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Diassociativity in conjugacy closed loops
 Comm. Algebra
"... Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is powerassociative and Q  is finite and relatively prime to 6, then Q is a group. ..."
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Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is powerassociative and Q  is finite and relatively prime to 6, then Q is a group. If Q is a finite nonassociative extra loop, then 16  Q. 1
Every diassociative Aloop is Moufang
 Proc. Amer. Math. Soc
"... Abstract. An Aloop is a loop in which every inner mapping is an automorphism. We settle a problem which had been open since 1956 by showing that every diassociative Aloop is Moufang. 1. ..."
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Abstract. An Aloop is a loop in which every inner mapping is an automorphism. We settle a problem which had been open since 1956 by showing that every diassociative Aloop is Moufang. 1.
Structural Interactions Of Conjugacy Closed Loops
"... We study conjugacy closed loops by means of their multi plication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication group, respectively. Put M = {a ∈ Q; La ∈ R}. We prove that the cosets of A agree with orbits of [L, R], ..."
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Cited by 8 (2 self)
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We study conjugacy closed loops by means of their multi plication groups. Let Q be a conjugacy closed loop, N its nucleus, A the associator subloop, and L and R the left and right multiplication group, respectively. Put M = {a ∈ Q; La ∈ R}. We prove that the cosets of A agree with orbits of [L, R], that Q/M ∼= (Inn Q)/L1 and that one can define an abelian group on Q/N ×L1. We also explain why the study of finite conjugacy closed loops can be restricted to the case of N/A nilpotent. Group [L,R] is shown to be a subgroup of a power of A (which is abelian), and we prove that Q/N can be embedded into Aut ([L, R]). Finally, we describe all conjugacy closed loops of order pq.
On Multiplication Groups of Left Conjugacy Closed Loops
, 2003
"... A loop Q is said to be left conjugacy closed (LCC) if the set fLx ; x 2 Qg is closed under conjugation. Let Q be such a loop, let L and R be the left and right multiplication groups of Q, respectively, and let Inn Q be its inner mapping group. Then there exists a homomorphism L ! Inn Q determined ..."
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Cited by 6 (1 self)
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A loop Q is said to be left conjugacy closed (LCC) if the set fLx ; x 2 Qg is closed under conjugation. Let Q be such a loop, let L and R be the left and right multiplication groups of Q, respectively, and let Inn Q be its inner mapping group. Then there exists a homomorphism L ! Inn Q determined by Lx 7! R x Lx , and the orbits of [L; R] coincide with the cosets of A(Q), the associator subloop of Q. All LCC loops of prime order are abelian groups.
POWERASSOCIATIVE, CONJUGACY CLOSED LOOPS
, 2005
"... Abstract. We study conjugacy closed loops (CCloops) and powerassociative CCloops (PACCloops). If Q is a PACCloop with nucleus N, then Q/N is an abelian group of exponent 12; if in addition Q is finite, then Q  is divisible by 16 or by 27. There are eight nonassociative PACCloops of order 16, ..."
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Abstract. We study conjugacy closed loops (CCloops) and powerassociative CCloops (PACCloops). If Q is a PACCloop with nucleus N, then Q/N is an abelian group of exponent 12; if in addition Q is finite, then Q  is divisible by 16 or by 27. There are eight nonassociative PACCloops of order 16, three of which are not extra loops. There are eight nonassociative PACCloops of order 27, four of which have the automorphic inverse property. We also study some special elements in loops, such as Moufang elements, weak inverse property (WIP) elements, and extra elements. In a CCloop, the set of WIP and the set of extra elements are normal subloops. For each c in a PACCloop, c 3 is WIP, c 6 is extra, and c 12 ∈ N. 1.
Automated theorem proving in loop theory
 proceedings of the ESARM workshop
, 2008
"... In this paper we compare the performance of various automated theorem provers on nearly all of the theorems in loop theory known to have been obtained with the assistance of automated theorem provers. Our analysis yields some surprising results, e.g., the theorem prover most often used by loop theor ..."
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In this paper we compare the performance of various automated theorem provers on nearly all of the theorems in loop theory known to have been obtained with the assistance of automated theorem provers. Our analysis yields some surprising results, e.g., the theorem prover most often used by loop theorists doesn’t necessarily yield the best performance. 1
Automated theorem proving in quasigroup and loop theory
 NORTHERN MICHIGAN UNIVERSITY, MARQUETTE, MI 49855 USA
"... We survey all known results in the area of quasigroup and loop theory to have been obtained with the assistance of automated theorem provers. We provide both informal and formal descriptions of selected problems, and compare the performance of selected stateofthe art first order theorem provers on ..."
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Cited by 5 (2 self)
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We survey all known results in the area of quasigroup and loop theory to have been obtained with the assistance of automated theorem provers. We provide both informal and formal descriptions of selected problems, and compare the performance of selected stateofthe art first order theorem provers on them. Our analysis yields some surprising results, e.g., the theorem prover most often used by loop theorists does not necessarily yield the best performance.
Gloops and permutation groups
 J. Algebra
, 1999
"... Abstract A Gloop is a loop which is isomorphic to all its loop isotopes. We apply some theorems about permutation groups to get information about Gloops. In particular, we study Gloops of order pq, where p! q are primes and p (q \Gamma 1). In the case p = 3, the only Gloop of order 3q is the gr ..."
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Abstract A Gloop is a loop which is isomorphic to all its loop isotopes. We apply some theorems about permutation groups to get information about Gloops. In particular, we study Gloops of order pq, where p! q are primes and p (q \Gamma 1). In the case p = 3, the only Gloop of order 3q is the group of order 3q. The notion &quot;Gloop &quot; splits naturally into &quot;left Gloop &quot; plus &quot;right Gloop&quot;. There exist nongroup right Gloops and left Gloops of order n iff n is composite and n? 5.
An holomorphic study of the Smarandache concept
 in loops, Scientia Magna
"... If two loops are isomorphic, then it is shown that their holomorphs are also isomorphic. Conversely, it is shown that if their holomorphs are isomorphic, then the loops are isotopic. It is shown that a loop is a Smarandache loop if and only if its holomorph is a Smarandache loop. This statement is a ..."
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If two loops are isomorphic, then it is shown that their holomorphs are also isomorphic. Conversely, it is shown that if their holomorphs are isomorphic, then the loops are isotopic. It is shown that a loop is a Smarandache loop if and only if its holomorph is a Smarandache loop. This statement is also shown to be true for some weak Smarandache loops(inverse property, weak inverse property) but false for others(conjugacy closed, Bol, central, extra, Burn, A, homogeneous) except if their holomorphs are nuclear or central. A necessary and sufficient condition for the Nuclearholomorph of a Smarandache Bol loop to be a Smarandache Bruck loop is shown. Whence, it is found also to be a Smarandache Kikkawa loop if in addition the loop is a Smarandache Aloop with a centrum holomorph. Under this same necessary and sufficient condition, the Centralholomorph of a Smarandache Aloop is shown to be a Smarandache Kloop. 1