable in
Section 3, including those of order 8t with an R∗-sequenceable subgroup
of order t with t congruent to 1, 2, 3 or 4 (mod 7).
In the non-abelian case several infinite families of R-sequenceable
groups are known, including dihedral and dicyclic groups D2n and Q4n
when n is even and n > 2. See [11] for a recent survey of results. More are
added in Section 3, including groups of the form H1×H2× · · · ×Hs×K,
where each Hi is one of, Z2
2, Z3
2, D8, Z6 × Z2, D12 or A4 and K is R∗-
sequenceable.
Again, let G be a group of order n, but now let a = (a1, a2, . . . , an) be
a linear arrangement of the elements of G. Define b = (b1, b2, . . . , bn−1)
by bi = a−1
i ai+1 for each i. If b contains one occurrence of each involution
of G and exactly two occurrences of elements from each set
{g, g−1 : g2 6= e}
then a is a terrace for G and b is its associated 2-sequencing.
Example 2. The following is a terrace for Z11:
(0, 2, 1, 8, 10, 5, 6, 9, 3, 7, 4).
Its 2-sequencing is
(2, 10, 7, 2, 6, 1, 3, 5, 4, 8).
Terraces were introduced by Bailey [4] as a tool for constructing quasi-
complete Latin squares; similar ideas had been used earlier by Williams [21]
(restricted to cyclic groups) and Gordon [7] (in the case of directed terraces;
those whose 2-sequencings have no repeated entries, in which case they
are called sequencings). Bailey’s Conjecture is that all groups other than
non-cyclic elementary abelian 2-groups are terraced (it is known that
non-cyclic elementary abelian 2-groups cannot be terraced [4]). This was
proven for abelian groups in [16] and many nonabelian groups are known
to have terraces. See [11] for more details on these topics. In Section 4 we
add more groups, including direct products comprised of arbitrarily many
non-cyclic, non-dicyclic groups of order 12, an R∗-sequenceable group and,
optionally, a group of odd order.
300 R-sequencings and terraces
For our constructions we are interested in direct and central factors of
a group. If a group G can be written as a direct product H ×K then H
is a direct factor of G. More generally, suppose H E G and let
CG(H) = {g ∈ G : gh = hg for all h ∈ H}
be the centralizer of H in G. If G = HCG(H) then H is a central factor
of G. Direct factors are also central factors but central factors are not
necessarily direct factors.
In the next section we give the main construction on which all the re-
sults rely. In Section 3 we see how it can be used to produce R-sequencings
and in Section 4 we consider how it can be adapted to produce terraces.
2. The construction
We present the main construction for a circular sequence of the non-
identity elements of our target group G, which has order n = 4mt and is
of the form H ×K with |H| = 4m and |K| = t.
Given a circular sequence a = (a1, a2, . . . , a4m−1 ←֓) of the non-
identity elements of H, a permutation σ ∈ S4m−1, and a circular sequence
k = (k1, k2, . . . , kt−1 ←֓) of the non-identity elements of K with kt−1k2 =
k1 = k2kt−1, we construct a sequence in H×K from 4m+1 subsequences.
Let b = (b1, b2, . . . , b4m−1 ←֓) be the quotients associated with a; that
is, bi = a−1
i ai+1 for each i. Similarly, let ℓ = (ℓ1, ℓ2, . . . , ℓt−1 ←֓) be the
quotients associated with k; so ℓi = k−1
i ki+1 for each i.
In practice, a will always be a directed R-terrace. In the next section k
will be a directed R∗-terrace and in Section 4 it will be a weaker object,
an “R-terrace".
The first three subsequences each have distinct characteristics. These
are followed by 2m − 1 that follow one pattern and then 2m − 2 that
follow a slightly different one. The final subsequence has just one element.
We define them in turn, noting the internal quotients that they generate
as we go, and then consider the quotients generated at the joins.
Note that in calculating the quotients we make use of the condition
kt−1k2 = k1 = k2kt−1. In particular, we use that k2 = ℓt−1 and k−1
t−1 = ℓ1.
Also, recall that for circular lists subscripts are calculated modulo the
length of the list.
The first subsequence is
(e, k1), (e, k2), . . . , (e, kt−2)
M. A. Ollis 301
which has internal quotients
(e, ℓ1), (e, ℓ2), . . . , (e, ℓt−3).
The second subsequence is
(aσ(1)−2m, kt−1), (aσ(1)−2m+1, k1), (aσ(1)−2m+2, k1),
(aσ(1)−2m+3, k1), . . . , (aσ(1), k1), (aσ(1)+1, k1),
(aσ(1)+2, k2), (aσ(1)+3, k3), . . . , (aσ(1)+t−2, kt−2)
which has internal quotients
(bσ(1)−2m, ℓt−1), (bσ(1)−2m+1, e), (bσ(1)−2m+2, e),
(bσ(1)−2m+3, e), . . . , (bσ(1), e), (bσ(1)+1, ℓ1),
(bσ(1)+2, ℓ2), (bσ(1)+3, ℓ3), . . . , (bσ(1)+t−3, ℓt−3).
