On elementary domains of partial projective representations of groups
We characterize the finite groups containing only elementary domains of factor sets of partial projective representations. A condition for a finite subset A of a group G, which contains the unity of the group, to induce an elementary partial representation, of G whose (idempotent) factor set is tot...
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irk-123456789-1522642019-06-10T01:25:53Z On elementary domains of partial projective representations of groups Pinedo, H. We characterize the finite groups containing only elementary domains of factor sets of partial projective representations. A condition for a finite subset A of a group G, which contains the unity of the group, to induce an elementary partial representation, of G whose (idempotent) factor set is total is given. Finally, we characterize the elementary partial representation of abelian groups of degrees ≤ 4 with total factor set. 2013 Article On elementary domains of partial projective representations of groups / H. Pinedo // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 63–82. — Бібліогр.: 9 назв. — англ. 1726-3255 2010 MSC:Primary 20C25; Secondary 20M18. http://dspace.nbuv.gov.ua/handle/123456789/152264 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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We characterize the finite groups containing only elementary domains of factor sets of partial projective representations. A condition for a finite subset A of a group G, which contains the unity of the group, to induce an elementary partial representation, of G whose (idempotent) factor set is total is given. Finally, we characterize the elementary partial representation of abelian groups of degrees ≤ 4 with total factor set. |
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Pinedo, H. |
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Pinedo, H. On elementary domains of partial projective representations of groups Algebra and Discrete Mathematics |
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Pinedo, H. |
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Pinedo, H. |
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On elementary domains of partial projective representations of groups |
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On elementary domains of partial projective representations of groups |
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On elementary domains of partial projective representations of groups |
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On elementary domains of partial projective representations of groups |
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On elementary domains of partial projective representations of groups |
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on elementary domains of partial projective representations of groups |
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Інститут прикладної математики і механіки НАН України |
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On elementary domains of partial projective representations of groups / H. Pinedo // Algebra and Discrete Mathematics. — 2013. — Vol. 15, № 1. — С. 63–82. — Бібліогр.: 9 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT pinedoh onelementarydomainsofpartialprojectiverepresentationsofgroups |
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2025-07-13T02:41:28Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 15 (2013). Number 1. pp. 63 – 82
c© Journal “Algebra and Discrete Mathematics”
On elementary domains
of partial projective representations of groups
Hector Edonis Pinedo Tapia1
Communicated by B. V. Novikov
Abstract. We characterize the finite groups containing
only elementary domains of factor sets of partial projective re-
presentations. A condition for a finite subset A of a group G, which
contains the unity of the group, to induce an elementary partial
representation of G whose (idempotent) factor set is total is given.
Finally, we characterize the elementary partial representation of
abelian groups of degrees ≤ 4 with total factor set.
Introduction
Elementary partial representations were introduced and studied by
M. Dokuchaev and N. Zhukavets in [6]. It turned out that together with
the irreducible (indecomposable) representations they are elementary
blocks from which the irreducible (indecomposable) partial representa-
tions can be constructed. In particular, given a finite group G and a
field K, whose characteristic does not divide the order of G, every partial
K-representation of G can be obtained from the elementary partial repre-
sentations of G and the (usual) irreducible representations of subgroups
of G.
Any elementary partial representation Γ is considered as a partial
projective representation with idempotent factor set σ, and the domain of
1This work was supported by FAPESP of Brazil.
2010 MSC: Primary 20C25; Secondary 20M18.
Key words and phrases: elementary partial representation, partial projective
representation, elementary domain, total factor set.
64 On elementary domains
such a σ is called elementary domain. It is known (see [3]), that for any
finite group G the domain of any partial factor set of G can be constructed
by taking unions of some elementary domains and, conversely, the union
of any collection of elementary domains is a domain for the factor set of
some partial projective representation of G. The main purpose of this
article is to characterize the finite groups which have only elementary
domains.
This paper is structured as follows. After the introduction, we recall
in Section 1 some background and give a few preliminary results. In
Section 2, elementary partial representations are treated and an algorithm
for finding them is given. Those considerations allow us in Section 3, to
calculate all the elementary domains for cyclic groups of order ≤ 5 and
with the help of Theorem 3.2 we conclude that any cyclic group of prime
order p > 5 has non-elementary domains. With the results of Section 3,
we guarantee that the finite groups containing only elementary domains
have order 2m3n, for some m,n ∈ N. In Sections 4 and 5, it is proved that
the only possibilities for m,n are (m,n) = (0, 0), (0, 1) or (1, 0).
In Section 6 a condition for a set A ∈ P1(G) to induce an elementary
partial representation with total factor set is given. Finally, in the last
section we use the description given in [6], of the elementary partial
representations of abelian groups of degrees less than or equal to 4, to
find the sets A ∈ P1(G) that induce elementary partial representations
whose factor sets are total.
1. Basic definitions and some results
For any group G with identity 1 and any field K, the group algebra
KG governs the theory of K-representations of G; in an analogous way
the partial group algebra KparG, which is the semigroup algebra KE(G),
controls the partial representations of G. We recall that E(G) is the
monoid generated by the symbols {[x] |x ∈ G} with defining relations:
[x−1][x][y] = [x−1][xy], [x][y][y−1] = [xy][y−1] and [x][1] = [x],
(it follows that [1][x] = [x]). This monoid was defined by R.Exel in [7]
(see the semigroup S(G)). The construction of Exel was completed in [8]
as follows. Denote by P1(G) the set of all finite subsets of G containing 1.
Put
G̃R = {(A, g) ∈ P1(G) ×G | g ∈ A},
H. Pinedo 65
with multiplication (A, a)(B, b) = (A ∪ aB, ab); G̃R is the Birget-Rhodes
expansion of G (see [9]), and as it was shown in [8] the map
E(G) ∋ [x] → ({1, x}, x) ∈ G̃R,
determines an isomorphism from E(G) to G̃R.
We shall identify the elements of E(G) with their images in G̃R. For
an element x = (A, a) ∈ E(G), the set A will be called the support of x.
