Finite local nearrings on metacyclic Miller-Moreno p-groups
In this paper the metacyclic Miller-Moreno p-groups which appear as the additive groups of finite local nearrings are classified.
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Інститут прикладної математики і механіки НАН України
2012
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| Цитувати: | Finite local nearrings on metacyclic Miller-Moreno p-groups / I.Yu. Raievska, Ya.P. Sysak // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 111–127. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859660093954457600 |
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| author | Raievska, I.Yu. Sysak, Ya.P. |
| author_facet | Raievska, I.Yu. Sysak, Ya.P. |
| citation_txt | Finite local nearrings on metacyclic Miller-Moreno p-groups / I.Yu. Raievska, Ya.P. Sysak // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 111–127. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | In this paper the metacyclic Miller-Moreno p-groups which appear as the additive groups of finite local nearrings are classified.
|
| first_indexed | 2025-11-30T09:36:50Z |
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h.Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 13 (2012). Number 1. pp. 111 – 127
c© Journal “Algebra and Discrete Mathematics”
Finite local nearrings on metacyclic
Miller-Moreno p-groups
I. Yu. Raievska and Ya. P. Sysak
Communicated by G. Pilz
Abstract. In this paper the metacyclic Miller-Moreno p-
groups which appear as the additive groups of finite local nearrings
are classified.
Introduction
Nearrings are a generalization of associative rings in the sense that
with respect to addition they need not be commutative and only one
distributive law is assumed. In this paper the concept "nearring" means
a left distributive nearring with a multiplicative identity. The reader is
referred to the books by J. Meldrum [9] or G. Pilz [11] for terminology,
definitions and basic facts concerning nearrings.
A nearring R is called local if the set of all non-invertible elements of
R forms a subgroup of the additive group of R. A study of local nearrings
was initiated by Maxson [5] who defined a number of their basic properties
and proved in particular that the additive group of a finite zero-symmetric
local nearring is a p-group. It follows from [3] that local nearrings with
cyclic additive group are commutative local rings. In [6] Maxson described
all non-isomorphic zero-symmetric local nearrings with non-cyclic additive
group of order p2 which are not nearfields. He also shown in [7] that
every non-cyclic abelian p-group of order pn > 4 is the additive group
of a zero-symmetric local nearring which is not a ring. This result was
2000 Mathematics Subject Classification: 16Y30.
Key words and phrases: Nearring with identity, local nearring, additive group,
Miller-Moreno group.
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112 Local nearrings
extended in [4] to infinite abelian p-groups of finite exponent. However
in the case of finite non-abelian p-groups the situation is different. For
instance, neither a generalized quaternion group nor a non-abelian group
of order 8 can be the additive group of a local nearring, as it was noted in
[8] (see also [2]). Some other examples of finite p-groups with this property
can be found in [4]. On the other hand, it was proved by Maxson in [8]
that each metacyclic Miller-Moreno group of order pn and exponent pn−1
with n ≥ 3 if p > 2 and n ≥ 4 if p = 2 can be the additive group of a local
nearring. The purpose of our paper is to give a full classification of the
metacyclic Miller-Moreno p-groups which appear as the additive groups
of finite local nearrings. Moreover, if G is such an additive group, then
we describe all possible multiplications ” · ” on G for which the system
(G,+, ·) is a local nearring.
1. Preliminaries
Recall first some concepts concerned nearrings.
Definition 1. A set R with two binary operations + and · is called a
(left) nearring if the following statements hold:
1. (R,+) = R+ is a (not necessarily abelian) group with neutral
element 0;
2. (R, ·) is a semigroup;
3. x(y + z) = xy + xz for all x, y, z ∈ R.
If R is a nearring, then the group R+ is called the additive group of
R. As it follows from statement 3, for each subgroup M of R+ and each
element x ∈ R the set xM = {x · y|y ∈ M} is a subgroup of R+ and,
in particular, x · 0 = 0. If in addition 0 · x = 0, then the nearring R is
called zero-symmetric, and if the semigroup (R, ·) is a monoid, i.e. it has
an identity element i, then R is a nearring with identity i. In the latter
case the group R∗ of all invertible elements of the monoid (R, ·) is called
the multiplicative group of R. A subgroup M of R+ is called R∗-invariant
if rM ≤ M for each r ∈ R∗, and M is an (R,R)-subgroup, if xMy ⊆ M
for arbitrary x, y ∈ R.
Definition 2. A nearring R with identity is said to be local if the set
L = R \R∗ of all non-invertible elements of R is a subgroup of R+.
Some basic properties of local nearrings are described in the following
lemma (see [1], Lemmas 3.2, 3.4 and 3.9).
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I . Yu. Raievska, Ya. P. Sysak 113
Lemma 1. Let R be a local nearring with identity i and L its subgroup
of all non-invertible elements of R+. Then the following statements hold:
1) L is an (R,R)-subgroup of R+;
2) each proper R∗-invariant subgroup of R+ is contained in L;
3) if R is finite, then R+ is a p-group for some prime p, the subgroup L
is normal in R+ and the factor group R+/L is elementary abelian.
Definition 3. A finite non-abelian group whose proper subgroups are
abelian is called a Miller-Moreno group.