The third subsequence is
(aσ(2)−2m+1, kt−1), (aσ(1)−2m+2, e), (aσ(1)−2m+3, e),
(aσ(1)−2m+4, e), . . . , (aσ(2), e), (aσ(2)+1, e),
(aσ(2)+2, k2), (aσ(3)+3, k3), . . . , (aσ(2)+t−2, kt−2)
which has internal quotients
(bσ(2)−2m+1, ℓ1), (bσ(2)−2m+2, e), (bσ(2)−2m+3, e),
(bσ(2)−2m+4, e), . . . , (bσ(2), e), (bσ(2)+1, ℓt−1),
(bσ(2)+2, ℓ2), (bσ(2)+3, ℓ3), . . . , (bσ(2)+t−3, ℓt−3).
For i in the range 4 6 i 6 2m + 2, the ith subsequence is
(aσ(i−1), kt−1), (aσ(i−1)+1, e), (aσ(i−1)+2, k2),
(aσ(i−1)+3, k3), (aσ(i−1)+4, k4), . . . , (aσ(i−1)+t−2, kt−2)
which has internal quotients
(bσ(i−1), ℓ1), (bσ(i−1)+1, ℓt−1), (bσ(i−1)+2, ℓ2),
(bσ(i−1)+3, ℓ3), (bσ(i−1)+4, ℓ4), . . . , (bσ(i−1)+t−3, ℓt−3).
302 R-sequencings and terraces
For i in the range 2m + 3 6 i 6 4m, the ith subsequence is
(aσ(i−1), kt−1), (aσ(i−1)+1, k1), (aσ(i−1)+2, k2),
(aσ(i−1)+3, k3), (aσ(i−1)+4, k4), . . . , (aσ(i−1)+t−2, kt−2)
which has internal quotients
(bσ(i−1), ℓt−1), (bσ(i−1)+1, ℓ1), (bσ(i−1)+2, ℓ2),
(bσ(i−1)+3, ℓ3), (bσ(i−1)+4, ℓ4), . . . , (bσ(i−1)+t−3, ℓt−3).
Note that the only difference in the structure of these subsequences
compared to the previous ones is in the second coordinate of the second
element, meaning that the only changes in structure in the quotients are
in the second coordinates of the first and second elements. Also note that
there are no subsequences of this form when m = 1.
The final subsequence consists of the single element (e, kt−1). Of course,
this gives rise to no internal quotients.
The quotients generated where the subsequences join are
(aσ(1)−2m, ℓt−2), (a−1
σ(1)+t−1aσ(2)−2m+1, ℓt−2),
(a−1
σ(2)+t−1aσ(3), ℓt−2), (a−1
σ(3)+t−1aσ(4), ℓt−2), . . . ,
(a−1
σ(4m−2)+t−1aσ(4m−1), ℓt−2), (a−1
σ(4m−1)+t−1, ℓt−2)
(the fourth to the penultimate one, inclusive, are excluded when m = 1).
Finally, (e, ℓt−1) is the quotient generated between the last subsequence
and the first.
When we come to prove that the main construction gives directed
R∗-terraces and other similar objects, we will see that the permutation σ
is responsible for lining up the subsequences in such a way that all of the
properties we need are satisfied. In order to do this successfully, we also
need constraints on the permutation.
Say that σ ∈ S4m−1 is admissible if σ(2) = σ(1)− 2m and
{σ(3), σ(4), . . . , σ(2m + 1)} = {σ(2) + 1, σ(2) + 2, . . . , σ(2) + 2m− 1}
where all calculations are performed modulo 4m− 1.
For a positive integer t, the pair a and σ are t-compatible if the following
4m elements are distinct (i.e. are all of H):
aσ(1)−2m, a−1
σ(1)+t−1aσ(2)−2m+1, a−1
σ(4m−1)+t−1
M. A. Ollis 303
and
a−1
σ(i)+t−1aσ(i+1)
for each i with 1 < i < 4m− 1.
3. R-sequencings
We can now prove the main results for R-sequencings. Theorem 1
gives the case where G has a direct factor of order a multiple of 4, which
is sufficient for the abelian group case, and Theorem 4 gives the variant
for a central factor.
Theorem 1. Let G = H ×K with |H| = 4m and |K| = t. If H has an
R-sequencing with a t-compatible σ ∈ S4m−1 and K is R∗-sequenceable
then G is R∗-sequenceable.
Proof. Let a be the directed R-terrace of H and k be the directed R∗-
terrace of K, with the usual notation for their elements and quotients.
Apply the main construction to get a circular sequence of elements in G
and their quotients. We check that all elements of H appear with each
element of K (with the exception that (e, e) does not appear) in each of
the sequence and its quotients.
The elements that appear with k1 in the sequence are:
e, aσ(1)−2m+1, aσ(1)−2m+2, . . . , aσ(1), aσ(1)+1,
aσ(2m+2)+1, aσ(2m+3)+1, . . . , aσ(4m−1)+1.