We shall need to work with a quotient semigroup E3 of E(G). Denote
by N3 the ideal {(A, a) ∈ E(G) | a ∈ A ⊆ G, |A| ≥ 4} and by E3 the
factor semigroup E(G)/N3. Evidently,
E3 = {(A, a) ∈ E(G) | |A| ≤ 3} ∪ 0.
Given a semigroup S, the Green relation J is defined for S, as follows:
(x, y) ∈ J ⇐⇒ S1xS1 = S1yS1.
In particular, for any (two sided) ideal I of S, we get I =
⋃
x ∈I
Jx.
We recall some results about the J -classes of E(G). For sets
A,B ∈ P1(G), define A � B ⇐⇒ ∃x ∈ G xA ⊆ B.
Lemma 1.1. [3] If (A, a), (B, b) ∈ E(G), then
A � B ⇐⇒ (B, b) ∈ E(G)(A, a)E(G).
The next result gives a characterization of the J -classes of E(G).
Proposition 1.2. [3] Let (A, a) ∈ E(G). Then
J(A,a) = { (B, b) ∈ E(G) | ∃x ∈ G xA = B}. (1)
It follows that J(A, a) does not depend on a ∈ A, so we shall write JA
instead of J(A, a). Further, if A = {1, a} we shall denote JA by Ja and if
A = {1, a, b}, JA will be denoted by Ja, b.
Definition 1.3. A set A ∈ P1(G) is called an n-set if |A| = n.
Now we determine the non-trivial J -classes of the semigroup E3.
The J - class of a 2-set
Let A = {1, a} be a 2-set. Condition (1) means that (B, b) ∈ Ja if
and only if xA = B, for some x ∈ G. Then if B = {1, b} (b 6= 1) we have
{x, xa} = {1, b}, which implies x = 1 or xa = 1. From this we conclude
b = a or b = a−1. Consequently, B = A or B = {1, a−1}, and we obtain
Ja = Jb ⇐⇒ b ∈ {a, a−1}. (2)
66 On elementary domains
The J - class of a 3-set
Consider now a 3-set A = {1, a1, a2}. We know that (B, b) ∈ Ja1, a2
exactly when {x, xa1, xa2} = B for some x ∈ G. Since 1 ∈ B, we have
the following cases
(i) If x = 1, then A = B,
(ii) if xa1 = 1, then {x, xa1, xa2} = {1, a−1
1 , a−1
1 a2} = B,
(iii) if xa2 = 1, then {x, xa1, xa2} = {1, a−1
2 a1, a
−1
2 } = B.
Hence
Ja1, a2 = Ja−1
1 , a−1
1 a2
= Ja−1
2 a1, a−1
2
. (3)
For n ∈ N, we denote by Cn the cyclic group of order n.
Now we compute the ideal generated by an idempotent element whose
support is given by a 2-set in the cyclic group.
Proposition 1.4. For 0 < s < t the ideal Is = 〈({1, as}, 1)〉 of E3(Ct) is
of the form Is = Jas ∪
t−1
⋃
m=1
m6=s
Jam, as ∪ 0.
Proof. For 1 ≤ s ≤ t − 1 and 1 ≤ m ≤ n ≤ t − 1 we want to find the
triples {1, am, an} (including {1, am} = {1, am, am} when m = n) such
that ({1, am, an}, 1) ∈ Is. By Lemma 1.1, this occurs exactly when there
exists x ∈ Ct such that {x, xas} ⊆ {1, am, an}. We consider the possible
cases for x.
Case 1 x = 1. Then {x, xas} = {1, as} ⊆ {1, am, an}. Hence m = s
or n = s, and we get the triples {1, am, as}, where 1 ≤ m ≤ t− 1.
Case 2 x = am. Then we have {x, xas} = {am, am+s} ⊆ {1, am, an}.
Therefore am+s ∈ {1, an}, then m = t − s or n = m + s, and the
triples are {1, at−s, an} and {1, am, am+s}. By (3) Jat−s, an = Jas, an+s and
Jam, am+s = Jat−m, as .
Case 3 x = an. In this case we obtain {x, xas} = {an, an+s} ⊆
{1, am, an}, and consequently an+s ∈ {1, am}, this implies n = t − s or
m = n+s−t. In this case the triples are {1, am, at−s} and {1, an+s−t, an}.
On the other hand by (3) Jam,at−s = Jas+m, as and Jan+s−t,an = Jas,at−n .
Then the triples appearing in the last two cases are such that their
J -classes are equal to J -classes of triples in Case 1. From this we conclude
that Is = Jas ∪
t−1
⋃
m=1
m6=s
Jam, as ∪ 0.
Corollary 1.5. With the notations of the Proposition 1.4, the only J -
class of a 2-set contained in Is is Js.
H. Pinedo 67
Given a subgroup H of G, we denote by NG(H) the normalizer of
H in G, and by Mn(KH) the ring of (n × n)-matrices over the group
algebra of KH. We have the next.
Theorem 1.6. [5] Let K be a field, G a finite group and let C denote
a full set of representatives of the conjugacy classes of subgroups of G.
Then the partial group algebra of G over K is of the form:
KparG ∼=
⊕
H∈C
1 ≤m ≤(G : H)
cm(H)Mm(KH), (4)
where cm(H) =
1
m
(G : NG(H))
(
(G : H) − 1
m− 1
)
−
∑
H< B ≤G
(B : H) | m
m/(B : H)cm/(B : H)(B)
(G : NG(B))
and cm(H)Mm(KH) means the direct sum of cm(H) copies of Mm(KH).
We shall need some examples from [1] which are simple consequences
of Theorem 1.6.
Corollary 1.7. (i) For any prime number p,
Kpar(Cp) ∼=
p−1
⊕
m=1
1
m
(
p− 1
m− 1
)
Mm(K) ⊕KCp,
(ii) Kpar(C9) ∼= K⊕4M2(K)⊕14M4(K)⊕14M5(K)⊕4M7(K)⊕M8(K)⊕
KC3 ⊕M2(KC3) ⊕ 9M3(K) ⊕ 9M6(K) ⊕KC9,
(iii) Kpar(C3 ×C3) ∼= K ⊕ 4M2(K) ⊕ 14M4(K) ⊕ 14M5(K) ⊕ 4M7(K) ⊕
M8(K) ⊕ 4KC3 ⊕ 4M2(KC3) ⊕ 8M3(K) ⊕ 8M6(K) ⊕K(C3 × C3),
(iv) Kpar(C2 × C2) ∼= K ⊕M3(K) ⊕ 3KC2 ⊕K(C2 × C2),
(v) Kpar(S3) ∼= K ⊕ M2(K) ⊕ 3M3(K) ⊕ M4(K) ⊕ M5(K) ⊕ 3KC2 ⊕
3M2(KC2) ⊕KS3.