Henceforth for each prime number p and positive integers m,n with
m ≥ 2 let G(pm, pn) denote the semidirect product <a> ⋊ <b> of two
cyclic groups <a> and <b> of orders pm and pn, respectively, in which
b−1ab = a1+pm−1
. It is well-known that each metacyclic Miller-Moreno
p-group is isomorphic either to a quaternion group Q8 or to a group
G(pm, pn) (see [12]).
Recall that the exponent of a finite p-group is the maximal order of
its elements. The following assertion is easily verified.
Lemma 2. The exponent of G(pm, pn) is equal to pm for m > n and to
pn for m ≤ n. Moreover, if x is an element of maximal order in G(pm, pn),
then there exist generators a, b of this group such that either a = x or
b = x and the relations ap
m
= bp
n
= 1 and b−1ab = a1+pm−1
hold.
Lemma 3. Let the group G(pm, pn) be additively written. Then for any
natural numbers r, s, t the equalities bs + ar = ar(1− spm−1) + bs and
(ar + bs)t = ar(t− s
(
t
2
)
pm−1) + bst hold.
Proof. Let q = 1 − pm−1. Since −b + a + b = a(1 + pm−1) and m ≥ 2,
then b+ a = aq + b, so bs+ ar = arqs + bs for arbitrary integers r ≥ 0
and s ≥ 0. Taking into consideration, that
qs = (1− pm−1)s ≡ 1− spm−1 ( mod pm)
by binomial’s formula, giving bs+ ar = ar(1− spm−1) + bs. Next, (ar +
bs)t = ar(1 + qs + · · · + qs(t−1)) + bst by induction on t. Therefore,
1 + qs + · · · + qs(t−1) ≡ 1 + (1 − spm−1) + · · · + (1 − s(t − 1)pm−1) =
t− s
(
t
2
)
pm−1 ( mod pm), thus (ar + bs)t = ar(t− s
(
t
2
)
pm−1) + bst.
Lemma 4. Put G = G(pm, pn) where m ≤ n. If A = AutG is the group
of all automorphisms of G and <x> is a cyclic subgroup of G, then the
following statements hold:
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114 Local nearrings
1) if m = n and <x> is a normal subgroup of order pm in G, then
|xA| ≤ p2m−2(p− 1);
2) if p > 2 and < x > is a non-normal subgroup of order pn, then
x−1 /∈ xA.
Proof. If < x > is a normal subgroup of order pm in G , then either
< x > is a central subgroup of G or ap
m−1
∈ < x >. Since the center
Z(G) =< ap > × < bp > and the derived subgroup G′ =< ap
m−1
> are
characteristic subgroups of G, it follows that in the first case xA ⊆ Z(G)
and so |xA| ≤ p2m−2(p− 1) and in the second case G =<x> ⋊ <b> and
ap
m−1
∈ <xα> for each α ∈ A. The latter implies
| <x><xα> | =
|x||xα|
| <x> ∩ <x>α |
≤ p2m−1,
so that the subgroup <x><xα> is abelian and thus xα ∈ CG(x). Since
CG(x) =<x> × <bp>, for m = n the number of all elements of order
pm in CG(x) is equal to p2m−2(p− 1) and hence |xA| ≤ p2m−2(p− 1), as
desired.
Now let < x > be a non-normal subgroup of order pn in G. Then
< a> ∩ <x>= 1 and so x = asbt for some natural numbers s, t such
that (t, p) = 1. In particular, G =< a> ⋊ < x> and [a, x] = [a, b]t =
atp
m−1
. Assume that xα = x−1 for some automorphism α ∈ A. Since
< ap
m−1
>α=< ap
m−1
>, it follows that aα = arxu for some natural
numbers r, u such that (r, p) = 1 and p is a divisor of u. Therefore
artp
m−1
= (atp
m−1
)α = [aα, xα] = [arxu, x−1] = [a, x−1]r = a−rtpm−1
and
hence a2rtp
m−1
= 1. But then pm divides 2rtpm−1 what is impossible if
p> 2. Therefore x−1 /∈ xA, as claimed.
Lemma 5. The exponent of the additive group of a finite nearring R with
identity i is equal to the additive order of i which coincides with additive
order of every invertible element of R.
Proof. Indeed, if i·k = 0 for some natural number k, then x·k = (x·i)·k =
x · (i · k) = x · 0 = 0 for each x ∈ R. On the other hand, if r · l = 0 for
r ∈ R∗ and for a natural number l, then i · l = r−1(r · l) = 0, so that the
additive orders of r and i coincide.
The following assertion is a direct consequence of Lemmas 2 and 5.
Corollary 1. Let R be a nearring with identity i whose additive group
R+ is isomorphic to a group G(pm, pn). Then there exist generators a, b
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I . Yu. Raievska, Ya. P. Sysak 115
of R+ satisfying relations apm = bpn = 0 and a + b = b + a(1 + pm−1)
such that either a or b coincides with i.
Lemma 6. Let R be a local nearring whose additive group R+ is isomor-
phic to a group G(pm, pn) and let L be the subgroup of all non-invertible
elements of R. Then L is a subgroup in R+ of index |R+ : L| = p.
Proof. Assume that |R+ : L| = pk with k ≥ 2. Since R = R∗ ∪ L and
R∗ ∩ L = ∅, we have |R∗| = pm+n − pm+n−k = pm+n−k(pk − 1). In
particular, the order of R∗ is divisible by pk − 1.
