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EXTRACTA

MATHEMATICAE

Volumen 33, Número 2, 2018

instituto de investigación de matemáticas de la

universidad de extremadura

EXTRACTA MATHEMATICAE

Vol. 37, Num. 2 (2022), 185 – 194

doi:10.17398/2605-5686.37.2.185

Available online April 25, 2022

On a class of power associative LCC-loops

O.O. George 1,@, J.O. Olaleru 1

J.O. Adéńıran 2, T.G. Jaiyéo. lá
3

1 Department of Mathematics, University of Lagos, Akoka, Nigeria
2 Department of Mathematics, Federal University of Agriculture

Abeokuta 110101, Nigeria
3 Department of Mathematics, Obafemi Awolowo University

Ile Ife 220005, Nigeria

oogeorge@unilag.edu.ng , jolaleru@unilag.edu.ng , ekenedilichineke@yahoo.com

adeniranoj@unaab.edu.ng , jaiyeolatemitope@yahoo.com , tjayeola@oauife.edu.ng

Received August 20, 2021 Presented by A. Turull
Accepted April 7, 2022

Abstract: Let LWPC denote the identity (xy · x) · xz = x((yx · x)z), and RWPC the mirror identity.
Phillips proved that a loop satisfies LWPC and RWPC if and only if it is a WIP PACC loop. Here, it

is proved that a loop Q fulfils LWPC if and only if it is a left conjugacy closed (LCC) loop that fulfils
the identity (xy · x)x = x(yx · x). Similarly, RWPC is equivalent to RCC and x(x · yx) = (x · xy)x.
If a loop satisfies LWPC or RWPC, then it is power associative (PA). The smallest nonassociative
LWPC-loop was found to be unique and of order 6 while there are exactly 6 nonassociative LWPC-

loops of order 8 up to isomorphism. Methods of construction of nonassociative LWPC-loops were

developed.

Key words: left (right) conjugacy closed loop, power associativity, LWPC-loop, RWPC-loop.

MSC (2020): 20N02, 20N05.

1. Introduction

A quasigroup (Q, ·) consists of a non-empty set Q with a binary operation
(·) on Q such that given x,y ∈ Q, the equations a ·x = b and y ·a = b have
unique solutions x,y ∈ Q respectively. We shall sometimes refer to (Q, ·) as
simply Q. It is usual to set x = a\b and y = b/a.

A loop is a quasigroup with a two-sided neutral element 1. We write xy for
x ·y and stipulate that · have lower priority than juxtaposition among factors
to be multiplied-for instance, x ·yz stands for x(yz). For an overview on loop
theory, see [1, 7, 11].

If a is an element of a loop Q, then La : x 7→ ax permutes Q and is
called the left translation of a. Similarly, Ra : x 7→ xa is called the right
translation of a. The loop Q is said to be a left conjugacy closed (LCC) if the

@ Corresponding author

ISSN: 0213-8743 (print), 2605-5686 (online)

© The author(s) - Released under a Creative Commons Attribution License (CC BY-NC 3.0)

https://doi.org/10.17398/2605-5686.37.2.185
mailto:oogeorge@unilag.edu.ng
mailto:jolaleru@unilag.edu.ng
mailto:ekenedilichineke@yahoo.com
mailto:adeniranoj@unaab.edu.ng
mailto:jaiyeolatemitope@yahoo.com
mailto:tjayeola@oauife.edu.ng
mailto:oogeorge@unilag.edu.ng
https://publicaciones.unex.es/index.php/EM
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186 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru

left translations are closed under conjugation (i.e., for all x,y ∈ Q, there exists
z ∈ Q such that LxLyL−1x = Lz). Similarly, right conjugacy closed (RCC)
loops are those in which the right translations are closed under conjugation.
A loop is said to be conjugacy closed (CC) if it is both LCC and RCC.