Denote by ψ : KparG → ⊕Ml(KH) the isomorphism established in
the proof of Theorem 1.6 (see Section 2). Let Pr = Prl be the projection
of ⊕Ml(KH) onto the matrix algebra Ml(KH). Consider also the map
[ ] : G ∋ g 7→ [g] ∈ KparG. A function of the form
Γ = Pr ◦ ψ ◦ [ ] : G → Ml(KH) (5)
68 On elementary domains
is called an elementary partial representation of G and we shall say that
the set D = {(x, y) ∈ G×G | Γ(x) Γ(y) 6= 0} is an elementary domain.
To define partial projective representations we need the concept of
K-cancellative monoid. A K-semigroup is a semigroup S with 0 together
with a map K×S ∋ (α, x) → αx ∈ S, satisfying α(βx) = (αβ)x, α(xy) =
(αx)y = x(αy), 1Kx = x and 0Kx = 0 for any α, β ∈ K,x, y ∈ S.
By a K-cancellative semigroup we mean a K-semigroup S such that for
any α, β ∈ K and 0 6= x ∈ S one has αx = βx =⇒ α = β.
Example 1. For any group algebra KG and n ∈ N, Mn(KG) is a K-
cancellative monoid.
Definition 1.8. [2, Theorem 3] Let M be a K-cancellative monoid. A
map Γ: G → M is a partial projective representation of G if and only if:
• For all x, y ∈ G,
Γ(x−1)Γ(xy) = 0 ⇔ Γ(x)Γ(y) = 0 ⇔ Γ(xy)Γ(y−1) = 0;
• There exists a unique partially defined map σ : G×G → K∗, with
domain: dom σ = {(x, y) | Γ(x)Γ(y) 6= 0}, such that for all (x, y) ∈
dom σ
Γ(x−1)Γ(x)Γ(y) = σ(x, y)Γ(x−1)Γ(xy),
Γ(x)Γ(y)Γ(y−1) = σ(x, y)Γ(xy)Γ(y−1).
The map σ is called a factor set of Γ or a partial factor set of G. It shall
be convenient to set σ(x, y) = 0 for each (x, y) ∈ G×G with Γ(x)Γ(y) = 0,
and maintain the notation dom σ for the set of pairs (x, y) ∈ G×G with
Γ(x)Γ(y) 6= 0.
The partial factor sets σ of G form a commutative inverse semigroup
which we denote by pm(G) (see [2]), and its quotient semigroup by a
natural congruence ∼ is the partial Schur multiplier pM(G) of G (see
[2, p. 259]). The semigroups pm(G) and pM(G) are disjoint unions of
abelian groups called components; described as follows.
Consider the next transformations in G×G
g : (x, y) 7→ (xy, y−1), h : (x, y) 7→ (y−1, x−1) and t : (x, y) 7→ (x, 1).
(6)
The maps in (6) satisfy the relations
g2 = h2 = 1, (gh)3 = 1, t2 = t, gt = t, tght = thgh, tht = 0, (7)
H. Pinedo 69
where 0 is the map (x, y) 7→ (1, 1). Take the abstract semigroup T
generated by the symbols g, h, t with the relations (7). The maps in (6)
determine an action of T on G×G. Denote by C(G) the semilattice of all
non-empty T -subsets of G×G with respect to the set theoretic inclusion
and intersection.
Theorem 1.9. [2] The semigroups pm(G) and pM(G) are semilattices
of abelian groups
pm(G) =
⋃
X∈C(G)
pmX(G), pM(G) =
⋃
X∈C(G)
pMX(G),
where pmX(G) = {σ ∈ pm(G) | dom σ = X} and pMX(G) =
pmX(G)
∼
.
Denote by Y ∗(E3) the semilattice of two sided ideals of E3 different
from E3 and ∅, with respect to the set theoretic union and inclusion. The
following result shall be useful.
Proposition 1.10. [4] The semigroups (C(G),∩) and (Y ∗(E3),∪) are
isomorphic.
The next assertion states that every domain of a partial factor set of
G can be constructed from the elementary ones.
Theorem 1.11. [3] Let G be a finite group. The domain of any partial
factor set of G is a union of elementary domains and, conversely, any
union of elementary domains is a domain for the factor set of a partial
projective representation of G.
Theorem 1.11 shows the importance of the elementary domains and
suggests the problem of a characterization of those finite groups which
have only elementary domains.
2. On the behavior of elementary partial representations
In order to study the behavior of the elementary partial representations
of G, we will consider a groupoid β(G) associated to G.
The groupoid β(G) is the small category with objects (A, 1) and
morphisms (A, g), where g ∈ G and A is a finite subset of G containing 1
and g−1. The composition (A, g) · (B, h) in β(G) is defined for the pairs
(A, g) and (B, h), such that A = hB, in which we define (hB, g) · (B, h) =
(B, gh).
70 On elementary domains
Clearly the identity morphisms in β(G) are (A, 1) with A ⊆ G, and the
inverse of (A, g) is (gA, g−1).
It is useful to represent the groupoid β(G) by an oriented graph Eβ(G),
whose vertexes are the identity morphisms (A, 1), and each morphism in
β(G) gives an oriented arrow (A, g) : (A, 1) −→ (gA, 1), from the vertex
s(A, g) = (A, 1) to the vertex r(A, g) = (gA, 1). Note that each connected
component of Eβ(G) is a subgroupoid (with the same composition).
We conclude that the vertexes belonging to the connected component
of (A, 1) are of the form (g−1
i A, 1), where A =
⋃m
i=1 (StA)gi is a disjoint
union of cosets and StA = {g ∈ G | gA = A}. In particular, the number
of vertexes in the connected component of (A, 1) is m = |A|
|St A| .