On the other hand, for each element r ∈ R∗ the mapping x 7→ rx
with x ∈ R is an automorphism of R+ because r(x + y) = rx + ry for
every x, y ∈ R. Therefore R∗ can be viewed as a subgroup of the group of
automorphisms AutR+. As it follows from [10], Corollary of Theorem 2,
the order of the group AutG(pm, pn) and so AutR+ is a number of the
form pl(p− 1) for some l ≥ m+ n, and therefore it is divisible by pk − 1
only if k = 1. This contradiction shows that |R+ : L| = p.
2. Nearrings with identity on metacyclic Miller-Moreno
p-groups
As it was mentioned in the Introduction, the quaternion group Q8
cannot be isomorphic to the additive group of a nearring R with identity.
Therefore in what follows the additive group of R is isomorphic to a group
G(pm, pn). Thus it follows from Corollary 1 that R+ =<a> + <b> with
elements a, b one of which coincides with identity element of R and the
relations apm = bpn = 0 and a+ b = b+ a(1 + pm−1) are valid. Moreover,
each element x ∈ R is uniquely written in the form x = ax1 + bx2 with
coefficients 0 ≤ x1 < pm and 0 ≤ x2 < pn.
Consider first the case when a coincides with identity element of R, so
that xa = ax = x for each x ∈ R. Then R+ is of exponent pm by Lemma
5 and so m ≥ n. Furthermore, for each x ∈ R there exist integers α(x)
and β(x) such that xb = aα(x) + bβ(x). It is clear that modulo pm and
pn, respectively, these integers are uniquely determined by x and so some
mappings α : R → Zpm and β : R → Zpn are determined.
Lemma 7. Let x = ax1 + bx2 and y = ay1 + by2 be elements of R. If a
coincides with identity element of R, then
xy = a[(x1y1 + α(x)y2)− (x1x2
(
y1
2
)
+ α(x)x2y1y2
+ α(x)β(x)
(
y2
2
)
)pm−1] + b(x2y1 + β(x)y2).
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116 Local nearrings
Moreover, for the mappings α : R → Zpm and β : R → Zpn the following
statements hold :
(0) α(0) = β(0) = 0 if and only if the nearring R is zero-symmetric;
(1) α(a) = 0 and β(a) = 1;
(2) α(x) ≡ 0 ( mod pm−n), except the case p = 2 = m and n = 1;
(3) x1(β(x)− 1) ≡ x2α(x) ( mod p );
(4)
α(xy) = x1α(y) + α(x)β(y)
− [x1x2
(
α(y)
2
)
+ α(x)x2α(y)β(y) + α(x)β(x)
(
β(y)
2
)
]pm−1;
(5) β(xy) = x2α(y) + β(x)β(y).
Proof. By the left distributive law, we have
xy = (xa)y1 + (xb)y2 = (ax1 + bx2)y1 + (aα(x) + bβ(x))y2.
Furthermore, Lemma 3 implies that
(ax1 + bx2)y1 = ax1(y1 − x2
(
y1
2
)
pm−1) + bx2y1,
(aα(x) + bβ(x))y2 = aα(x)(y2 − β(x)
(
y2
2
)
pm−1) + bβ(x)y2
and
bx2y1 + aα(x)(y2 − β(x)
(
y2
2
)
pm−1)
= aα(x)(y2 − β(x)
(
y2
2
)
pm−1)(1− x2y1p
m−1) + bx2y1.
Thus
(∗)
xy = a[(x1y1 + α(x)y2)− (x1x2
(
y1
2
)
+ α(x)x2y1y2
+ α(x)β(x)
(
y2
2
)
)pm−1] + b(x2y1 + β(x)y2),
as desired.
As 0 · a = a · 0 = 0, the nearring R is zero-symmetric if and only if
0 = 0 · b = aα(0) + bβ(0) whence α(0) = β(0) = 0. Similarly, from the
equality b = ab = aα(a) + bβ(a) it follows that α(a) = 0 and β(a) =
1. Since (xb)pn = x(bpn) = 0 and xb = aα(x) + bβ(x), we have 0 =
(aα(x) + bβ(x))pn = aα(x)(pn − β(x)
(
pn
2
)
pm−1) by Lemma 3 and hence
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I . Yu. Raievska, Ya. P. Sysak 117
α(x) ≡ 0 ( mod pm−n), except the case p = 2 = m and n = 1 in which
the group R+ is dihedral of order 8. Next, if y = a(1− pm−1) + b, then
xy = a[α(x) + x1 − (x1 + α(x)x2)p
m−1] + b(x2 + β(x))
by formula (*). On the other hand , y = b+ a and so
xy = xb+ x = a(α(x) + x1 − x1β(x)p
m−1) + b(x2 + β(x))
by Lemma 3. Comparing both results for xy, we obtain
(x1 + α(x)x2)p
m−1 ≡ x1β(x)p
m−1( mod pm).
Thus x1(β(x)− 1) ≡ x2α(x) ( mod p ) and so statement (3) holds.
Finally, the associativity of multiplication in R implies that
x(yb) = (xy)b = aα(xy) + bβ(xy).
Furthermore, substituting yb = aα(y) + bβ(y) instead of y in formula (*),
we also have
x(yb) = a[(x1α(y) + α(x)β(y))− (x1x2
(
α(y)
2
)
+ α(x)x2α(y)β(y)
+ α(x)β(x)
(
β(y)
2
)
)pm−1] + b(x2α(y) + β(x)β(y)).
