The study of Central loops (C-loops) has received attention from authors likes Fenyves in (1968) and (1969), Kenneth Kunen (1996), Philips and Vojtechovsky in (2006) “C-loops:
Introduction” , Adeniran, Oyebo and Mohammed (to appear) studied some autotopic maps of C-loops , Solarin and Chiboka (1995, 1993), and Jaiyeola and Adeniran (2006), and Jaiyeola in (2009) . LC-loops, RC-loops, C-loops and their isotopes were studied by Jaiyeola and Adeniran (2009) and it was shown that isotopes exist under triples of the form (A,B,B) and (A,B,A). Our aim in this section is to investigate and find out under what conditions the derivatives of a Central loop can become Central loops?
Let briefly recall that:
If ( ) is an inverse property loop, then the inverse mapping of ( ) is defined by for every . Moreover we have: (it is the identity mapping of ( ) ) we also have ( ) ( ) ( ) and ( ) ( ) ( ) .
An ordered triple ( ) of permutations of the set , is called an autotopism of the loop ( ) if and only if ( ) . Note that the set of all autotopisms of ( ) forms a group under componantwise multiplication:
( ) ( ) ( ). The unit element of this group is ( ), and ( ) ( ).
A permutation of a set is said to be a pseudo-automorphism of a loop ( ) if and only if there exists an element of , called a companion of , such that ( ( ) ( )) is an autotopism of ( ).
78
An element of a loop ( ) is called a Moufang element of ( ) if and only if we
have ( ( ) ( ) ( ) ( )) is an autotopism of ( ).
A loop ( ) is said to be a Moufang loop, if every is Moufang element of ( ).
The following results are well-known: [See (Bruck, 1958), Chapter VII, lemma 2.1
and theorem 2.3] (1 is the unit element of ( )).
Lemma 4.3.2
If ( ) is an autotopism of an inverse property loop ( ), then (i) ( ) and ( ) are autotopisms of ( ), (ii) and are Moufang elements of ( ).
Lemma 4.3.3
If ( ) is an autotopism of the inverse property loop ( ) and if , then is a pseudo-automorphism of ( ) with companion .
Remark: 4.3.3
A set of permutations on a G is the representation of a loop ( ) if and only if:
(i) (identity mapping)
(ii) is transitive on G ( that is for all , there exists a unique such that ) (iii) If and fixes one element of G, then .
The left and right representation of a loop G is denoted by ( ) ( ) and ( ) ( )
Definition 4.3.9
Central loop is also one of the loops of Bol-Moufang type but the least studied.
Closely related to them are Left-Central loops (“LC-loops” for short) and Right-Central loops (“RC-loops”). The identities and the equivalent ones that describe these loops (C,RC,LC) are available in [Fenyves 1969] and we give them below for a loop ( ).
(( ) ) ( ( ) central identity (4.3.11) ( ) ( ) ( ( )) left central identity (4.3.12) ( ( )) ( ( )) left central identity (4.3.13)
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(( ) ) ( ( )) left central identity (4.3.14) ( ) ( ) (( ) ) right central identity (4.3.15) (( ) ) (( ) ) right central identity (4.3.16) (( ) ) ( ( )) right central loop (4.3.17) Definition 4.3.10
If a loop ( ) satisfies a ,L, R identity for all , then it is called a
C-,LC-,RC- loop respectively.
Theorem 4.3.19
If ( ) is an LC-loop, then
(a) ( ) has the left inverse property,
(b) ( ) is left alternative,
(c) is in the left nucleus of ( ) for all .
(Analogous results hold for RC-loop.)
Proof.
(a) Let ( ) be an LC-loop, and define by for all . Substituting in (4.3.13) ( ( )) ( ( )) we get ( ( )), whence ( ) i.e. (a) holds. Moreover we see
from this (by ) that ( ) also holds.
(b) (4.3.13) reduces with to the left alternative identity ( ).
(c) Applying (b) to (4.3.13) we obtain ( ) ( ). This completes the proof. Theorem 4.3.20 (Jaiyeola and Adeniran, 2006)
A loop is a C-loop if and only if it is both an RC- and LC-loop.