For an arbitrary groupoid β, denote by β(2) ⊆ β × β the set of all
composable pairs. The groupoid algebra Kβ is a vector space over K with
base β and multiplication given by
γ1γ2 =
{
γ1 · γ2 if (γ1, γ2) ∈ β(2),
0 if (γ1, γ2) /∈ β(2),
extended by linearity to Kβ.
Proposition 2.1. [1] Let β be a groupoid such that Eβ is connected and
has m vertexes. Let H = {γ ∈ β | s(γ) = r(γ) = x1} be the isotropy group
of a vertex x1 ∈ Eβ. Then Kβ ∼= Mm(KH) as K-algebras.
The isomorphism given in [1] is as follows:
Set ξ1, i = {γ ∈ β | s(γ) = x1, r(γ) = xi}, where 1 ≤ i ≤ m. Fix γi ∈ ξ1, i,
then any γ ∈ β with s(γ) = xi and r(γ) = xj can be written uniquely in
the form γ = γjhγ
−1
i , with h ∈ H. The map
Kβ ∋ γ 7→ ej,i(h) ∈ Mm(KH) (8)
extended by linearity, gives the desired isomorphism.
Remark 1. If a groupoid β is a finite union of subgroupoids βi, the
groupoid algebra Kβ is a direct sum ⊕iKβi. Thus by Proposition 2.1,
there is an isomorphism ⊕φi : Kβ → ⊕Mmi
(KHi).
In the groupoid β(G), when identifying the set A with the vertex
(A, 1) of the graph Eβ(G), and the arrow (A, g) of Eβ(G) with the element
g ∈ G, we obtain that StA coincides with the isotropy group of (A, 1).
From now on we always suppose that the group G is finite.
H. Pinedo 71
In [1] it was shown that the mapping λp : G → Kβ(G) defined by:
λp(g) =
∑
1,g−1∈A
(A, g) (9)
is a partial representation of G. We also recall the next.
Theorem 2.2. [1] For a finite group G and any field K, there is a K-
algebra isomorphism α : KparG → Kβ(G) such that α([g]) = λp(g).
From Remark 1 and Theorem 2.2, we obtain an isomorphism ⊕φi ◦
α : Kpar(G) → ⊕Mmi
(KHi). Thus the elementary partial representations
of G have the form
Γ = (Prj ◦ ⊕φi) ◦ (α ◦ [ ]) = φj ◦ λp : G → Mmj
(KHj). (10)
Let A ∈ P1(G) such that H = Hj = StA and let m = mj be the
index of H in A, write A =
⋃m
i=1Hgi as union of disjoint cosets (g1 = 1)
and φ = φj . Then by (8)
Γ (g) =
∑
g−1∈g−1
i
A
φ((g−1
i A, g)) =
∑
g=g−1
t ht, igi
et, i(ht, i). (11)
It shall be convenient to state an algorithm that helps us to determine
elementary partial representations. For any g ∈ G, consider the set:
Ig = {i ∈ {1, . . . ,m} | gig ∈ A}. (12)
The map Γ can be defined as follows:
If Ig = ∅ set:
Γ(g) = 0. (13)
If Ig 6= ∅, then for any i ∈ Ig there exists a unique j = ji,g in
{1, . . . ,m} and h = hi,g ∈ H such that gig = hgj . By (11) we have
Γ(g) =
∑
i∈Ig
ei,j(h). (14)
Remark 2. Every elementary partial representation, Γ : G → Mm(KH)
is monomial over H. That is, for every g ∈ G each row and each column
of the matrix Γ(g) contains at most one non-zero entry, which belongs to
H.
72 On elementary domains
Summarizing, in order to obtain the elementary domains of a finite
group G, we need to find all its elementary partial representations. For this,
we consider the groupoid β(G) and take X ⊂ β(G) a connected component
of some vertex A with stabilizer H. If X has m vertexes, formula (10) tells
us that we must compose the map λp defined in (9) with the isomorphism
φ : Kβ(G) → Mm(KH) established in Proposition 2.1. The isomorphism
φ depends on the choice of the arrows (A, g1), . . . , (A, gm). The next
lemma tells us that the elementary domain associated to Γ = φ ◦ λp does
not depend on the choice of the labeling elements g1, . . . , gm.
Lemma 2.3. [6] Let X be a connected component of β(G) with m ver-
texes. Fix a vertex A with stabilizer H and pick two different collections
{g1, . . . , gm}, {g′
1, . . . , g
′
m} such that the set {g1A, . . . , gmA} coincides
with {g′
1A, . . . , g
′
mA} and gives all the vertexes of X .
Let φ1 : KX → Mm(KH) and φ2 : KX → Mm(KH) be the isomor-
phisms determined by {g1, . . . , gm} and {g′
1, . . . , g
′
m} respectively. Then
there exists an invertible matrix C ∈ Mm(KH) whose non-zero entries
are all in H such that φ1(x) = C−1φ2(x)C, for all x ∈ Kβ.
We denote by CD(G) the set formed by the elementary domains. Using
Lemma 2.3, Theorems 1.6,1.9 and Proposition 1.10 we obtain the next.
Theorem 2.4. For a finite group G, the following inequalities hold
|CD(G)| ≤
∑
H ∈ C
1 ≤m ≤(G : H)
cm(H) ≤ |C(G)| = |Y ∗(E3)|, (15)
where C, H and cm(H) are as in Theorem 1.6.
3. Elementary domains and cyclic groups
We start this section by calculating all the elementary domains of the
cyclic groups of orders ≤ 5. In particular, we shall see that C4 and C5
have non-elementary domains.
We denote by Di = {(x, y) ∈ G×G | Γi(x)Γi(y) 6= 0} the elementary
domain associated to the elementary partial representation Γi and make
the “identifications" A ≡ (A, 1) and g ≡ (A, g).
The cyclic group C2 = 〈a | a2 = 1〉. In this case the groupoid is
β(C2) = {({1}, 1), ({1, a}, 1), ({1, a}, a)} whose vertexes are ({1}, 1) and
({1, a}, 1)}. We have the arrows
H. Pinedo 73
1: {1} → {1}, 1: {1, a} → {1, a} and a : {1, a} → {1, a}. Hence Eβ
has two connected components,
Eβ1 = {({1}, 1)} and Eβ2 = {({1, a}, 1), ({1, a}, a)}.