Kinyon and Kunen [8] thoroughly analyzed power associative CC-loops
(PACC-loops). A loop is power associative (PA) if each of its elements gener-
ates a (cyclic) subgroup. By [8], the structure of PACC-loops heavily depends
upon the structure of WIP elements. That motivated Phillips [12] to find a
short equational basis for WIP PACC loops.

Now, WIP stands for Weak Inverse Property. A loop Q is said to be a
WIP-loop if it satisfies the equivalent identities

x(yx)ρ = yρ and (xy)λx = yλ,

where x ·xρ = 1 = xλ ·x for all x ∈ Q.
By [12], a loop Q is a WIP PACC-loop if and only if it fulfils the laws

(xy ·x) ·xz = x((yx ·x)z) , (LWPC)
zx · (x ·yx) = (z(x ·xy))x. (RWPC)

The purpose of this paper is to initiate the study of loops that fulfil only
one of the latter identities. Our main result is Theorem 2.2 that proves that
an LWPC loop is power associative, and that LWPC loops are exactly the
LCC loops in which (xy · x)x = x(yx · x). A mirror result holds for RWPC
loops.

During our preliminary search for LWPC-loops, we found two LCC-loops
(that are LWPC-loops) of orders 6 and 8 with the property that their right
nuclei are abelian groups which are of index two. A general construction
of such loops can be found in Drápal [4]. These loops were constructed by
means of an arbitrary abelian group and two permutations that satisfy some
constraints (cf. Proposition 5.1). We shall be adopting this construction to
show that an infinite series of LWPC-loop is feasible.

If Q is a loop and α,β and γ permute Q, then (α,β,γ) is said to be an
autotopism of Q if α(y)β(z) = γ(yz) for all y,z ∈ Q. Autotopisms of Q can be
composed through componentwise multiplication, and thus they form a group
called the autotopism group denoted by Atp(Q).

The fact that left translations are closed under conjugation can be ex-
pressed equationally by the law x ·y(x\z) = xy/x · z (cf. [6]). This law may
also be written as x ·yz = (xy/x) ·xz. Hence, Q is LCC if and only if(

R−1x Lx, Lx,Lx
)
∈ Atp(Q) for all x ∈ Q. (1)



on a class of power associative lcc-loops 187

Writing LWPC as (x((y/x)/x) ·x) ·xz = x ·yz implies that this law holds if
and only if (

RxLxR
−2
x ,Lx,Lx

)
∈ Atp(Q) for all x ∈ Q. (2)

Characterizations of LCC and LWPC by autotopisms are of crucial impor-
tance for the proof of the main result.

2. LWPC-loop, RWPC-loop and their properties

The following criterion for power associativity will be useful to establish
the main result in Theorem 2.2.

Lemma 2.1. Let x be an element of a loop Q. Suppose that xλ = xρ,
and denote the latter element by x−1. Suppose that for each i ≥ 1 any
bracketing of i occurrences of x yields the same element, and denote this
element by xi. Similarly, let any bracketing of i occurrences of x−1 yield
an element x−i. Set x0 = 1 = (x−1)0 and (x−i)−1 = xi. Finally, suppose
that yjy−i = yj−i = y−iyj whenever j ≥ i ≥ 1 and y ∈ {x, x−1}. Then x
generates a subgroup of Q, and the element xi attains the usual meaning of
the ith power, for every integer i.

Proof. First note that xjx−i = xj−i = x−ixj holds for any positive integers
i and j since if j < i, then y−jyi = yi−j = yiy−j may be used, with y = x−1.
We have to show that xi ·xjxk = xixj ·xk for any i,j and k. If any of them
is zero, the corresponding power is equal to 1 and the equality holds. If all
of i,j and k are positive, then the equality follows from the assumption on
bracketing. If two or more exponents are negative, replace x with y = x−1.
Thus only the case with exactly one of exponents negative needs to be solved.
This means to verify that each of the ensuing triples associates, under the
assumption that all of i,j and k are positive integers,(

xi,xj,x−k
)
,
(
xi,x−j,xk

)
and

(
x−i,xj,xk

)
.