When m > 1, the admissibility of σ implies that the two sets
{σ(2m + 2), σ(2m + 3), . . . , σ(4m− 1)}
and
{σ(1) + 1, σ(1) + 2, . . . , σ(1) + 2m− 2}
are equal as each has all of the numbers from 1 to 4m− 1 except for
{σ(2), σ(2) + 1, σ(2) + 2, . . . , σ(2) + 2m}
(recall that these calculations are performed modulo 4m− 1). Applying
this to the last 2m−2 elements we see that the sequence contains all of the
elements of H. When m = 1 we have the elements aσ(1)−1, aσ(1), aσ(1)+1
which are distinct.
304 R-sequencings and terraces
The elements that appear with kj , for 2 6 j 6 t− 2 are:
e, aσ(1)+j , aσ(2)+j , . . . , aσ(4m−1)+j
which comprise all of the elements of H.
The elements that appear with kt−1 are:
aσ(1)−2m, aσ(2)−2m+1, aσ(3), aσ(4), . . . , aσ(4m−1).
Applying the first clause of the admissibility definition to the first two
elements we see that these are all of the non-identity elements of H.
The elements that appear with e are:
aσ(2)−2m+2, aσ(2)−2m+3, aσ(2)−2m+4, . . . , aσ(2),
aσ(2)+1, aσ(3)+1, aσ(4)+1, . . . , aσ(2m+1)+1.
Applying the second clause of the admissibility definition to the last 2m−1
elements we see that these are all of the non-identity elements of H.
Turning to the sequence of quotients, the elements that appear with
ℓ1 are:
e, bσ(1)+1, bσ(2)−2m+1, bσ(3), bσ(4), . . . , bσ(2m+1),
bσ(2m+2)+1, bσ(2m+3)+1, . . . , bσ(4m−1)+1.
The admissibility of σ implies that
{σ(2), σ(2m + 2), σ(2m + 3), . . . , σ(4m− 1)} =
{σ(1) + 1, σ(2m + 2) + 1, σ(2m + 3) + 1, . . . , σ(4m− 1) + 1}.
Coupled with the first clause of the admissibility definition applied to the
third element we see that the sequence contains all of the elements of H.
The elements that appear with ℓj , for 2 6 j 6 t− 3 are:
e, bσ(1)+j , bσ(2)+j , . . . , bσ(4m−1)+j .
These are all of the elements of H.
The elements that appear with ℓt−2 are:
aσ(1)−2m, a−1
σ(1)+t−1aσ(2)−2m+1, a−1
σ(2)+t−1aσ(3),
a−1
σ(3)+t−1aσ(4), . . . , a−1
σ(4m−2)+t−1aσ(4m−1), aσ(4m−1)+t−1.
M. A. Ollis 305
As a and σ are t-compatible, these are all of the elements of H.
The elements that appear with ℓt−1 are:
bσ(1)−2m, bσ(2)+1, bσ(3)+1, . . . , bσ(2m+1)+1,
bσ(2m+2), bσ(2m+3), . . . , bσ(4m−1), e.
We again use that
{σ(2), σ(2m + 2), σ(2m + 3), . . . , σ(4m− 1)} =
{σ(1) + 1, σ(2m + 2) + 1, σ(2m + 3) + 1, . . . , σ(4m− 1) + 1}
and the first clause of the admissibility definition, this time applied to
the first element. Doing so, we see that the sequence contains all of the
elements of H.
The elements that appear with e are:
bσ(1)−2m+1, bσ(1)−2m+2, . . . , bσ(1), bσ(2)−2m+2, bσ(2)−2m+3, . . . , bσ(2).
Using the first clause of the admissibility definition we see that these are
all of the non-identity elements of H.
This shows that our sequence is a directed R-terrace. Finally, observe
that the first two elements of our sequence are (e, k1) and (e, k2) and the
last is (e, kt−1). Therefore, that k is a directed R∗-terrace of K implies
that our sequence is a directed R∗-terrace of G.
Theorem 2. Let A be an abelian group such that A ≡ S × T where S
is a Sylow 2-subgroup that is isomorphic to Z4 × Z2 and T has order
congruent to 1, 2, 3 or 4 (mod 7). If T is R∗-sequenceable then A is
R∗-sequenceable.
Proof. To apply Theorem 1 for each desired value of t we require an
R-sequencing for Z4 × Z2 along with a t-compatible σ. The following
sequences and permutations do what is required:
t ≡ 1 (mod 7), σ = (1, 6, 7),
a = (0, 1), (2, 1), (1, 0), (2, 0), (3, 1), (3, 0), (1, 1).
t ≡ 2 (mod 7), σ = (1, 4)(2, 7, 5),
a = (1, 0), (2, 0), (1, 1), (0, 1), (2, 1), (3, 0), (3, 1).