Observe that St {1} = {1} and St {1, a} = {1, a}. Therefore using (13)
and (14) we obtain the elementary partial representations of C2 :
Γ1 = G → K{1} and Γ2 = G → K{1, a},
where Γ1(1) = 1, Γ1(a) = 0, Γ2(1) = 1, Γ2(a) = a. Then D1 = {(1, 1)}
andD2 = C2×C2. By Theorem 1.11,C2 contains only elementary domains.
The cyclic group C3 = 〈a | a3 = 1〉. The associated groupoid is
β(C3) = {({1}, 1), ({1, a}, 1), ({1, a}, a2), ({1, a2}, 1), ({1, a2}, a),
({1, a, a2}, 1), ({1, a, a2}, a), ({1, a, a2}, a2)}.
The non-trivial oriented arrows of Eβ are:
a2 : {1, a} → {1, a2}, a : {1, a2} → {1, a}, a, a2 : C3 → C3.
Therefore there are three connected components of Eβ, as follows: Eβ1 =
{({1}, 1)}, Eβ2 = {({1, a}, 1), ({1, a}, a2), ({1, a2}, 1), ({1, a2}, a)} and
Eβ3 = {(C3, 1), (C3, a), (C3, a
2)}.
Set A1 = {1}, A2 = {1, a} and A3 = C3, then StAi = Ai, where
i ∈ {1, 3} and StA2 = 1. By (13) and (14) the elementary partial repre-
sentations of C3 are:
Γ1 : C3 → K1, Γ2 : C3 → M2(K1) and Γ3 : C3 → KC3, given by :
Γ1 : 1 7→ 1, a 7→ 0, a2 7→ 0, Γ3 : 1 7→ 1, a 7→ a, a2 7→ a2, and
Γ2 : 1 7→
(
1 0
0 1
)
, a 7→
(
0 1
0 0
)
, a2 7→
(
0 0
1 0
)
,
Finally we get
D1 = {(1, 1)}, D2 = {(1, 1), (1, a), (1, a2), (a, 1), (a2, 1), (a, a2), (a2, a)}
and D3 = C3 × C3. Since D1 ⊂ D2 ⊂ D3, Theorem 1.11 implies that C3
contains only elementary domains.
The cyclic group C4 = 〈a | a4 = 1〉. The non-trivial oriented arrows
are:
a3 : {1, a} → {1, a3}, a : {1, a3} → {1, a},
a2 : {1, a2} → {1, a2},
a2 : {1, a, a2} → {1, a2, a3}, a2 : {1, a2, a3} → {1, a, a2},
74 On elementary domains
a : {1, a, a3} → {1, a, a2}, a3 : {1, a, a2} → {1, a, a3},
a : {1, a2, a3} → {1, a, a3}, a3 : {1, a, a3} → {1, a2, a3}.
Set A1 = {1}, A2 = {1, a}, A3 = {1, a2}, A4 = {1, a, a3} and A5 = C4.
For Hi = StAi, we obtain H1 = H2 = H4 = 1, H3 = A3 and H5 = C4.
Hence there are five elementary partial representations of C4, as follows:
Γ1 : C4 → K, 1 7→ 1K , a 7→ 0, a2 7→ 0, a3 7→ 0,
Γ2 : C4 → M2(K), 1 7→ Id, a 7→ e12, a
2 7→ 0, a3 7→ e21,
Γ3 : C4 → KC2, 1 7→ 1, a 7→ 0, a2 7→ 1, a3 7→ 0,
Γ4 : C4 → M3(K), 1 7→ Id, a 7→ e21 + e13, a2 7→ e23 + e32,
a3 7→ e12 + e31,
Γ5 : C4 → KC4, 1 7→ 1, a 7→ a, a2 7→ a2, a3 7→ a3,
and the elementary domains of C4 are:
D1 = {(1, 1)}, D2 = {(1, 1), (1, a), (1, a3), (a, 1), (a3, 1), (a, a3), (a3, a)},
D3 = {(1, 1), (1, a2), (a2, 1), (a2, a2)} and D4 = D5 = C4 × C4.
Observe that D = D2 ∪D3 is a non-elementary domain of C4.
The cyclic group C5 = 〈a | a5 = 1〉. We can show that the elemen-
tary domains of C5 are:
D1 ={(1, 1)}, D2 = {(1, 1), (1, a), (1, a4), (a, 1), (a4, 1), (a, a4), (a4, a)},
D3 ={(1, 1), (1, a2), (1, a3), (a2, 1), (a3, 1), (a3, a2), (a2, a3)},
D4 ={(1, C5), (C5, 1), (a, a), (a, a3), (a, a4), (a4, a), (a3, a), (a3, a2),
(a2, a3), (a2, a4), (a4, a2), (a4, a4)},
D5 ={(1, C5), (C5, 1), (a, a2), (a2, a), (a, a4), (a4, a), (a3, a2), (a2, a2),
(a2, a3), (a3, a4), (a4, a3), (a3, a3)} and D6 = D7 = C5 × C5.
Note that D = D2 ∪D3 is not an elementary domain of C5.
By our calculations, the groups C1, C2 and C3 contain only elementary
domains but for the groups C4 and C5 there are domains that are not
elementary. We shall prove that for any prime number p greater than 5
the cyclic group Cp contains non-elementary domains. First, we give a
technical lemma.
Lemma 3.1. Let p be a prime number, p > 5. Then
2
(p−1)(p−2)
6 + 2
p−1
2 − 1 >
p−1
∑
m=1
1
m
(
p− 1
m− 1
)
+ 1.
H. Pinedo 75
Proof. It is enough to prove that 2
(p−1)(p−2)
6 +2
p−1
2 −3 >
p−1
∑
m=2
1
m
(
p− 1
m− 1
)
.
Since
p−1
∑
m=2
1
m
(
p− 1
m− 1
)
<
1
2
p−1
∑
m=2
(
p− 1
m− 1
)
= 2p−2 − 1, it is sufficient to
establish the inequality 2
(p−1)(p−2)
6 + 2
p−1
2 − 3 > 2p−2 − 1.