The leftmost and the rightmost triples are mirror symmetric. Hence, only the
first two triples will be considered. Now,

xi ·xjx−k = xixj−k = xi+j−k = xi+jx−k = xixj ·x−k

and
xi ·x−jxk = xix−j+k = xi−j+k = xi−jxk = xix−j ·xk.



188 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru

Theorem 2.2. Let Q be a loop. Then

(i) Q fulfills LWPC ⇔ Q is LCC and (xy ·x)x = x(yx ·x)︸ ︷︷ ︸
Pλ(x,y)

for all x,y ∈ Q.

(ii) Q fulfills RWPC ⇔ Q is RCC and x(x ·yx) = (x ·xy)x︸ ︷︷ ︸
Pρ(x,y)

for all x,y ∈ Q.

If Q fulfills LWPC or RWPC, then Q is power associative.

Proof. Setting z = 1 in the LWPC law yields (xy · x)x = x(yx · x). To
prove (i) thus means to show LWPC ⇔ LCC under the assumption of R2xLx =
LxR

2
x, for all x ∈ Q. The equivalence of LCC and LWPC follows immediately

from the expression of the identities via autotopisms as in (1) and (2), since
R2xLx = LxR

2
x implies that RxLxR

−2
x = RxR

−2
x Lx = R

−1
x Lx . Nothing else is

needed to get (i).
Point (ii) follows by a mirror argument. Let us assume that Q fulfills

LWPC. To prove that Q is power associative, let us start by showing that for
k ≥ 1 any bracketing of i occurrences of x yields the same element. Proceed
by induction. Cases k = 1 and k = 2 are clear. The case k = 3 follows
from LWPC by setting y = z = 1. Assume k ≥ 4. We need to verify that
xixj = xxk−1 whenever i ≥ 2 and i + j = k. If i ≥ 3, express xi as xxi−2 ·x
and set z = xj−1. LWPC yields

xixj = (xxi−2 ·x) ·xxj−1 = x((xi−2x ·x) ·xj−1) = xxk−1.

Assume i = 2 and set y = 1 in LWPC. Then, x2 · xxj−1 = x(x2xj−1) =
xxk−1. LWPC with z = 1 implies xx = (xxρ · x)x = x(xρx · x). Hence
x = xρx ·x and 1 = xρx. Therefore, xρ = xλ = x−1.

By Lemma 2.1 it remains to verify that xjx−i = xj−i = x−ixj whenever
j ≥ i ≥ 1. Proceed by outer induction on j ≥ 2 and inner induction on i ≥ 1.
To get the case i = 1, start from

xxj−1 = xj = (xxj−2 ·x) ·xx−1 = x · (xj−2x ·x)x−1 = x ·xjx−1.

By cancelling, xjx−1 = xj−1. Therefore

(x−1xj ·x−1)x−1 = x−1(xjx−1 ·x−1) = xj−3 = xj−2x−1,

and that yields x−1xj ·x−1 = xj−2 = xj−1x−1. Thus, x−1xj = xj−1. Assume
i ≥ 2. Note that xx−i = x−(i−1) follows from the induction assumption since



on a class of power associative lcc-loops 189

this is the same as y−1yi = yi−1, where y = x−1. Hence

x ·xj−i = xj−i+1 = xjx−(i−1)

= (xxj−2x) ·xx−i = x · (xj−2x ·x)x−i = x ·xjx−i.

By cancelling, xj−i = xjx−i. To finish up , first observe that x−i ·xj−ix−i =
xj−3i = xj−ix−i ·x−i. Indeed, if j − i ≤ 0, this follows from the earlier part
of the proof. If j − i > 0, then xj−ix−i = xj−2i = x−ixj−i by the induction
assumption (where a switch to y = x−1 is needed if j−i < i). Since j−2i may
be treated similarly as j−i, the expression of xj−3i follows. Now, x−ixj = xj−i
may be obtained by cancellation from (x−ixj ·x−i)x−i = x−i(xjx−i ·x−i) =
x−i ·xj−ix−i = xj−ix−i ·x−i.