Proof.
Suppose that ( ) is a C-loop. This means that
(( ) ) ( ( ) (4.3.18)
holds for all . First of all, we prove that ( ) has the inverse property.
Define and by and by respectively for all . Then for any given elements , using the solution of the equations and , and (4.3.18), we obtain
( ) ( ( )) .( ) /
80 and
( ) (( ) ) ( ( )) . Therefore ( ) is an inverse property loop, and also holds for each . So (4.3.18) implies that ( ( ) ( ) ( ) ( ) ) is an autotopism of ( ) for all . Thus, by Theorem 4.1.2 we have,
( ( ) ( ) ( ) ( )) ( ( ) ( ) ( ) ( )) and
( ( ) ( ) ( ) ( ) ) ( ( ) ( ) ( ) ( )) are also autotopisms of ( ). And therefore the RC-identity (4.3.16) and the LC-identity (4.3.13) holds for ( ).
Conversely, if ( ) is both an LC- and RC-loop, then by theorem , ( ) is an inverse property loop, It follows from (4.3.13) that ( ( ) ( ) ( ) ( )) is an autotopism of ( ) for all , and so, by inverse property of ( ), ( ( ) ( ) ( ) ( ) )
( ( ) ( ) ( ) ( ) ) is also an autotopism. This shows that ( ) satisfies the C-
identity (4.3.18). The theorem is proved.
b-) Central loops and commutativity.
Lemma 4.3.4 (Jaiyeola and Adeniran, 2006)
Let ( ) be a commutative loop.( ) is a LC(RC)-loop ( ) is a RC(LC)-loop.
Proof.
Let ( ) be a commutative LC(RC)-loop, then by the identity (4.3.13)[(4.3.16)]:
( ( )) ( ( )) ( ( )) ( ( )) ( ( )) ( ( )) (( ) ) (( ) ) is an RC(LC)-loop by identity
(4.3.16) [(4.3.13)]. Hence the proof is complete.
Corollary: 4.3.1 (Jaiyeola and Adeniran, 2006)
A commutative loop is a RC(LC)-loop it is a C-loop.
Proof.
Recall that by theorem 4.3.20 a loop is a C-loop if and only if it is both an RC-loop and an LC-loop. Using that fact alone, the sufficient part of is proved. The necessary part
follows by this fact and Lemma 4.3.4.
81 Theorem 4.3.21 (Jaiyeola and Adeniran, 2006)
Let ( ) be a LC-loop with identity e, if ( ) is a (A,I,A)-loop isotope of ( ) such that ( ), then:
for a fixed . .
The system { + of isotopic loop obey a form of generalised left distributive law for all fixed .
Proof.
(1) If ( ) is an isotopism, then ( ) . ( ) such
that ( ). Thus
( ) ( ) ( ) ( ( ) ( )) ( ( ) ( ) ) ( ( ) ( ) ) .
(2) Recall that ( ( ) ( ) ). Let , then we have
( ( ) ( ) ) ( ( ) ). ( ( ) ( ) ) ( ( ) ) ( ) ( ( ) ) . Thus , hence
.
(3) From (2), if ( ) then ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). Hence the system * + of isotopic loops obeys a form of generalised left distributive law for all fixed .
Theorem: 4.3.22 (Jaiyeola and Adeniran, 2006)
Let ( ) be a RC-loop with identity e, if ( ) is a (I,B,B)-loop isotope of ( ) such that ( ), then:
for a fixed . .
The system { + of isotopic loop obeys a form of generalised right distributive
law for all fixed .
82
Proof.
(1) If ( ) is an isotopism, then ( ) . ( ) ( ) such that ( ), thus
( ) ( ) ( ) ( ( ) ( )) ( ( ) ( ) )
( ( ) ( ) ) .
(2) Recall that ( ( ) ( ) ). Let then
( ( ) ( ) ) ( ( ) ). ( ( ) ( ) ) ( ( ) ) ( ) ( ( ) ) . Thus is an
isomorphism hence we have .