306 R-sequencings and terraces
t ≡ 3 (mod 7), σ = (1, 6, 7),
a = (0, 1), (1, 0), (2, 0), (1, 1), (3, 0), (3, 1), (2, 1).
t ≡ 4 (mod 7), σ = (1, 7)(2, 3, 6),
a = (1, 0), (1, 1), (2, 0), (3, 0), (2, 1), (0, 1), (3, 1).
We can therefore construct the R-sequencing for A.
A computer search has shown that there are no satisfactory a and σ
for other values of t (mod 7). We can now find R-sequencings for new
families of abelian groups whose Sylow 2-subgroups are non-cyclic of
order 8, the only open cases in the even-order question for abelian groups:
Corollary 1. Let K be an abelian group with |K| > 5. If |K| is congruent
to 1, 3, 9 or 11 (mod 14) and the Sylow 3-subgroups of K are isomorphic
to Z
α
3 × Z
α
9 × Z
β
27 × Z
γ
81 or Z
α
3 × Z
α
9 × Z
β
27 × Z
γ
81 × Z3t, where t > 1 and
t ≡ α + β (mod 2), then Z4 × Z2 ×K is R∗-sequenceable. In particular,
if |K| is congruent to one of 1, 11, 17, 23, 25, 29, 31, or 37 (mod 42)
then Z4 × Z2 ×K is R∗-sequenceable.
Proof. The group K is R∗-sequenceable [5,10,16] and hence we can apply
Theorem 1 with H = Z4×Z2. The last sentence describes the cases where
the Sylow 3-subgroups of K are trivial.
Further, any progress on finding R∗-sequencings for odd-order groups
with Sylow 3-subgroups other than those described in Corollary 1 can
now be translated directly into solving more even-order cases by the same
method. For example, it is known that for any abelian 3-group T there are
infinitely many R∗-sequenceable abelian groups whose Sylow 3-subgroups
are isomorphic to T [10].
Theorem 1 generalises the methods of [8] and [16] which are limited to
the cases H = Z
2
2 and H = Z
3
2. However, when m = 1 and t ≡ 0 (mod 3)
it is impossible to achieve t-compatibility.
In this case, Headley [8] uses a slightly different construction which
also works in our more general set-up; we will refer to this as the Headley
construction. Given a circular sequence a = [a1, a2, a3] of the non-identity
elements of H and a circular sequence k = [k1, k2, . . . , kt−1] of the elements
of K with kt−1k2 = k1 = k2kt−1, we again construct a sequence in H ×K.
The first line, the second and third line combined, and the fourth line
each have t elements and the fifth line has t− 1. Recall that subscripts
M. A. Ollis 307
in a are calculated modulo 3; we leave them unreduced in order to make
the structure clearer:
(e, k1), (e, k2), . . . , (e, kt−2), (a2, kt−1), (a1, e),
(a3, k2), (a1, k3), (a2, k4), . . . , (at−6, kt−4),
(a1, kt−3), (a3, kt−2), (a3, kt−1), (a2, k1), (a1, k1),
(a3, k1), (a2, k2), (a3, k3), . . . , (at−3, kt−3), (a2, kt−2), (a1, kt−1), (a3, e),
(a2, e), (a1, k2), (a2, k3), . . . , (a2, kt−3), (a1, kt−2), (e, kt−1).
Theorem 3. Let G = H × K with |H| = 4 and |K| = t, where t ≡ 0
(mod 3). If H has an R-sequencing and K is R∗-sequenceable then G is
R∗-sequenceable.
Proof. Headley’s construction as described above gives the required di-
rected R∗-terrace when a is a directed R-terrace for H and k is a directed
R∗-terrace for K. Checking the sequence and the quotients is a similar
(but more straightforward) process to the proof of Theorem 1.
Following the approach of [15], we may relax the condition that H be
a direct factor to it being a central factor if we add in conditions on the
directed R∗-terrace of its quotient group.
Theorem 4. Let G be a group of order 4mt with central factor H of
order 4m. If H has a directed R-terrace a with a t-compatible σ ∈ S4m−1
and G/H has a directed R∗-terrace [K1, K2, . . . , Kt−1] such that there
are elements k2 ∈ K2 and kt−1 ∈ Kt−1 that commute, then G is R∗-
sequenceable. If m = 1 and t ≡ 0 (mod 3) then the requirement for a
t-compatible permutation σ may be dropped.
Proof. Let k1 = k2kt−1 and for each i with 3 6 i 6 t − 2 choose ki ∈
Ki ∩CG(H) (as HCG(H) = G, the set Ki ∩CG(H) must be non-empty).
Each element of G is expressible in the form hki for a unique h ∈ H and
we have that kih = hki for all i and all h ∈ H.
Now apply the main construction or Headley’s construction as appro-
priate to a and [k1, k2, . . . , kt−1], with elements (h, k) of G replaced with
hk throughout. Thanks to the commutativity of the elements of H with
the ki, the argument goes through exactly as in Theorem 1 or 3.