Corollary 2.3. A CC-loop is a power associative WIP-loop if and only
if it fulfills the laws (xy ·x)x = x(yx ·x) and x(x ·yx) = (x ·xy)x.

Proof. This follows from Theorem 2.2.

Corollary 2.4. A CC-loop is a power associative WIP-loop if and only
if it is a WIP LWPC-loop (alternatively, a WIP RWPC-loop).

Proof. This follows from Theorem 2.2 and the fact that LCC and RCC
are equivalent in a WIP-loop.

Corollary 2.5. (i) A loop is a LWPC-loop if and only if the conju-
gates of its left translations are left translations and any left translation
commutes with the square of its corresponding right translation.

(ii) A loop is a RWPC-loop if and only if the conjugates of its right trans-
lations are right translations and any right translation commutes with
the square of its corresponding left translation.

(iii) A loop is a WIP PACC-loop if and only if the conjugates of both its
left translations and right translations are left translations and right
translations respectively and left and right translations commute with
the squares of their corresponding right and left translations respectively.

Proof. This follows from Theorem 2.2.



190 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru

3. Construction of LWPC-loops

Suppose that G and R are abelian groups and that f : G × G → R is
a mapping. Call it zero preserving if f(x, 0) = 0 = f(0,x) for all x ∈ G.
Say that f is additive on the right if f(x,y + z) = f(x,y) + f(x,z) for all
x,y,z ∈ G. Say that f is additive if it is both right and left additive. Say
that f is quadratically triadditive on the left if

g : G×G×G −→ R
(x,y,z) 7−→ f(x + y,z) −f(x,z) −f(y,z)

is a triadditive symmetric mapping (symmetric means that permuting x,y
and z has no influence upon the value g(x,y,z)).

Define the radical Rad(f) as the set of all x ∈ G such that f(x,y) = 0 =
f(y,x) for all y ∈ G.

Theorem 3.1. Let R be a subgroup of an abelian group G, and let f :
G×G → R be such that Rad(f) ≤ R, f is zero preserving and right additive.
Then

x ·y = x + y + f(x,y)

defines upon G an LCC loop. This loop is associative if and only if f is
biadditive, and conjugacy closed if and only if f is quadratically triadditive
on the left. The LWPC law is fulfilled if and only if

f(2x + y,x) = 2f(x,x) + f(y,x) for all x,y ∈ G. (3)

If G is an elementary abelian 2-group, then (3) always holds, while for G of
odd order, (3) is equivalent to

f(x + y,x) = f(x,x) + f(y,x) for all x,y ∈ G.

Proof. By [3, Theorem 5.3] and [5, Corollary 2.2], only the part relating
to (3) needs to be considered. Since f is additive on the right,

(xy ·x)x = 3x + y + f(x,y) + f(x + y,x) + f(2x + y,x) ,
x(yx ·x) = 3x + y + f(y,x) + f(x + y,x) + f(x, 2x + y) .

Hence (xy · x)x = x(yx · x) if and only if (3) holds. By Theorem 2.2, this
means that (3) characterizes the LWPC loops. Let it be satisfied. If 2z = 0
for all z ∈ G, then (3) is trivially true. Furthermore, setting y = 0 implies
that 2f(x,x) = f(2x,x).



on a class of power associative lcc-loops 191

By adding with itself both the left and the right hand sides and using the
additivity on the right, we obtain

f(2x + y, 2x) = f(2x, 2x) + f(y, 2x) for all x,y ∈ G.

If G is of odd order, then 2x may be replaced by x.

Theorem 3.1 provides a tool how to construct LWPC loops that are not
conjugacy closed. By [5, Theorem 4.6], the construction of Theorem 3.1 covers
all LCC loops Q for which there exists a prime p and a central subloop Z such
that |Z| = p and Q/Z is an elementary abelian p- group. Classification of
such loops up to isomorphism was considered in [5, Section 5], while Section
8 of the same paper proves that all (left) Bol loops of order 8 are LCC, and
that all of them may be obtained by the construction of Theorem 3.1. By [8],
a nonassociative WIP PACC loop is of order divisible by 16.