(3) From (2), if ( ) then ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). Hence the system * + of isotopic loops obeys a form of generalised right distributive law for all fixed . Proposition 4.3.1 (Jaiyeola and Adeniran, 2006)
Let ( ) ( ) be a loop with identity e and ( ) ( ) be a (I,B,B)-commutative loop isotope of ( ) such that ( ) If ( ) is a LC-loop, RC-loop or a C-loop then { + and { } are systems of isotopic commutative C-loop that obeys a form of generalised distributive law.
Proof.
Form the results in theorem 4.3.36, is a commutative C-loop for any fixed . : Hence and are commutative C-loops.
( ( ) ( ) ), thus ( ( ) ) ( ) ( ) (( ) ) ( ) , Thence; if ( ), ( ) . But ( ) ( ), thus we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) the system
* + obeys a form of the generalised left distributive law for all fixed . Also ( ) ( ) ( ) ( ) ( ) ( ) ( ) * + obeys a form of the generalised right distributive law for all . Therefore the system
* + obeys a form of the generalised distributive law for all fixed .
On the other hand ( ( ) ( ) ), thus
83
( ( ) ) ( ) ( ) (( ) )
( ) . Thence, if
( ) ( ) . But ( ) ( ), thus ( ) ( ) ( ) ( ) ( ) ( ) ( ) the system * + obeys a form of the generalised left distributive law for all fixed . Also, ( ) ( ) ( ) ( ) ( ) ( ) ( ) * + obeys the generalised right distributive law for all fixed . Therefore the system * + obeys a form of the generalised distributive law for all fixed . This completes the proof. c-) Derivatives of Central loops:
Theorem: 4.3.23 (Jaiyeola and Adeniran, 2006)
Let ( ) be a commutative LC-loop with identity element e. If ( ) is a (I,B,B)-commutative loop isotope of ( ) such that ( ), then
F is a commutative C-loop.
is a commutative C-loop for any fixed .
; hence and are commutative C-loop.
Proof.
(1) F is a commutative C-loop follows immediately from corollary 4.3.1 (2) If ( ) is an isotopism, then ( ) . With ( ) we have ( ) ( ) hence ( ) ( ) ( ) ( ( ) ( )) is an isotopism.
F is an RC-loop ( ) ( ) . Thus ( ( ) ( ) ) ( ( ) ( ) ) . ( ) is an isotopism implies that ( ( ) ( )) is an isotopism which implies ( ( ) ( ) ) is an isotopism which also implies ( ( ) ( ) ) . Whence we obtain . Applying the isotopic characterisation of RC-loop, is an RC-loop. By hypothesis, is a commutative, hence by lemma 4.3.4, is a commutative C-loop.
(3) Recall that ( ( ) ( ) ). Let , , then we clearly have ( ( ( ) ( ) ) ( ( ) ). But ( ( ) ( ) ), thus
84
( ( ) ) ( ) ) ( ) .
Similarly, if , then ( ( ) ( ) ) ( ( ) ).
But ( ( ) ( ) ), thus ( ( ) ) ( )
( ) . From (2), , hence we also obtain that . is a commutative C-loop, therefore and are C-loops.
Commutativity is an isomorphic invariant property, hence we conclude that and
are commutative.
Remark: 4.3.4
This result is also true if ( ) is a RC-loop.
Theorem: 4.3.24 (Jaiyeola and Adeniran, 2006)
Let ( ) be a commutative C-loop with identity element e. If ( ) is a (B,I,B)-commutative loop isotope of ( ) such that ( ), then
is a commutative C-loop for any fixed .
; hence and are commutative C-loop.
Proof.
A C-loop is an LC-loop. Hence the result follows from Theorem 4.3.23. Remark: 4.3.5
The sets as we have used above here for a loop is not a magma. But Capodaglio Di Cocco (1993, 2003) gives examples of regular permutation sets that form loops. General identity is due to R-schauffer, who introduced the concept in connection problems of the coding, these are mention by Jaiyeola and Adeniran in “On isotopic Characterisation of C-loops”.