Comparing the coefficients under a and b in two expressions obtained for
x(yb), we derive statements (4) and (5) of the lemma.
Consider now the case when b coincides with identity element of R
and so xb = bx = x for each x ∈ R. Then R+ is of exponent pn by Lemma
5 and thus m ≤ n. As above, there exist integers α(x) and β(x) which
are uniquely determined by x modulo pm and pn, respectively, such that
xa = aα(x) + bβ(x).
Lemma 8. If b coincides with identity element in R, then
xy = a[α(x)y1 + x1y2 − x1x2
(
y2
2
)
pm−1] + b[β(x)y1 + x2y2].
Also for the mappings α : R → Zpm and β : R → Zpn the following
statements hold:
(0) α(0) = β(0) = 0 if and only if the nearring R is zero-symmetric;
(1) α(b) = 1 and β(b) = 0;
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118 Local nearrings
(2) β(x) ≡ 0 ( mod pn−m+1);
(3) α(x)(1− x2) ≡ 0 ( mod p );
(4) α(xy) = α(x)α(y) + x1β(y)− x1x2
(
β(y)
2
)
pm−1;
(5) β(xy) = β(x)α(y) + x2β(y).
Proof. Clearly R is zero-symmetric if and only if 0 = 0 ·a = aα(0)+ bβ(0),
whence α(0) = β(0) = 0. From the equality a = ba = aα(b) + bβ(b) it
follows that α(b) = 1 and β(b) = 0. Using the left distributive law, we
have also
xy = (xa)y1 + (xb)y2 = (aα(x) + bβ(x))y1 + (ax1 + bx2)y2.
Next, Lemma 3 implies that
(aα(x) + bβ(x))y1 = aα(x)(y1 − β(x)
(
y1
2
)
pm−1) + bβ(x)y1,
(ax1 + bx2)y2 = ax1(y2 − x2
(
y2
2
)
pm−1) + bx2y2
and
bβ(x)y1 + ax1(y2 − x2
(
y2
2
)
pm−1)
= ax1(y2 − x2
(
y2
2
)
pm−1)(1− β(x)y1p
m−1) + bβ(x)y1.
Therefore
(∗∗)
xy = a[(α(x)y1 + x1y2)− (α(x)β(x)
(
y1
2
)
+ x1x2
(
y2
2
)
+
+ x1β(x)y1y2)p
m−1] + b(β(x)y1 + x2y2).
Substituting y = a(1− pm−1) + b in formula (**), we derive
xy = a[α(x)(1− pm−1) + x1 − (α(x)β(x)
(
1−pm−1
2
)
+ x1x2
(
1
2
)
+ x1β(x)(1− pm−1))pm−1] + b(β(x)(1− pm−1) + x2).
In the case p > 2 or p = 2 and m ≥ 3 this implies
(i) xy = a[x1 +α(x)− (α(x) + x1β(x))p
m−1] + b(β(x)(1− pm−1) + x2).
If p = 2 and m = 2, then
(ii) xy = a[x1 − α(x) + 2(α(x)β(x) + x1β(x))] + b(x2 − β(x)).
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I . Yu. Raievska, Ya. P. Sysak 119
On the other hand, y = b+ a and thus
(iii) xy = x+ xa = a[x1 + α(x)− x2α(x)p
m−1] + b(x2 + β(x)).
Comparing formulas (i) and (ii) with formula (iii), we obtain in the first
case the congruences
(α(x) + x1β(x))p
m−1 ≡ x2α(x)p
m−1( mod pm)
and
β(x)pm−1 ≡ 0( mod pn)
and in the second case the congruences
−α(x) + 2α(x)β(x) + 2x1β(x) ≡ α(x)− 2x2α(x)( mod 22)
and
β(x) ≡ −β(x)( mod 2n).
In both cases this gives the congruences
α(x)(1− x2) ≡ 0 ( mod p ) and β(x) ≡ 0 ( mod pn−m+1),
i.e. statements (3) and (2), respectively. But then
(α(x)β(x)
(
y1
2
)
+ x1β(x)y1y2)p
m−1 ≡ 0 ( mod pm)
and so formula (**) becomes the equality
xy = a[(α(x)y1 + x1y2)− x1x2
(
y2
2
)
pm−1] + b[β(x)y1 + x2y2],
as desired. Finally, replacing in this equality the element y by ya =
aα(y) + bβ(y) and using that x(ya) = (xy)a, we have got the expressions
for α(xy) and β(xy) contained in statements (4) and (5) of the lemma.
3. Local nearrings on metacyclic Miller-Moreno p-groups
Let R be a local nearring whose additive group R+ is isomorphic
to a group G(pm, pn) and let L denote the subgroup in R+ of all non-
invertible elements from R. As it was mentioned above, R = R∗ ∪ L
and R+ =< a > + < b > with elements a, b satisfying the relations
apm = bpn = 0 and a+ b = b+ a(1 + pm−1). Furthermore, < a > is a
normal subgroup of R+ and each element x ∈ R is uniquely written in
the form x = ax1 + bx2 with coefficients 0 < x1 < pm and 0 < x2 < pn.
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120 Local nearrings
Lemma 9. 1) If a ∈ R∗, then pm+n > 8 and m> n.
2) If b ∈ R∗, then p = 2 and m ≤ n.