By the library of LOOPS package [9] of GAP [10] there are, up to isomor-
phism, 19 left conjugacy closed loops of order 8 that are not right conjugacy
closed. Six of them are Bol loops discovered by Burn [2] and described in
[5, Section 8]. We have verified that of the remaining 13 loops none fulfills the
LWPC law, and exactly one fulfills the law x(x ·yx) = (x ·xy)x.

Note that if Q is a loop from Theorem 3.1, then the law x(x·yx) = (x·xy)x.
holds if and only if

3f(x,x) + f(x,y) + f(y,x) = f(x,x) + f(x,y) + f(2x + y,x).

In characteristic 2, this is always true. The following thus holds:

Proposition 3.2. A loop of order 8 is LWPC if and only if it is left
Bol. Each such loop fulfills the law x(x · yx) = (x · xy)x. There are exactly
six isomorphism classes of nonassociative LWPC loops of order 8. None of
them is conjugacy closed or Moufang, and in each of them the left nucleus Nλ
is of order 4.

Consider now a loop that possesses a right nucleus of index two, and
suppose that the nucleus is isomorphic to an abelian group (G, +). It is easy
to see (cf. [4, Proposition 4.2]) that such a loop is isomorphic to a loop G[f,g]
that is defined upon {0, 1}×G by

(0,x)(0,y) = (0,x + y) , (0,x)(1,y) = (1,g(x) + y) ,

(1,x)(0,y) = (1,x + y) , (1,x)(1,y) = (0,f(x) + y),

for all x,y ∈ G, where f and g are permutations of G such that g(0) = 0. By



192 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru

[4, Proposition 5.1], such a loop G[f,g] is LCC if and only if g2 = idG and
both

y + f(y + z) = f
(
z + g(y)

)
+ g(y) ,

x + f(y + z) = f−1
(
z + f(y)

)
+ f(y)

(4)

are true for all x,y,z ∈ G.

Proposition 3.3. Let f and g be permutations of G, where G is an
abelian group. Suppose that g(0) = 0, g2 = idG and that (4) holds. The loop
G[f,g] fulfills the LWPC law if and only if

g
(
f(x + y) + x

)
= x + f

(
g(y) + x

)
,

f
(
g
(
f(x) + y

)
+ x
)

= f(x) + g
(
f(y) + x

) (5)
for all x,y ∈ G.

Proof. By Theorem 2.2, the only step to do is to verify that the two equal-
ities hold if and only if (ab ·a)a = a(ba ·a) whenever a = (ε,x) and b = (η,y)
where x,y ∈ G and ε,η ∈{0, 1}. The case ε = η = 0 is clear.

Assume ε = 0 and η = 1. Then ab ·a = (1,x + g(x) + y) and (ab ·a)a =
(1, 2x + g(x) + y), while ba ·a = (1, 2x + y) and a(ba ·a) = (1, 2x + y + g(x)) =
(ab ·a)a.

Thus, a = (1,x) may be assumed. Suppose first that η = 0. Then ab ·a =
(0,f(x + y) + x) and (ab · a)a = (1,g(f(x + y) + x) + x), while ba · a =
(0,f(x + g(y)) + x) and a(ba ·a) = (1,f(x + g(y)) + 2x). Thus, the equality
holds if and only if g(f(x + y) + x) = f(x + g(y)) + x.

Suppose now that η = 1. Then, ab ·a = (1,g(f(x) + y) + x) and (ab ·a)a =
(0,f(g(f(x) + y) + x) + x), while

ba ·a =
(
1,g
(
f(y) + x

)
+ x
)
,

a(ba ·a) =
(
0,f(x) + g

(
f(y) + x

)
+ x
)
,

yielding thus the second equality of (5).