C- Conjugacy Closed Loops
The notion of a conjugacy closed loop (CC-loop) is due to Goodaire and Robinson (1982), and independently to Сойкис (1970), with somewhat different terminology.
Following, approximately (Goodaire and Robinson, 1982), these loops have a number of interesting properties. For example, by Goodaire and Robinson (1982), the left and right inner mappings are automorphisms. These properties allow a rather detailed structural analysis to
85
be made; in particular, all CC-loops of orders and (for primes ) are known (see Kunen (2000)). The study of CC-loops has taken off in the last few years because of the dissemination of Basarab‟s Theorem (The quotient of a CC-loop by its nucleus is an abelian group). There are probably no interesting results about the class of all loops, since it is to broad; for example, there are already 109 loops of order six (Bruck, 1946). However, there has been much study of specific classes of loops. Most well-known are the groups, which are associative loops. For these, there are many structure theorems, which enable one to enumerate easily the groups of small orders; for example, there are only two groups of order six. This section yields additional structural information about CC-loops. Previous work on conjugacy closed loops done by: Goodaire and Robinson (1982, 1990), and (1990) showed why this particular variety of loop is interesting. Another motivation for studying CC-loops is that they arise naturally in the study of isotopy and the CCloops forms a natural variety of -loops. (=isotopy-isomorphy loops).
Definition 4.3.11
A loop is left (resp. right) conjugacy closed if its left (resp. right) translations maps are closed under composition. That is a loop ( ) is left (resp. right) conjugacy closed if and only if for all , we have ( ) ( ) ( ) (resp. ( ) ( ) ( ) ); where ( ) ( ) are left and right translation maps respectively; and * ( ) + and
* ( ) +.
A loop that is both left and right conjugacy closed is called conjugacy closed. It is common to refer to these loops as LCC-, RCC-,and CC-loops respectively.
Definition 4.3.12
A loop ( ) is a conjugacy closed (or a CC-loop) if and only if there are functions such that for all :
RCC: ( ) ( ) ( ) LCC: ( ) ( ) ( ).
Lemma 4.3.5
A loop ( ) is a CC-loop if and only if there exist functions such that ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( ). Proof.
RCC and LCC assert that ( ) ( ) ( ) ( ) and ( ) ( ) ( ) ( )
86
By the results of Kunen (2000), for every odd prime there are precisely 3 nonassocitive conjugacy closed loops of order . Сsörgö and Drápal (2009) described these 3 loops by multiplicative formulas on and .
Case : Let be the smallest positive integer relatively prime to and such that is a square modulo (i.e., ). Define multiplication on by . Case : Let be the smallest positive integer relatively prime to and such that is not a square modulo . Define multiplication on by . Case : Define multiplication on by ( ) ( ) ( ).
Moreover, Wilson (1975) constructed a nonassociative CC-loop of order for every prime , and Kunen (2000) showed that there are no other nonassociative CC-loop of this
order. Here is the construction:
Let be an additive cyclic group of order , . Let be the additive cyclic group of order 2. Define multiplication on as follows:
( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ).
For , this only non-group CC-loop of order is displayed in Table 1.
Table 7
By another result of Wilson (1975), the only -loop, and hence the only CC-loop, of prime order is a cyclic group of order . The structure theory for CC-loops uses combinatorial arguments similar to those used in the proof of the Sylow theorems.
Definition 4.3.13
For a loop ( ) with a two-sided neutral element 1 and for :
1) Define and by: .
1 2 3 4 5 6
1 1 2 3 4 5 6
2 2 3 1 5 6 4
3 3 1 2 6 4 5
4 4 6 5 2 1 3
5 5 4 6 3 2 1
6 6 5 4 1 3 2
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2) is power-associative if and only if the subloop is a group. ( ) is power associative if and only if every element is power-associative.
3) The CC-loops which are also diassociative (that is every is a group) are the Extra loops introduced by Fenyves (1968, 1969).