We now turn to which groups it is possible to use in the role of H
in Theorems 1 and 4. We present here one possible pair of directed R-
terrace a and permutation σ for the values of t modulo 4m− 1 for which
they exist for the groups D8, Z6 × Z2, D12 and A4. The group Z
2
2 with
308 R-sequencings and terraces
t 6≡ 0 (mod 3) is covered in [8] and Z
3
2 is covered in [16]. Recall that
cyclic groups of even order and Q8 and Q12 are not R-sequenceable.
These directed R-terraces and permutations were found using the
group-theory software package GAP [6].
The case H = D8.
t ≡ 0 (mod 7), σ = (1, 6), a = u, v, u3, u2, u2v, uv, u3v.
t ≡ 1 (mod 7), σ = (1, 6), a = v, u3v, u2, u, uv, u3, u2v.
t ≡ 2 (mod 7), σ = (1, 7)(2, 3, 6), a = u, v, u2, u3, u3v, uv, u2v.
t ≡ 3 (mod 7), σ = (1, 6, 7), a = v, u3v, u2, u, uv, u3, u2v.
t ≡ 4 (mod 7), σ = (1, 7)(2, 3, 6), a = v, u, u2v, u2, u3, uv, u3v.
t ≡ 5 (mod 7), σ = (1, 5, 2)(3, 4), a = v, u2v, u3v, u2, u3, uv, u.
t ≡ 6 (mod 7), σ = (1, 5, 2), a = v, u, u2v, u2, u3, uv, u3v.
The case H = Z6 × Z2.
t ≡ 0 (mod 11), σ = (1, 7, 2)(4, 5, 6)(10, 11),
a = (2, 0), (4, 0), (2, 1), (3, 0), (5, 1), (3, 1), (4, 1), (1, 0), (1, 1), (0, 1), (5, 0).
t ≡ 1 (mod 11), σ = (1, 6, 3, 2, 11, 8, 9, 7)(4, 5),
a = (1, 0), (0, 1), (2, 1), (5, 1), (2, 0), (4, 1), (3, 1), (4, 0), (5, 0), (3, 0), (1, 1).
t ≡ 2 (mod 11), σ = (1, 2, 7, 8, 6)(3, 10, 5, 11, 4, 9),
a = (1, 0), (2, 1), (2, 0), (1, 1), (5, 1), (3, 0), (5, 0), (4, 0), (0, 1), (3, 1), (4, 1).
t ≡ 3 (mod 11), σ = (1, 6, 5, 2, 11, 9, 8, 10, 7, 3),
a = (1, 0), (4, 0), (2, 1), (3, 1), (5, 0), (0, 1), (4, 1), (3, 0), (2, 0), (5, 1), (1, 1).
t ≡ 4 (mod 11), σ = (2, 6, 7, 8)(3, 9)(4, 11)(5, 10),
a = (1, 0), (5, 0), (0, 1), (4, 0), (1, 1), (3, 1), (4, 1), (3, 0), (2, 0), (2, 1), (5, 1).
t ≡ 5 (mod 11), σ = (1, 10, 3, 8, 11)(2, 4, 9)(5, 7),
a = (0, 1), (1, 0), (2, 0), (4, 1), (4, 0), (3, 1), (2, 1), (5, 1), (1, 1), (5, 0), (3, 0).
t ≡ 6 (mod 11), σ = (1, 8, 9, 10, 11)(3, 7, 6),
a = (1, 0), (2, 0), (1, 1), (5, 0), (4, 0), (0, 1), (2, 1), (5, 1), (3, 1), (3, 0), (4, 1).
M. A. Ollis 309
t ≡ 7 (mod 11), σ = (1, 7, 5, 2)(3, 6, 4)(8, 11),
a = (0, 1), (2, 0), (1, 0), (3, 0), (3, 1), (1, 1), (4, 1), (5, 1), (4, 0), (2, 1), (5, 0).
t ≡ 8 (mod 11), σ = (1, 3, 9, 5, 2, 8, 7, 11, 4)(6, 10),
a = (1, 0), (0, 1), (5, 1), (3, 0), (4, 0), (1, 1), (4, 1), (2, 1), (2, 0), (3, 1), (5, 0)
t ≡ 9 (mod 11), σ = (1, 3, 2, 8, 4, 10, 6)(5, 9)(7, 11),
a = (1, 0), (4, 0), (2, 1), (3, 1), (5, 0), (0, 1), (4, 1), (3, 0), (2, 0), (5, 1), (1, 1).
t ≡ 10 (mod 11), σ = (1, 2, 7, 8, 6, 9, 3, 11, 5)(4, 10),
a = (2, 0), (2, 1), (0, 1), (3, 0), (4, 0), (5, 1), (4, 1), (1, 1), (5, 0), (1, 0), (3, 1).