Assume now that f(x) = −x. Then (5) holds if and only if g(−x) =
−g(x) for every x ∈ G. Proposition 3.3 together with [4, Proposition 5.7]
immediately yield the following statement:



on a class of power associative lcc-loops 193

Theorem 3.4. Let g be a permutation of an abelian group G, g(0) = 0,
such that g2(x) = x and g(−x) = −g(x) for every x ∈ G. Suppose also that
g(x) 6= −x for at least one x ∈ G. Define an operation · upon {0, 1}×G by

(0,x)(η,y) =
(
η,gη(x) + y

)
and (1,x)(η,y) =

(
η, (−1)ηx + y

)
for all x,y ∈ G and η ∈{0, 1}.

The operation · describes a nonassociative LWPC loop in which the right
nucleus is equal to {(0,x); x ∈ G} and the left nucleus is equal to

{
(0,x) :

g(x + y) = g(x) −y for every y ∈ G
}

.

Surprisingly, the case g = idG fulfills the assumptions of Theorem 3.4. For
each n ≥ 3 the operation

(ε,x)(η,y) =
(
ε + η, (−1)εηx + y

)
thus yields an LWPC loop upon Z2 × Zn. Such a loop is never conju-
gacy closed since the left nucleus is trivial, while the right nucleus coincides
with {0}×Zn.

The least nonassociative LWPC loop is of order 6. Up to isomorphism this
is the only nonassociative LWPC loop of order 6. The latter fact has been
verified by using the LOOPS package [9] of GAP [10].

Questions 3.5. (a) Can WIP LWPC loops be described (as a loop
variety) by equations that do not use division and/or inverses?

(b) What is the least odd order for which there exists a nonassociative
LWPC loop?

Acknowledgements

We acknowledge the valuable suggestions and recommendations of
the anonymous referee which have improved the presentation and struc-
tural arrangements of this work.

References

[1] R.H. Bruck, “ A survey of Binary Systems ”, Springer-Verlag, Berlin-
Gottingen-Heidelberg, 1958.

[2] R.P. Burn, Finite Bol loops, Math. Proc. Cambridge Philos. Soc. 84 (3)
(1978), 377 – 385.

[3] P. Csörgo, A. Drápal, Left conjugacy closed Loops of nilpotency class
two, Results Math. 47 (2005), 242 – 265.



194 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru

[4] A. Drápal, On left conjugacy closed loops with a nucleus of index two, Abh.
Math. Sem. Univ. Hamburg 74 (2004), 205 – 221.

[5] A. Drápal, On extraspecial left conjugacy closed loops, J. Algebra 302
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[6] E.G. Goodaire, D.A. Robinson, A class of loops which are isomorphic
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[7] T.G. Jaiyéo. lá, “ A Study of New Concepts in Smarandache Quasigroups
and Loops ”, InfoLearn (ILQ), Ann Arbor, MI, 2009.

[8] M.K. Kinyon, K. Kunen, Power-associative, conjugacy closed loops, J.
Algebra 304 (2006), 671 – 711.

[9] G.P. Nagy, P. Vojtěchovský, The LOOPS Package, Computing with
quasigroups and loops in GAP 3.4.1.
https://www.gap-system.org/Manuals/pkg/loops/doc/manual.pdf

[10] The GAP Group, GAP - Groups, Algorithms, Programming, Version 4.11.0.
http://www.gap-system.org

[11] H.O. Pflugfelder, “ Quasigroups and Loops: Introduction ”, Sigma Series
in Pure Mathematics, 7, Heldermann Verlag, Berlin, 1990.

[12] J.D. Phillips, A short basis for the variety of WIP PACC - loops, Quasigroups
Related Systems 14 (2006), 73 – 80.

https://www.gap-system.org/Manuals/pkg/loops/doc/manual.pdf
http://www.gap-system.org

	Introduction
	LWPC-loop, RWPC-loop and their properties
	Construction of LWPC-loops