In a quasigroup (loop), we may define left division , and right division, , by :
( ) ( ) (4.3.19).
By cancellation, and setting or , we have also
( ) ( ) (4.3.20)
4) A loop ( ) is conjugacy closed (a CC-loop) if and only if it satisfies both of the following equations: ( ) ( ) ( ( )) (RCC) ( ) (( ) ) ( ) (LCC) As usual, equations written this way with variables are understood to be universally quantified. 1) and 2) of this definition are related by:
Lemma 4.3.6 (Kunen, 2000)
Let be an element of a CC-loop ( ). Then
is power associative . Power associative CC-loops (PACC-loops) were thoroughly analysed by Kinyon and Kunen (2006). Central to their analysis is the notion of weak inverse property elements.
Definition 4.3.14
An element of a loop ( ) is a weak inverse property (WIP) element if and only if
( ) ( ) (4.3.21)
A loop ( ) is then a weak inverse property loop if all of its elements have the weak inverse property.
Kinyon and Kunen (2006) showed that if ( ) is a PACC-loop and if , then
( ), the nucleus of ( ); they prove this by showing that is a WIP element and is an extra loop. Recall, that an element of a CC-loop is a Moufang element if and only if for all , ( ( )) ( ) ( ) ( see (Bruck, 1971)), and is an extra element if and only if ( ( )) (( ) ) Kinyon and Kunen (2006).
There is thus, a chain of five prominent varieties of CC-loops: (1) groups, (2) Extra loops, (3) WIP PACC-loops, (4) PACC-loops, (5) CC-loops.
This study shall give a surprisingly short and elegant basis for the variety of WIP PACC-loops.
88 Theorem 4.3.25
In the variety of loops, WIP PACC-loops are axiomatized by the following two identities: (( ) ) ( ) .(( ) ) / (LWIPC) (4.3.22)
( ) ( ( )) . ( ( ))/ (RWIPC) (4.3.23) Proof.
We first show that WIP PACC-loops satisfy both LWPC and RWPC. We proceed in four steps. Firstly, we note that in WIP PACC-loops, it is easy to see that both ( ) ( ) (( ) ) and ( ) ( ) hold, and that together they imply
(( ) ) (( ) ( )) ( ). Secondly, we note that in WIP PACC loops, we have ( ) ( ) (( ) ) ( ) ( ) , and ( ) ( ) hold, and that together they imply ( ) ( ) ( ). Thirdly, we note that in WIP PACC-loops we have ( ) ( ) ( ) ( ) ( ), and we also have ( ) (( ) ( ) ( )), that together they imply ( ) (( ) ) . Fourthly, and finally, we note that in WIP PACC-loops, it is easy to see that both identities ( ) ( ) and ( ) (( ) ) hold and that together the three identities just establish, implies LWPC. Of course, dualize this proof to obtain RWPC.
We now show that loops satisfying both LWPC and RWPC are, in fact, WIP PACC loops. In LWPC, replace with to obtain (( ) ) .(( ) ) ( )/.
Now replace with ( ) to obtain ( (( ) ) ) ( ( )) In other words, there exists a function ( ) such that ( ) ( ) ( ) ( ( )). Thus, in the variety of loops, LWPC implies LCC. Similarly, RWPC implies RCC. Also, it is easy to see that LWPC implies , which by Lemma 5 guarantees power associativity. Recall, that an element of a CC-loop is a Moufang element if and only if in the loop we have ( ( )) ( ) ( ) (see Bruck, 1971), and is an Extra element if and only if ( ( )) (( ) ) (see Kinyon and Kunen, 2006).
Here is a very useful associator identity.
Theorem 4.3.26 (Phillips, 2006)
Let be a Moufang element in a CC-loop ( ). Then , we have
( ) ( ).
Corollary 4.3.2
89
Let be a Moufang element in a WIP CC-loop. Then we have ( ) Proof.
In WIP CC-loops, squares are nuclear (Kinyon, Kunen and Phillips, 2004). Now use
theorem 4.3.26.