The case H = D12.
t ≡ 0 (mod 11), σ = (1, 7, 5, 4, 6, 3, 2)(8, 9, 11, 10),
a = v, u, u4v, uv, u5v, u3, u4, u2, u2v, u3v, u5.
t ≡ 1 (mod 11), σ = (1, 4, 2, 9, 7, 11, 8, 5, 10, 6),
a = u3, v, u, u3v, uv, u2v, u4, u2, u5, u5v, u4v.
t ≡ 2 (mod 11), σ = (2, 6, 10, 5, 11, 3, 8)(4, 7, 9),
a = u2, u, u5, v, u4v, u3v, u3, u5v, u2v, u4, uv.
t ≡ 3 (mod 11), σ = (1, 6, 2, 11, 7, 5, 3, 4)(8, 10),
a = v, u4v, u, u2v, u4, u5, u2, uv, u3v, u3, u5v.
t ≡ 4 (mod 11), σ = (1, 5)(2, 10, 9, 7, 4, 3, 11, 6),
a = v, u5v, u2, u4, u3, u2v, u4v, uv, u, u3v, u5.
t ≡ 5 (mod 11), σ = (1, 10, 3, 6, 8, 11, 2, 4, 9),
a = v, u2, u5, u5v, u4, u3, u2v, u4v, u, u3v, uv.
t ≡ 6 (mod 11), σ = (1, 10, 11, 3, 6, 5, 9, 2, 4, 8),
a = u2, u, u3, v, u4, u5v, u5, u3v, u2v, u4v, uv.
310 R-sequencings and terraces
t ≡ 7 (mod 11), σ = (2, 6, 9, 4, 8)(3, 10)(5, 11),
a = v, u5v, uv, u3, u3v, u2, u4v, u2v, u5, u4, u.
t ≡ 8 (mod 11), σ = (2, 6, 10)(3, 11, 4, 9, 5, 8),
a = u, v, u2v, uv, u4v, u2, u5v, u4, u3, u3v, u5.
t ≡ 9 (mod 11), σ = (1, 4, 10, 8, 5, 11, 6, 3, 2, 9, 7),
a = u2, u4, v, u5, u3v, u2v, u3, u, uv, u4v, u5v.
t ≡ 10 (mod 11), σ = (2, 6, 8)(3, 9, 5, 7, 10)(4, 11),
a = u3, v, u4v, uv, u5, u5v, u, u2v, u3v, u4, u2.
The case H = A4.
t ≡ 0 (mod 11), σ = (1, 6, 5)(2, 11, 7, 3, 4)(8, 10, 9),
a = (2, 3, 4), (1, 2)(3, 4), (1, 3, 2), (1, 3)(2, 4), (1, 4, 2), (1, 3, 4),
(1, 2, 3), (1, 4, 3), (1, 2, 4), (2, 4, 3), (1, 4)(2, 3).
t ≡ 1 (mod 11), σ = (1, 8)(4, 5, 6)(9, 10, 11),
a = (2, 3, 4), (1, 2, 4), (1, 4, 3), (1, 3, 4), (1, 3, 2), (1, 2)(3, 4),
(1, 2, 3), (1, 3)(2, 4), (2, 4, 3), (1, 4, 2), (1, 4)(2, 3).
t ≡ 2 (mod 11), σ = (1, 3)(2, 8, 4, 9, 6, 10, 7)(5, 11),
a = (1, 2)(3, 4), (2, 3, 4), (1, 3)(2, 4), (1, 3, 4), (1, 4, 2), (1, 2, 3),
(1, 4, 3), (1, 4)(2, 3), (1, 2, 4), (1, 3, 2), (2, 4, 3).
t ≡ 3 (mod 11), σ = (1, 9, 2, 3, 7, 4, 8, 11, 10)(5, 6),
a = (2, 3, 4), (2, 4, 3), (1, 2, 4), (1, 4, 2), (1, 2, 3), (1, 4, 3),
(1, 3, 4), (1, 3, 2), (1, 4)(2, 3), (1, 3)(2, 4), (1, 2)(3, 4).
M. A. Ollis 311
t ≡ 4 (mod 11), σ = (1, 5, 2, 10, 8, 6, 11, 7)(3, 4),
a = (2, 3, 4), (1, 2, 4), (1, 3, 2), (1, 3)(2, 4), (1, 2, 3), (1, 4, 2),
(1, 4, 3), (1, 3, 4), (1, 4)(2, 3), (2, 4, 3), (1, 2)(3, 4).
t ≡ 5 (mod 11), σ = (1, 3)(2, 8, 4, 9, 7)(5, 11)(6, 10),
a = (2, 3, 4), (1, 2, 3), (1, 3, 4), (1, 2)(3, 4), (2, 4, 3), (1, 3, 2),
(1, 4, 2), (1, 4, 3), (1, 2, 4), (1, 4)(2, 3), (1, 3)(2, 4).
t ≡ 6 (mod 11), σ = (1, 2, 7, 9, 4, 8, 6, 10, 3)(5, 11),
a = (2, 3, 4), (1, 2, 4), (1, 2, 3), (1, 3, 4), (2, 4, 3), (1, 4, 3),
(1, 3)(2, 4), (1, 3, 2), (1, 4)(2, 3), (1, 4, 2), (1, 2)(3, 4).