Proof. If a ∈ R∗, then R+ is a group of exponent pm by Lemma 5 and so
m ≥ n. It was shown in [8] that the dihedral group of order 8 is not the
additive group of a local nearring, so that pm+n > 8. Assume that m = n.
As it was shown in the proof of Lemma 6, the multiplicative group R∗ can
be viewed as a subgroup of AutR+ such that R∗x ⊆ xAutR+
for each x ∈ R
and, in particular, R∗ = R∗a ⊆ aAutR+
. Since |aAutR+
| ≤ p2m−2(p − 1)
by Lemma 4, it follows that |R∗| ≤ p2m−2(p− 1). On the other hand, the
subgroup L is of index p in R+, so that |R∗| = |R| − |L| = |R| − 1
p
|R| =
p2m−p2m−1 = p2m−1(p−1), contrary to the latter inequality. Thus m> n,
proving statement 1).
Now let b ∈ R∗. Then R+ is a group of exponent pn and so m ≤ n.
Since the subgroup <b> is non-normal in R+, it follows that −b 6∈ bAutR+
by Lemma 4. As R∗ = R∗b ⊆ bAutR+
, this means that −b 6∈ R∗. But then
−b ∈ L and if p > 2, then b ∈ L, contrary to the assumption. Thus p = 2
and m ≤ n, proving statement 2).
Theorem 1. Let R be a local nearring whose additive group R+ is isomor-
phic to a group G(pm, pn). Then R+ = <a> + <b>, one of the elements
a, b coincides with identity element of R and the following statements hold:
1) apm = bpn = 0 and a+ b = b+ a(1 + pm−1);
2) if a coincides with identity element of R, then pm+n > 8, m > n,
L = <ap> + <b> and R∗ = {ax1 + bx2 | x1 6≡ 0 (mod p)};
3) if b coincides with identity element of R, then p = 2, m ≤ n, L =
<a> + <b2> and R∗ = {ax1 + bx2 | x2 ≡ 1 ( mod 2 )}.
Proof. By Corollary 1, there exists the decomposition R+ =<a> + <b>
in which one of the elements a or b coincides with identity element of R
and statement 1) holds.
If a coincides with identity element, then m> n and b ∈ L by Lemma
9, so that L = <ap> + <b>. Since xL ⊆ L for each x ∈ R by Lemma
1, we have xb = aα(x) + bβ(x) ∈ L whence α(x) ≡ 0 ( mod p ) for each
x ∈ R. Furthermore, R = R∗∪L and so an element x = ax1+ bx2 belongs
to R∗ if and only if x1 6≡ 0 ( mod p ).
Similarly, if b coincides with identity element of R, then p = 2, m ≤ n
and a ∈ L by Lemma 9. Therefore L = <a> + <b2> and β(x) ≡
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I . Yu. Raievska, Ya. P. Sysak 121
0 ( mod 2 ) for each x ∈ R. Thus an element x = ax1 + bx2 belongs to
R∗ if and only if x2 ≡ 1 ( mod 2 ).
Applying statements 2) and 3) of Theorem 1 to Lemmas 7 and 8,
respectively, we obtain the following formulas for multiplying elements
x = ax1+bx2 and y = ay1+by2 in the local nearring R whose the additive
group is isomorphic to a group G(pm, pn).
Corollary 2. If a coincides with identity element of R and xb = aα(x) +
bβ(x), then
xy = a(x1y1 + α(x)y2 − x1x2
(
y1
2
)
pm−1) + b(x2y1 + β(x)y2),
where the mappings α : R → Zpm and β : R → Zpn satisfy the conditions:
(0) α(0) = β(0) = 0 if and only if the nearring R is zero-symmetric;
(1) α(a) = 0 and β(a) = 1;
(2) α(x) ≡ 0 ( mod pm−n);
(3) x1(β(x)− 1) ≡ 0 ( mod p );
(4) α(xy) = x1α(y) + α(x)β(y)− x1x2
(
α(y)
2
)
pm−1;
(5) β(xy) = x2α(y) + β(x)β(y).
Corollary 3. If b coincides with identity element of R and xa = aα(x) +
bβ(x), then
xy = a(α(x)y1 + x1y2 − x1x2
(
y2
2
)
2m−1) + b(β(x)y1 + x2y2),
where mappings α : R → Z2m and β : R → Z2n satisfy the conditions:
(0) α(0) = β(0) = 0 if and only if the nearring R is zero-symmetric;
(1) α(b) = 1 and β(b) = 0;
(2) β(x) ≡ 0 ( mod 2n−m+1);
(3) α(x) ≡ 0 ( mod 2 ) if and only if x2 ≡ 0 ( mod 2 );
(4) α(xy) = α(x)α(y) + x1β(y)− x1x2
(
β(y)
2
)
2m−1;
(5) β(xy) = β(x)α(y) + x2β(y).
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122 Local nearrings
4. Groups G(pm, pn) as the additive groups of local near-
rings
The following theorem gives sufficient conditions for existing a finite
local nearring whose additive group is isomorphic to a group G(pm, pn).
Together with Theorem 1 and remarks mentioned in the Introduction
this completes our classification of all metacyclic Miller-Moreno p-groups
which are the additive groups of local nearrings.
Theorem 2. For an arbitrary prime p and natural numbers m,n such
that either pm+n > 8 and m > n or p = 2 and 1 < m ≤ n there exists
a local nearring R whose additive group R+ is isomorphic to a group
G(pm, pn).