Buchsteiner loops (Buchsteiner, 1976) are those loops that satisfy the Buchsteiner law
(( ) ) ( ( )) .
For every Buchsteiner loop ( ) we have ( ) ( ) ( ) ( ) and is abelian for the proof see Csorgo, Drapal and Kinyon (2009). Hence we have:
Theorem 4.3.27 (Csorgo and Drapal, 2009)
Let ( ) be a Buchsteiner loop. Then ( ) is a conjugacy closed loop. What are the non-group CC-loop of order seven or less? By Wilson (1974) as we said earlier, these cannot have prime order, and it is easy to see by inspection that all loops of order four are commutative, and hence groups if they are CC, so that leaves order six. In that case, we have the CC-loop from Table 1, and that is the only one, as is true in general for orders ,
where is an odd prime, by the following Theorem
Theorem 4.3.28
If is an odd prime, then there are exactly three CC-loops of order , exactly two of
which are groups.
Proof. (See Kunen, 2000)
The proof of the following theorem is patterned after Goodaire and Robinson (1982,
1990), from which (1) is immediate.
Theorem 4.3.29
Suppose that are odd primes and is not divisible by . Then:
1) : The only CC-loop of order is .
2) : There are exactly three CC-loops of order besides and . Any finite loop of order is a group and, hence, must also be a -loop. Wilson (1969, 1974, 1975) proved that a finite -loop of prime order is necessarily a group; he also exhibited for each even integer a -loop of order which is not associative and then raised questions concerning the existence of finite -loop which are not groups for every possible composite order . For our purposes the most relevant feature of conjugacy closed loops is that they are all -loops as we shall see.
90 -isotopes. In practice, however, one never needs to examine all of the principal -isotopes of a loop ( ) to see whether or not ( ) is a -loop. In fact, if denotes the identity element of the loop ( ), then a necessary and sufficient condition for ( ) to be a -loop is that ( ) be isomorphic merely to its principal and principal -isotopes. (For justification of this last result see [Wilson (1966) and also Bryant and Hans (1966)].
Some useful characterizations of CC-loops are provided by the following Theorem 4.3.30
91
We now turn to the algebraic properties of conjugacy closed loops with the following Theorem 4.3.31
92
( ) ( ) or ( ) ( ) , respectively, is an autotopism of ( ) where denotes the identity map on . Since
( ) ( ) ( ) ( ) ( ) and
( ) ( ) ( ) ( ) ( ) , it is then clear that is in all three of the nuclei mentioned above whenever it is in any one of the three nuclei. Hence, the nuclei of ( ) coincide.
Consequently, because of the equivalence of statements (i) and (ii) of Theorem 4.3.30, every subloop of ( ) must also be a conjugacy closed loop.
93 homomorphic image of a conjugacy closed loop ( ) is a loop. (In general, a homomorphic image of a loop need not be a loop.) Any homomorphic image of a conjugacy closed loop information from Wilson (1966) and Osborn (1960), one can show that a loop ( ) satisfies Wilson‟s identity (4.3.24) for all if and only if ( ) is a conjugacy closed loop and
satisfies the weak inverse property.
Remark 4.3.7
From Theorem 4.3.31 we have the following result: If ( ) is a conjugacy closed loop with nucleus , then is a normal subloop of ( ) and the quotient loop is also closed loop.
From now we shall focus on the constructions of conjugacy closed loops. Our purpose is to present results and constructions which can be used to produce examples of conjugacy closed loops (and, hence, also G-loops) which are not associative.
94
The preceding result reinforces something already mentioned in the literature: any loop with index 2 nucleus is a -loop. (See Belousov (1967) and Basarab (1968)). It is also of interest to view this in the context of cyclic loop-extensions (see Goodaire and Robinson, 1982).
The next result describes another procedure for obtaining conjugacy closed loops.
Theorem 4.3.33
Let and be rings with commutative and associative and let be a homomorphism of ( ) into ( ). For ( ) and ( ) in where define by
( ) ( ) ( ( ) .