t ≡ 7 (mod 11), σ = (1, 8, 11)(3, 7, 5)(4, 6),
a = (2, 3, 4), (1, 2, 4), (1, 4, 2), (1, 2)(3, 4), (2, 4, 3), (1, 3)(2, 4),
(1, 2, 3), (1, 3, 4), (1, 4, 3), (1, 4)(2, 3), (1, 3, 2).
t ≡ 8 (mod 11), σ = (1, 5, 3)(2, 10, 6, 11, 8, 9, 7),
a = (2, 3, 4), (2, 4, 3), (1, 2, 4), (1, 4, 2), (1, 3)(2, 4), (1, 3, 4),
(1, 2, 3), (1, 4, 3), (1, 3, 2), (1, 4)(2, 3), (1, 2)(3, 4).
t ≡ 9 (mod 11), σ = (1, 6, 2, 11, 10, 7, 4, 3),
a = (1, 2)(3, 4), (2, 3, 4), (1, 3, 2), (1, 4)(2, 3), (1, 2, 3), (1, 4, 3),
(2, 4, 3), (1, 2, 4), (1, 3, 4), (1, 4, 2), (1, 3)(2, 4).
t ≡ 10 (mod 11), σ = (1, 7, 3, 4, 2)(9, 10),
a = (1, 2)(3, 4), (2, 3, 4), (1, 3)(2, 4), (1, 3, 4), (2, 4, 3), (1, 2, 3),
(1, 4, 3), (1, 4)(2, 3), (1, 3, 2), (1, 2, 4), (1, 4, 2).
These directed R-terraces along with Theorems 1, 3 and 4 allow us to
show the R-sequenceability of many new groups. The following result of
Wang and Leonard extends the scope further still:
312 R-sequencings and terraces
Theorem 5. [19] If K is an R∗-sequenceable group of even order and N
is a nilpotent group of odd order then K ×N is R∗-sequenceable.
Proof. Follows immediately from Corollaries 2 and 6 of [19].
Theorem 6. Groups of the form H1 ×H2 × · · · ×Hs ×K ×N , where
each Hi is one of the groups Z
2
2, Z3
2, D8, Z6 × Z2, D12 or A4, the group
K is R∗-sequenceable and N is a nilpotent group of odd order, are R∗-
sequenceable.
Proof. Repeatedly apply Theorem 1 and/or 3 to construct a directed
R∗-terrace for H1 ×H2 × · · · ×Hs ×K. Apply Theorem 5 to complete
the proof.
Groups that are known to be R∗-sequenceable include: abelian groups
with non-trivial non-cyclic Sylow 2-subgroups of orders other than 8
[5, 8]; abelian groups of odd order or with Sylow 2-subgroups isomorphic
to Z
3
2 whose Sylow 3-subgroups are isomorphic to Z
α
3 × Z
α
9 × Z
β
27 or
Z
α
3 × Z
α+1
9 × Z
β
27 [5, 16]; the abelian groups described in Corollary 1;
nonabelian groups whose order is the product of two odd primes [20];
dihedral groups of order 4k, unless k < 4 or k ≡ 0 or 1 (mod 6) [18]; and
dicyclic groups of order congruent to 16 or 32 (mod 48) [18].
4. Terraces
In this section we follow the approach of [12, 15] whereby we relax
the requirement of directedness in the R∗-terrace for K and see that
R-terraces emerge from the construction. Further, if these R-terraces have
an additional property then we may turn them into terraces.
Let G be a group of order n and let a = (a1, a2, . . . , an−1, ←֓) be
a circular arrangement of the non-identity elements of G. Define b =
(b1, b2, . . . , bn−1 ←֓) by bi = a−1
i ai+1 for each i. If b contains one occur-
rence of each involution of G and exactly two occurrences of elements
from each set {g, g−1 : g2 6= e} then a is a rotational terrace or R-terrace
and b is a rotational 2-sequencing or R-2-sequencing.
As in the directed definition, if ai−1ai+1 = ai = ai+1ai−1 then a is
an R∗-terrace and b is an R∗-2-sequencing. By re-indexing if necessary,
we may assume this value of i in an R∗-terrace is 1, in which case the
R∗-terrace is standard.
Given a standard R∗-terrace with this notation, suppose there is a
value r such that br = a−1
r+1. Then r is a right match-point of b. We
M. A. Ollis 313
will require standard R∗-terraces whose associated R∗-2-sequencings have
a right match-point r with 2 6 r 6 n − 3. An equivalent object is an
extendable terrace: a basic terrace (e, a2, . . . , an) is extendable if an =
a2
2 and aj−1aj+1 = aj = aj+1aj−1 for some j with 5 6 j < n. The
circular sequence (a1, a2, . . . , an−1 ←֓) is a standard R∗-terrace whose
R∗-2-sequencing has a right match-point r where 2 6 r 6 n − 3 if and
only if
(e, ar+1, ar+2, . . . , an−1, a1, a2, . . . , ar)
is an extendable terrace [15]. This relationship is illustrated in Examples 1
and 2: the standard R∗-terrace in Example 1 has 5 as a match-point
which can be used to give the extendable terrace in Example 2.