Proof. Let R be an additively written group G(pm, pn) with generators
a, b satisfying the relations apm = bpn = 0 and a+ b = b+ a(1 + pm−1).
Then R = <a> + <b> and each element x ∈ R is uniquely written in
the form x = ax1+ bx2 with coefficients 0 < x1 < pm and 0 < x2 < pn. In
order to define a multiplication ” · ” on R in such a manner that (R,+, ·)
is a local nearring, we separately consider two cases.
1. The case pm+n > 8 and m > n.
For arbitrary elements x = ax1+ bx2 and y = ay1+ by2 of R we define
the multiplication ” · ” by the rule
x · y = a(x1y1 + α(x)y2 − x1x2
(
y1
2
)
pm−1) + b(x2y1 + β(x)y2),
where α : R → Zpm and β : R → Zpn are mappings satisfying conditions
(1) - (5) of Corollary 2. As an example, we can for instance take
α(x) = 0 for all x ∈ R and β(x) =
{
1 if x1 6≡ 0 ( mod p ),
0 if x1 ≡ 0 ( mod p ).
It is easy to see that the element a is a multiplicative identity for (R, ·)
and x ·b = aα(x)+bβ(x) for each x ∈ R. We show that with respect to the
operations ” + ” and ” · ” the system (R,+, ·) is a nearring with identity
element a. Clearly it suffices to check that if z = az1 + bz2 is an arbitrary
element of R, then x · (y + z) = x · y + x · z and (x · y) · b = x · (y · b).
Indeed, we have
x · z = a(x1z1 + α(x)z2 − x1x2
(
z1
2
)
pm−1) + b(x2z1 + β(x)z2)
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I . Yu. Raievska, Ya. P. Sysak 123
and
b[x2y1 + β(x)y2] + a[x1z1 + α(x)z2 − x1x2
(
z1
2
)
pm−1] =
= a[(1− (x2y1 + β(x)y2)p
m−1)(x1z1 + α(x)z2−
− x1x2
(
z1
2
)
pm−1)] + b[x2y1 + β(x)y2]
by Lemma 3. Since α(x) ≡ 0 ( mod p ) by condition (2) of Corollary 2
and so a(α(x)pm−1) = 0, this implies
x · y + x · z = a[x1(y1 + z1) + α(x)(y2 + z2)− (x1x2
(
y1
2
)
+ x1x2
(
z1
2
)
+
+ x1x2y1z1 + x1β(x)y2z1)p
m−1] + b[x2(y1 + z1) + β(x)(y2 + z2)].
On the other hand, y + z = (ay1 + by2) + (az1 + bz2) = a(y1 + z1 −
y2z1p
m−1) + b(y2 + z2) because by2 + az1 = a(1 − y2p
m−1)z1 + by2 by
Lemma 3 and b(y2z1p
m−1) = 0 because m > n. Therefore
x · (y + z) = a[x1(y1 + z1) + α(x)(y2 + z2)− (x1y2z1+
+ x1x2
(
y1+z1−y2z1p
m−1
2
)
)pm−1] + b[x2(y1 + z1) + β(x)(y2 + z2)]
and hence
x · y + x · z − x · (y + z) = a[−(x1x2
(
y1
2
)
+ x1x2
(
z1
2
)
+ x1x2y1z1+
+ x1β(x)y2z1)p
m−1 + (x1y2z1 + x1x2
(
y1+z1−y2z1p
m−1
2
)
)pm−1] =
= ax1(1− β(x))y2z1p
m−1.
Since x1(β(x)−1) ≡ 0 ( mod p ) by condition (3) of Corollary 2, it follows
that ax1(1 − β(x))y2z1p
m−1 = 0 and thus x · (y + z) = x · y + x · z, as
desired.
Next, y · b = aα(y) + bβ(y) and so x · (y · b) = xα(y) + (x · b)β(y).
Applying Lemma 3, we also have
xα(y) = (ax1 + bx2)α(y) = ax1(α(y)− x2
(
α(y)
2
)
pm−1) + bx2α(y),
(x · b)β(y) = (aα(x) + bβ(x))β(y) = aα(x)(β(y)− β(x)
(
β(y)
2
)
pm−1)+
+ bβ(x)β(y)
and
bx2α(y) + aα(x)(β(y)− β(x)
(
β(y)
2
)
pm−1) =
= aα(x)(β(y)− β(x)
(
β(y)
2
)
pm−1)(1− x2α(y)p
m−1) + bx2α(y).
As α(x) ≡ 0 ( mod p ), it follows that
α(x)β(x)
(
β(y)
2
)
pm−1) = x2α(x)α(y)β(y)p
m−1 =
= x2β(x)α(y)
(
β(y)
2
)
p2m−2 = 0,
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124 Local nearrings
whence
x·(y·b) = a[x1α(y)+α(x)β(y)−x1x2
(
α(y)
2
)
pm−1]+b[x2α(y)+β(x)β(y)].
However x1α(y) + α(x)β(y) − x1x2
(
α(y)
2
)
pm−1 = α(x · y) and x2α(y) +
β(x)β(y) = β(x · y) by conditions (4) and (5) of Corollary 2, so that
(x · y) · b = aα(x · y) + bβ(x · y) = x · (y · b). Thus, the system (R,+, ·) is
a nearring with identity element a.