Since p is a prime number greater than 5, we have (p−1)(p−2)
6 ≥ p − 2.
Consequently 2
(p−1)(p−2)
6 + 2
p−1
2 − 3 ≥ 2p−2 + 2
p−1
2 − 3 > 2p−2 − 1.
Theorem 3.2. If p is a prime number greater than 5, then there are
non-elementary domains for Cp.
Proof. By (i) of Corollary 1.7 and Theorem 2.4, it is enough to prove that
the cardinality of Y ∗(E3) is greater than
p−1
∑
m=1
1
m
(
p− 1
m− 1
)
+ 1.
For this, we calculate the number of the J -classes of E3(Cp), where
Cp = 〈a | ap = 1〉. Recall that by (2) Jam = Jap−m for 1 ≤ m ≤ p− 1.
Therefore there are p−1
2 different J -classes induced by the 2-sets. Let
I1 = 〈({1, a}, 1)〉, I2 = 〈({1, a2}, 1)〉, · · · , I p−1
2
= 〈({1, a
p−1
2 }, 1)〉,
be all the different ideals generated by idempotent elements of E3(Cp)
with 2-sets as supports. Using Corollary 1.5, it is seen that by taking
unions of those ideals we form 2
p−1
2 − 1 different ideals of E3(Cp).
With respect to the 3-sets, observe that there are
(
p− 1
2
)
sets of the
form {1 , am, an} with 1 ≤ m < n < p and by (3) Jam, an = Jap−m, an−m =
Jap+m−n, ap−n . Since p is a prime number greater than 5, the sets
{1, am, an}, {1, ap−m, an−m}, {1, ap+m−n, ap−n}
are all different. Thus there are (p−1)(p−2)
6 different J -classes of the 3-
sets. Now consider the ideals Im, n = 〈({1, am, an}, 1)〉. By Lemma 1.1
Im, n = Jam, an ∪0. Therefore using the J -classes of the 3-sets, we construct
2
(p−1)(p−2)
6 − 1 different ideals in E3(Cp), which are unions of the ideals
Im,n, (0 < m < n < p). Note that none of these ideals contains an
element of E3(Cp) with a 2-set as support. Then they are all different
from the ideals which correspond to the J -classes of the 2-sets. Since 0 is
also an ideal, there are at least 2
(p−1)(p−2)
6 + 2
p−1
2 − 1 nontrivial ideals in
E3(Cp). Finally, by Lemma 3.1, we conclude that Cp has non-elementary
domains.
76 On elementary domains
Now we prove the next.
Proposition 3.3. Let H be a proper subgroup of a finite group G. If
H contains non-elementary domains, then G contains non-elementary
domains.
Proof. Let D be a non-elementary domain of H. By Theorem 1.9, D is
a T -set of H ×H, where T is the semigroup generated by the symbols
g, h, t with defining relations given in (7). Since D ⊆ G × G is a T -set
(considering T acting on G×G), we have D ∈ C(G).
Suppose that D is an elementary domain of G and take an elementary
partial representation Γ: G → Ml(KH
′), where H ′ = StA for some
A ∈ P1(G) such that D = {(x, y) ∈ G×G | Γ(x)Γ(y) 6= 0}. If there exists
x ∈ A \H, by (14) Γ(x) 6= 0. On the other hand, the pair (x, 1) is not in
D which implies Γ(x) = 0. We conclude that A ⊆ H and H ′ = StA is a
subgroup of H. Thus the map Γ′ = Γ|H : H → Ml(KH
′) is an elementary
partial representation of H and
{(x, y) ∈ H×H | Γ′(x)Γ′(y) 6= 0} = {(x, y) ∈ G×G | Γ(x)Γ(y) 6= 0} = D.
Therefore D is an elementary domain of H, contradicting our hypothesis.
Hence D is not an elementary domain of G.
From Theorem 3.2 and Proposition 3.3 we obtain the next.
Proposition 3.4. Let G be a finite group such that there exists a prime
number p ≥ 5 dividing the order of G. Then G has non-elementary do-
mains. Equivalently, if a finite group G contains only elementary domains,
there exist m,n ∈ N such that |G| = 2m3n.
4. Elementary domains and abelian groups
The purpose of this section is to prove that the finite abelian groups
containing only elementary domains are C1, C2 and C3. That is, the only
possibilities for m and n in Proposition 3.4 are (m,n) = (0, 0), (1, 0) or
(m,n) = (0, 1). We give the next.
Lemma 4.1. The groups C9 and C3 × C3 contain non-elementary do-
mains.
Proof. We first check that there are non-elementary domains in C9. By
(ii) of Corollary 1.7 and Theorem 2.4 we only need to verify that the
H. Pinedo 77
number of non-trivial ideals of E3(C9) is greater than 59.
The J -classes of E3(C9) induced by the 3-sets are: Ja, a2 , Ja, a3 , Ja, a4 ,
Ja, a5 , Ja, a6 , Ja, a7 , Ja2, a4 , Ja2, a5 , Ja2, a6 and Ja3, a6 .
Consequently, by taking unions of the ideals induced by the 3-sets, we can
form 210 − 1 non-trivial ideals of E3(C9), and there exist non-elementary
domains in C9.
With respect to C3 × C3 = 〈a, b | a3 = b3 = [a, b] = 1〉, using (iii) of
Corollary 1.7 and Theorem 2.4, it is enough to prove that |Y ∗(E3)| > 63.
We note that the semigroup E3(C3 × C3) has 12 J -classes induced by
the 3-sets, which are: Ja, a2 , Jb, b2 , Ja2b, b2a, Jab, a2b2 , Ja, b, Ja, b2 , Ja, ab,
Ja, a2b, Ja, ab2 , Ja, a2b2 , Jb, ab2 , and Jb, a2b2 . Then the number of nontrivial
ideals of E3(C3 × C3) is greater than 63.
By Lemma 4.1 and Proposition 3.3, an abelian group containing only
elementary domains has order 2m3. We already know that C4 has non-
elementary domains. Then in order to prove that m ≤ 1 and (m,n) 6=
(1, 1), we give the next.