95 (Note that only the commutativity and the associativity of the ring and the homomorphism property for relative to the + operations were needed to perform these simplifications.) Hence, by Theorem 4.3.30, we conclude that ( ) is a conjugacy closed loop.
Now with ( ) we see that explicit description of , it is clear that is an abelian group and our proof is complete.
Corollary 4.3.4 (Goodaire and Robinson, 1982).
If and are positive integers with , then there is a conjugacy closed loop of
order which is not a group.
Proof.
Let ( ) and ( ) be the rings of integers modulo and , respectively. Employing the usual additive coset notation, we define
( ( )) ( ) for all ( ) . Note that for we have ( ) ( ) | | ( ) ( )
and so is well-defined. It is now easy to see that is a homomorphism of ( ) onto
96
( ) and that ( ) ( ) if and only if | . With , we see that and so ( ). It follows that the loop ( ) constructed from and
in Theorem 4.3.33 is a conjugacy closed loop which is not a group. Finally note that
| | | || | .
The loop constructed in the next result was originally offered by Belousov (1967, p.184)
as an example of -loop.
Although the computational details are quite different, we are able to proceed as in the proof of Theorem 4.3.33. Clearly ( ) is a loop. Its identity element is ( ) and for
97 Proof.
Let be the finite field of prime order . Then the multiplicative group of non-zero elements of is cyclic of order . Since | the cyclic group has a (unique) subgroup of order . Now let and note that is a subloop of the conjugacy closed loop ( ) of Theorem 4.3.34. As such, ( ) is also a conjugacy closed loop (see Theorem 4.3.31 (v)). Since there exist elements so that
( )( )( ) . Thus, ( ) is not associative (see the proof of
Theorem 4.2.34). Furthermore, | | | || | .
An obvious modification in our proof of the preceding corollary yields the following generalization: If is a prime, if and are positive integers with , and if | , then there is a conjugacy closed loop of order which is not a group.
Remark 4.3.8
In all the constructions we just did, the loop is an abelian group. This confirm the work of Csorgo and Drapal (2009) which assert that every Conjugacy closed loop ( ) is known to satisfy the followimg properties: ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) and is an abelian group.
The following two results are also from Csorgo and Drapal (2009).
Theorem 4.3.35
Let ( ) be loop such that ( ) ( ) ( ) ( ) ( ) and ( ) ( ) ( ). If is an abelian group, then ( ) is
conjugacy closed.
A loop ( ) is said to be an -loop if ( ) ( ). If ( ) is a LCC-loop, then we have ( ) ( ) ( ) if and only if . LCC A-loops thus coincide with groups.
We can hence conclude this section by
Theorem 4.3.36 (Сsögö and Drápal, 2009) Let
( ) be an -loop such that is an abelian group, then is a group. Theorem 4.3.37
(aсapa, 1991) If ( ) be
a CC-loop then is an abelian group. The proof of this Theorem 4.3.37 can be found in aсapa (1991) and in Kinyon, Kunen and Phillips (2004), although the first author studies in more detail those loops ( ) such that ( ) and all its loop isotopes satisfy RCC.
98
Recall that if is a subloop of a finite loop ( ) ( ) then the subloop of ( ) is means that | | divides | |. A finite loop has the weak Lagrange property if the order of any subloop divides the order of the loop, and a finite loop has a strong Lagrange property if every subloop has a weak Lagrange property. In general if is a normal subloop of ( ), and and both have the strong Lagrange property, then so does ( ) (see Bruck (1968) and Chein, Kinyon, Rajah and Vojtechovsky (2003)). It is immediate
from Theorem 4.3.19 that:
99 THEOREM: 4.3.39
If ( ) is a Commutative Extra loop then its Derivatives are both Moufang and Central loops for any fixed element of .
Proof.
Applying theorem 4.3.17 and theorem 4.3.31.
Remark 4.3.9
One of the very important topics in the theory of commutative Extra loop is the question of nilpotency. Although nilpotency theory can be developed for loops in general see
One of the very important topics in the theory of commutative Extra loop is the question of nilpotency. Although nilpotency theory can be developed for loops in general see