We show finally that R = (R,+, ·) is a local nearring in which the
subgroup L =<ap> + <b> consists of all non-invertible elements of R.
Obviously, it is enough to prove that the element x = ax1 + bx2 is right
invertible if and only if x1 6≡ 0 ( mod p ) because in this case the set R \L
coincides with the multiplicative group R∗ by Lemma 1.
Indeed, if x is right invertible, then there exists an element y = ay1+by2
such as x ·y = a. From our definition of x ·y it follows that x1y1+α(x)y2−
x1x2
(
y1
2
)
pm−1 ≡ 1 ( mod pm). But then x1y1 ≡ 1 ( mod p ) because
α(x) ≡ 0 ( mod p ) by condition (2) of Corollary 2, so that x1 6≡ 0 (
mod p ).
Conversely, if the latter holds, then β(x) ≡ 1 ( mod p ) by condition (3)
of Corollary 2 and so there exist natural numbers y1, y2 such that x1y1 ≡
1 ( mod pm) and x2y1 + β(x)y2 ≡ 0 ( mod pn). Taking y = ay1 + by2 and
z1 = 1 + α(x)y2 − x1x2
(
y1
2
)
pm−1, we have x · y = az1. Since α(x) ≡ 0 (
mod p ), there exists such a natural number s that z1s ≡ 1 ( mod pm),
whence x · (ys) = (x · y)s = (az1)s = a. Thus the element x is right
invertible and so the nearring R is local.
2. The case p = 2 and m ≤ n.
Define now the multiplication ” · ” by the rule
x · y = a(α(x)y1 + x1y2 − x1x2
(
y2
2
)
2m−1) + b(β(x)y1 + x2y2),
where the mappings α : R → Z2m and β : R → Z2n satisfy conditions (1)
- (5) of Corollary 3. In particular, we can put
α(x) =
{
1 if x2 6≡ 0 ( mod 2 ),
0 if x2 ≡ 0 ( mod 2 )
and β(x) = 0 for all x ∈ R.
Clearly the element b is a multiplicative identity of R and x · a = aα(x) +
bβ(x) for each x ∈ R. As in the above case, in order to show that the
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I . Yu. Raievska, Ya. P. Sysak 125
system (R,+, ·) is a nearring with identity element b, it is enough to verify
that x · (y + z) = x · y + x · z and (x · y) · a = x · (y · a).
Indeed,
x · z = a(α(x)z1 + x1z2 − x1x2
(
z2
2
)
2m−1) + b(β(x)z1 + x2z2)
by the definition and
b[β(x)y1 + x2y2] + a[α(x)z1 + x1z2 − x1x2
(
z2
2
)
2m−1] =
= a[(1− (β(x)y1 + x2y2)2
m−1)(α(x)z1 + x1z2 − x1x2
(
z2
2
)
2m−1)]+
+ b[β(x)y1 + x2y2]
by Lemma 3. Since β(x) ≡ 0 ( mod 2n−m+1) by condition (2) of Corollary
3, this implies that
x · y + x · z = a[α(x)(y1 + z1) + x1(y2 + z2)− (x1x2
(
y2
2
)
+ x1x2
(
z2
2
)
+
+ x2y2α(x)z1 + x1x2y2z2)2
m−1] + b[β(x)(y1 + z1) + x2(y2 + z2)].
On the other hand, y+ z = a(y1 + z1 − y2z12
m−1) + b(y2 + z2) and so
x · (y + z) = a[α(x)(y1 + z1 − y2z12
m−1) + x1(y2 + z2)−
− x1x2
(
y2+z2
2
)
2m−1] + b[β(x)(y1 + z1 − y2z12
m−1) + x2(y2 + z2)].
Thus
−x · (y + z) + x · y + x · z = −b[β(x)(y1 + z1 − y2z12
m−1)+
+ x2(y2 + z2)] + a[(α(x)y2z1 + x1x2
(
y2+z2
2
)
− x1x2
(
y2
2
)
− x1x2
(
z2
2
)
−
− α(x)x2y2z1 − x1x2y2z2)2
m−1] + b[β(x)(y1 + z1) + x2(y2 + z2)].
However
a[(α(x)y2z1 + x1x2
(
y2+z2
2
)
− x1x2
(
y2
2
)
− x1x2
(
z2
2
)
−
− α(x)x2y2z1 − x1x2y2z2)2
m−1] = aα(x)(1− x2)y2z22
m−1 = 0
by condition (3) of Corollary 3 and therefore
−x · (y + z) + x · y + x · z = −b[β(x)(y1 + z1 − y2z12
m−1)+
+ x2(y2 + z2)] + b[β(x)(y1 + z1) + x2(y2 + z2)] =
= bβ(x)y2z12
m−1 = 0,
as required.
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126 Local nearrings
Next, y · a = aα(y) + bβ(y) and so x · (y · a) = (x · a)α(y) + xβ(y) =
(aα(x) + bβ(x))α(y) + (ax1 + bx2)β(y). Applying Lemma 3, we have
(aα(x) + bβ(x))α(y) = aα(x)(α(y)− β(x)
(
α(y)
2
)
2m−1) + bβ(x)α(y),
(ax1 + bx2)β(y) = ax1(β(y)− x2
(
β(y)
2
)
2m−1) + bx2β(y)
and
bβ(x)α(y) + ax1(β(y)− x2
(
β(y)
2
)
2m−1) =
= ax1(β(y)− x2
(
β(y)
2
)
2m−1)(1− β(x)α(y)2m−1) + bβ(x)α(y).