Lemma 4.2. The groups C2 × C2 and C6 contain non-elementary do-
mains.
The proof of Lemma 4.2 is similar to that of Lemma 4.1. Finally we
obtain the next.
Theorem 4.3. The finite abelian groups which contain only elementary
domains are C1, C2 and C3.
5. Elementary domains and non-abelian groups
In this section we prove that any finite non-abelian group contains
non-elementary domains.
Let G be non-abelian group and m,n ∈ N such that the order of G is
2m3n. Suppose that G contains only elementary domains.
If m ≥ 2 or n ≥ 2, G would contain an abelian subgroup of order 4 or
9. By Proposition 3.3 and Theorem 4.3, there would be non-elementary
domains for G, which contradicts our assumption. Consequently m ≤ 1
and n ≤ 1 and since G is non-abelian, it must be isomorphic to S3.
We shall verify that S3 = 〈g, h | g2 = h2 = (gh)3 = 1〉 contains non-
elementary domains.
Indeed, by (v) of Corollary 1.7, we see that S3 contains at most 15 ele-
mentary domains. Note that the J -classes induced by the 3-element sets
78 On elementary domains
are Jg, h, Jg, (gh)2 , Jh, gh and Jgh, (gh)2 . From these J -classes we form 15
different ideals of E3(S3), and since 0 is also an ideal, we get that S3
contains non-elementary domains.
Then Theorem 4.3 can be extended to finite arbitrary groups as
follows.
Theorem 5.1. The finite groups containing only elementary domains
are C1, C2 and C3.
6. Total factor sets
A partial factor set σ of a group G is called total if domσ = G×G.
When the field K is algebraically closed, there is an epimorphism from
pMG×G(G) to any other component of pM(G) (see [4]). Hence in some
sense we can determine the structure of pM(G) if we know the structure
of pMG×G(G).
For any A ∈ P1(G), using (13) and (14), we produce an elementary
partial representation Γ : G → Ml(KH), where H = StA and l = |A|
|H| .
Our interest is to find the elements of P1(G) that determine elementary
partial representations whose (idempotent) factor sets are total. Such
sets will be also called total.
The next theorem gives us a condition for an element A of P1(G) to
be total.
Theorem 6.1. Let G be a group of order n and A ∈ P1(G). Suppose that
|A| = n− k for some 0 < k < n and that the stabilizer H of A has order
|H| = m. If n > k(2m+ 1) then A is total.
Proof. Let Γ: G → Mn−k
m
(KH) be the elementary partial representa-
tion corresponding to A. Write A =
⋃
n−k
m
i=1 Hgi, as a disjoint union of
cosets, where 1 = g1, g2, . . . , gn−k
m
∈ A. Further, for x ∈ G set A(x) =
{x, g2x, . . . , gn−k
m
x}. Note that the function f : Ix ∋ i 7→ gix ∈ A ∩ A(x)
is bijective, where Ix is as in (12).
Therefore |Ix| = |A∩A(x)| = |A|+ |A(x)|−|A∪A(x)| ≥ n−k+ n−k
m −n =
n−k(m+1)
m , and by hypothesis n > k(2m+ 1) we obtain |Ix| > n−k
2m , for all
x ∈ G. In particular, Γ(x) 6= 0 for any x ∈ G.
Thus, Γ(x) =
∑
i∈Ix
ei,j(h), where i ∈ Ix, j = ji,x ∈ {1, . . . , n−k
m } and
h = hi,x ∈ H satisfy gix = hgj . Since gjx
−1 = h−1gi, we get that
i ∈ Ix is equivalent to j = ji,x ∈ Ix−1 . Now take y ∈ G and write
H. Pinedo 79
Γ(y) =
∑
i∈Iy
es, t(h
′) . We have Γ(x)Γ(y) = 0 ⇔ Ix−1 ∩ Iy = ∅ ⇔
|Ix−1 | + |Iy| = |Ix−1 ∪ Iy|. Since Ix−1 ∪ Iy ⊆ {1, . . . , n−k
m } we obtain
|Ix−1 | + |Iy| ≤ n−k
m , which contradicts |Ia| > n−k
2m , for all a ∈ G. So we
have Γ(x)Γ(y) 6= 0 for all x, y ∈ G and A is total.
When we calculated the elementary partial representations of the
cyclic group G of order 2 or 3, it was verified that each A ∈ P1(G) with
|A| < |G| was not total. Therefore by Theorem 6.1 we obtain the next.
Corollary 6.2. Let G be a finite group and A ∈ P1(G) such that |A| =
|G| − 1, then A is total if and only if n > 3.
Unfortunately the converse of Theorem 6.1 is not true. We show this
with the next example.
Example 2. A counterexample for the converse of Theorem 6.1.
Consider the cyclic group 〈a | a8 = 1〉, and let A = {1, a, a3, a4, a5, a7},
then A ∈ P1(G) and H = StA = 〈a4〉.
Then A = 〈a4〉 ∪ 〈a4〉 a ∪ 〈a4〉 a3, using (13) and (14) we see that the
elementary partial representation induced by A is Γ: C8 → M3(KH)
given by:
Γ(1) = e11(1) + e22(1) + e33(1), Γ(a) = e12(1) + e31(a4),
Γ(a2) = e23(1) + e32(a4), Γ(a3) = e13(1) + e21(a4),
Γ(a4) = e11(a4) + e22(a4) + e33(a4), Γ(a5) = e31(1) + e12(a4),
Γ(a6) = e32(1) + e23(a4), Γ(a7) = e21(1) + e13(a4).
It is readily seen that the factor set of Γ is total.
7. Small degree elementary partial representations
Throughout this section G will denote a finite abelian group.
In [6] the authors gave a description of the elementary partial repre-
sentations of abelian groups of degrees ≤ 4. We will use that description
to identify the sets A ∈ P1(G) such that the induced elementary partial
representations Γ: G → Mm(KH), where m ≤ 4, have total factor set.
1 × 1 elementary partial representations. Here Γ: G → KH and
A = H. Then:
Γ: h 7→ h, g 7→ 0, for each h ∈ H, g ∈ G \H.
80 On elementary domains
Hence A is total if and only if A = G.