We can now give the main theorem for constructing terraces.
Theorem 7. Let G = H ×K with |H| = 4m and |K| = t. If H has a
directed R-terrace with a t-compatible σ ∈ S4m−1 and K has an extendable
terrace then G has an extendable terrace.
Proof. First, from the extendable terrace, construct an R∗-terrace k for K
that has a right match-point in position r, where 2 6 r 6 t− 3. Let a be
the directed R-terrace and apply the main construction to a and k. We
claim that this gives an R∗-terrace for G that has a right match-point in
position r and hence that G has an extendable terrace.
Compared to the proof of Theorem 1, all that changes is that some
non-involutions g ∈ K may appear as ℓi and ℓj , with i 6= j, in the R∗-2-
sequencing of K and, if this is the case for a given g, then g−1 does not
appear in the R∗-2-sequencing of K.
The consequence for our purported R-2-sequencing is that, for any
given non-involution h ∈ H, rather than having each of the four elements
of the form (h±1, g±1) once, we have (h, g) and (h−1, g) twice and neither
(h, g−1) nor (h−1, g−1) appears. This does not break the constraints of
being an R-2-sequencing. Similarly, if h ∈ H is an involution then we
have (h, g) twice and (h, g−1) does not appear.
Finally, as the first t− 2 elements of the R∗-terrace are
(e, k1), (e, k2), . . . , (e, kt−2),
the right match-point at position r of the R∗-2-sequencing is maintained.
As with the R-sequencing result, we have an analogue for central
factors:
314 R-sequencings and terraces
Theorem 8. Let G be a group of order 4mt with central factor H of
order 4m. If H has a directed R-terrace a with a t-compatible σ ∈ S4m−1
and G/H has an extendable terrace (H, K2, K3, . . . , Kt) with j as the
position of the element that is the product of its neighbours and such
that there are elements kj−1 ∈ Kj−1 and kj+1 ∈ Kj+1 that commute,
then G has an extendable terrace. If m = 1 and t ≡ 0 (mod 3) then the
requirement for a t-compatible permutation σ may be dropped.
Proof. Turn the extendable terrace for G/H into its equivalent standard
R∗-terrace and the argument then mirrors that of Theorem 4. It is possible
to ensure that the match-point condition is met by careful choice of the ki.
As in Theorem 7, the standard R∗-terrace for G that emerges has an
equivalent extendable terrace.
These results allow the construction of terraces for many infinite
families of groups for which terraces were not previously known, even
more so in conjunction with this powerful result for constructing new
terraces from existing ones:
Theorem 9. [2, 3] Let G be a group with a normal subgroup N . If N
has odd order and G/N has a terrace then G has a terrace. If N has odd
index and N has a terrace then G has a terrace.
For example:
Corollary 2. Let G be of the form H1 ×H2 × · · · ×Hs ×K ×N , where
each Hi is one of Z2
2, Z3
2, D8, Z6 × Z2, D12 or A4, the group K has an
extendable terrace, and |N | is odd. Then H1 ×H2 × · · · ×Hs ×K has an
extendable terrace and G has a terrace.
Proof. To show that H1 ×H2 × · · · ×Hs ×K has an extendable terrace,
repeatedly apply Theorem 7 or, if Hi
∼= Z
2
2, Theorem 8 when necessary.
Use Theorem 9 to complete the proof.
Groups that are known to have an extendable terrace include: Zs,
where s > 7 and s is not twice an odd number [12,14]; abelian 2-groups
of order at least 8 that are not elementary abelian [12,14]; Zs
2 ×Zp where
s > 2 and p is an odd prime [12–15]; non-abelian groups of order 12, 16
or 20 [15]; D8s for s > 1 [15]; and these two additional families of groups
of orders 8s, with s > 1 [15] (the first are the semidihedral groups; the
second don’t seem to have an accepted name in the literature):
SD8s = 〈u, v : u4s = e = v2, vu = u2s−1v〉,
M8s = 〈u, v : u4s = e = v2, vu = u2s+1v〉.
M. A. Ollis 315
In [15] it is suggested that groups with many involutions might be the
most promising place to look for a counterexamples to Bailey’s Conjecture.
Many new such groups are now known to be terraced; for example, Zr
2 ×
Ds
8 ×Dt
12 provided that r 6= 1 and t > 0.
Acknowledgements
I am grateful to Gage Martin and Devin Willmott for their insights
and commentary relating to this work.
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Contact information
M. A. Ollis Marlboro College,
P.O. Box A, Marlboro, Vermont 05344, USA
E-Mail(s): matt@marlboro.edu
Received by the editors: 25.11.2015
and in final form 04.12.2015.
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