From this and the congruence β(x) ≡ 0 ( mod 2n−m+1) it follows that
x·(y·a) = a[α(x)α(y)+x1β(y)−x1x2
(
β(y)
2
)
2m−1]+b[β(x)α(y)+x2β(y)].
In view of conditions (4) and (5) of Corollary 3, the coefficients in brackets
under a and b in the right part of the last equality coincide with α(x·y) and
β(x · y), respectively. Therefore x · (y ·a) = aα(x · y)+ bβ(x · y) = (x · y) ·a
and hence (R,+, ·) is a nearring with identity element b.
To complete the proof it remains to show that the set of all non-
invertible elements of (R,+, ·) coincides with L =<a> + <b2>. As in
the previous case, it suffices to prove that an element x = ax1 + bx2 is
right invertible if and only if x2 ≡ 1 ( mod 2 ).
Really, if there exists an element y = ay1 + by2 such that x · y = b,
then β(x)y1 + x2y2 ≡ 1 ( mod 2n) by our definition of x · y. But then
x2y2 ≡ 1 ( mod 2m−1) because β(x) ≡ 0 ( mod 2n−m+1) by condition (2)
of Corollary 3, whence x2 ≡ 1 ( mod 2 ).
Conversely, if the last congruence holds, then α(x) ≡ 1 ( mod 2 ) by
condition (3) of Corollary 3 and so there exist natural numbers y1, y2 such
that x2y2 ≡ 1 ( mod 2n) and α(x)y1 + x1y2 ≡ 0 ( mod 2m). Therefore
for y = ay1 + by2 and z2 = 1 + β(x)y1 we have x · y = bz2. Since
β(x) ≡ 0 ( mod 2n−m+1 ), there is a natural number s such that z2s ≡
1 ( mod 2n). But then x · (ys) = (x · y)s = (bz2)s = b and so the element
x is right invertible. Thus the nearring R is local and this completes the
proof.
References
[1] B. Amberg, P. Hubert, Ya. Sysak, Local nearrings with dihedral multiplicative group,
Journal of Algebra, 273, 2004, pp. 700-717.
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I . Yu. Raievska, Ya. P. Sysak 127
[2] J. R. Clay, C.J. Maxson The near-rings with identities on generalized quaternion
groups, Ist. Lombardo Accad. Sci. Lett. Rend., A 104, 1970, pp. 525–530.
[3] J. R. Clay, J.J. Malone, Jr. The near-rings with identities on certain finite groups,
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34, 2006, pp.743 - 747.
[5] C. J. Maxson, On local near-rings, Math. Zeitschr., 106, 1968, pp. 197-205.
[6] C. J. Maxson, Local nearrings of cardinality p
2, Canad. Math. Bull., 11, 1968, pp.
555-561.
[7] C. J. Maxson, On the construction of finite local near-rings (I): on non-cyclic
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[8] C. J. Maxson, On the construction of finite local near-rings (II): on non-abelian
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[9] J. D. P. Meldrum, Near-rings and their links with groups, Pitman, Boston, 1985,
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Contact information
I. Yu. Raievska Institute of Mathematics Ukrainian National
Academy of Science, Tereshchenkivska st. 3, Kiev,
01601, Ukraine
E-Mail: raemarina@rambler.ru
Ya. P. Sysak Institute of Mathematics Ukrainian National
Academy of Science, Tereshchenkivska st. 3, Kiev,
01601 Ukraine
E-Mail: sysak@imath.kiev.ua
Received by the editors: 23.09.2011
and in final form 30.03.2012.
I. Yu. Raievska, Ya. P. Sysak
|
| id | nasplib_isofts_kiev_ua-123456789-152191 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-30T09:36:50Z |
| publishDate | 2012 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Raievska, I.Yu. Sysak, Ya.P. 2019-06-08T11:09:34Z 2019-06-08T11:09:34Z 2012 Finite local nearrings on metacyclic Miller-Moreno p-groups / I.Yu. Raievska, Ya.P. Sysak // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 1. — С. 111–127. — Бібліогр.: 12 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16Y30. https://nasplib.isofts.kiev.ua/handle/123456789/152191 In this paper the metacyclic Miller-Moreno p-groups which appear as the additive groups of finite local nearrings are classified. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Finite local nearrings on metacyclic Miller-Moreno p-groups Article published earlier |
| spellingShingle | Finite local nearrings on metacyclic Miller-Moreno p-groups Raievska, I.Yu. Sysak, Ya.P. |
| title | Finite local nearrings on metacyclic Miller-Moreno p-groups |
| title_full | Finite local nearrings on metacyclic Miller-Moreno p-groups |
| title_fullStr | Finite local nearrings on metacyclic Miller-Moreno p-groups |
| title_full_unstemmed | Finite local nearrings on metacyclic Miller-Moreno p-groups |
| title_short | Finite local nearrings on metacyclic Miller-Moreno p-groups |
| title_sort | finite local nearrings on metacyclic miller-moreno p-groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/152191 |
| work_keys_str_mv | AT raievskaiyu finitelocalnearringsonmetacyclicmillermorenopgroups AT sysakyap finitelocalnearringsonmetacyclicmillermorenopgroups |