2 × 2 elementary partial representations. Write A = H ∪ aH as a
disjoint union of cosets. Suppose that A is total, then Ix 6= ∅ for all x ∈ G.
The latter implies
G = H ∪ aH ∪ a−1H.
Now we use (13) and (14) to determine Γ. For h ∈ H, Γ(h) = e11(h) +
e22(h). If ah = a−1h′ for some h′ ∈ H, then Γ(ah) = e12(h) + e21(h′) and
in the case in which ah /∈ a−1H, we obtain Γ(ah) = e12(h).
Analogously, if a−1h = ah′, then Γ(a−1h) = e12(h′) + e21(h), and if
a−1h /∈ aH, we get Γ(a−1h) = e21(h).
Thus for A being total we also need the condition aH = a−1H, but
this implies G = H ∪ aH = A which leads to H = StA = StG = G, and
this contradicts |A|
|H| = 2. Summarizing we conclude:
Proposition 7.1. If A ∈ P1(G) is such that its induced elementary
partial representation has degree 2, then A is not total.
3 × 3 elementary partial representations. Write A = H ∪ aH ∪ bH, as
a disjoint union of cosets. By [6, Theorem 3.2], there are 5 non-equivalent
elementary partial representations of G of degree 3. The possibly total
ones are given in the next two cases:
Case 1: Let a2 = h1 ∈ H, b2 = h2 ∈ H, ab /∈ H.
Then, Γ is equivalent to:
ϕH, a, b, 3 : h 7→ e11(h) + e22(h) + e33(h), ah → e12(h) + e21(h1h),
bh 7→ e13(h) + e31(h2h), abh 7→ e23(h1h) + e32(h2h),
g 7→ 0, if g /∈ H ∪ aH ∪ bH ∪ abH.
Since A = H ∪ aH ∪ bH as a disjoint union of cosets and ab /∈ H, we have
ab /∈ A. Therefore A is total if and only if G = A ∪ abH.
Case 2: Let a2 /∈ H, a4 = h1 ∈ H, ab = h2 ∈ H.
Observe that A = H ∪ aH ∪ bh2
−1H. By Lemma 2.3, we obtain an
elementary partial representation equivalent to Γ choosing the elements
g′
1 = 1, g′
2 = a−1 and g′
3 = b−1h2. Thus replacing b by bh2
−1, without loss
of generality we may suppose ab = 1.
Then, following the proof of [6, Theorem 3.2], Γ is equivalent to:
ϕH, a, b, 5 : h 7→ e11(h) + e22(h) + e33(h), ah → e12(h) + e31(h),
a−1h 7→ e21(h) + e13(h), a2h 7→ e23(h1h) + e32(h),
g 7→ 0, if g /∈ H ∪ aH ∪ a−1H ∪ a2H.
H. Pinedo 81
Since a /∈ H, a4 ∈ H and ab = 1, we get a2 /∈ A. Hence, A is total if and
only if G = A ∪ a2H.
4×4 elementary partial representations. Write A = H∪aH∪bH∪cH,
as a disjoint union of cosets. As it was seen in the proof of [6, Theorem
3.3], the elementary partial representations that may have totally defined
factor sets are given in the next three cases:
Case 1: Let ac = 1, a2 = b−1, a5 = h1 ∈ H. In this case Γ is
equivalent to:
ϕH, a, b, c, 2.1.1 : h 7→ e11(h) + e22(h) + e33(h) + e44(h),
ah 7→ e12(h) + e31(h) + e43(h),
a−1h → e21(h) + e13(h) + e34(h),
a2h 7→ e32(h) + e41(h) + e24(h1h),
a−2h 7→ e23(h) + e14(h) + e42(h1
−1h),
g 7→ 0, if g /∈ H ∪ aH ∪ a−1H ∪ a2H ∪ a−2H.
The relations ac = 1, a2 = b−1 and a5 = h1 ∈ H imply a2 /∈ A. Therefore
Γ is total if and only if G = A ∪ a2H.
Case 2: Let ac = 1, a2 = b−1, a6 = h1 ∈ H. Then, Γ is equivalent to:
ϕH, a, b, c, 2.1.2 : h 7→ e11(h) + e22(h) + e33(h) + e44(h), a2h 7→ e32(h) + e41(h),
a−1h → e21(h) + e13(h) + e34(h), a3h 7→ e42(h) + e24(h1h),
a−2h 7→ e23(h) + e14(h), ah 7→ e12(h) + e31(h) + e43(h),
g 7→ 0, if g /∈ H ∪ aH ∪ a−1H ∪ a2H ∪ a−2H ∪ a3H.
The conditions ac = 1, a2 = b−1 and a6 = h1 ∈ H imply a2, a3 /∈ A.
Then Γ is total if and only if G = A ∪ a2H ∪ a3H.
Finally, the last case is:
Case 3: Let ac = 1, a3 = h1 ∈ H, b2 = h2 ∈ H. Then Γ is equivalent
to:
ϕH, a, b, c, 2.2.1 : h 7→ e11(h) + e22(h) + e33(h) + e44(h), bh 7→ e14(h) + e41(h2h),
a−1h → e21(h) + e13(h) + e32(h1
−1h), abh 7→ e34(h) + e42(h2h),
a−1bh 7→ e24(h) + e43(h2h), ah 7→ e12(h) + e31(h) + e23(h1h),
g 7→ 0, if g /∈ H ∪ aH ∪ a−1H ∪ abH ∪ a−1bH ∪ bH.
Using ac = 1, a3 = h1 ∈ H and b2 = h2 ∈ H, we conclude that
ab, a−1b /∈ A. Therefore Γ is total if and only if G = A ∪ abH ∪ a−1bH.
82 On elementary domains
Acknowledgments. The author would like to thank professors Mikhailo
Dokuchaev and Boris Novikov for their many useful suggestions and
stimulating discussions.
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Contact information
H. Pinedo Instituto de Matemática e Estatística, Universi-
dade de São Paulo, Rua do Matão, 1010, 05508-
090 São Paulo, SP, Brasil.
E-Mail: hectorp@ime.usp.br
URL: www.ime.usp.br
Received by the editors: 12.03.2012
and in final form 06.06.2012.
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