Mathematics – Group Theory
Scientific paper
2005-07-14
J. Algebra 304 (2006), 679-711
Mathematics
Group Theory
31 pages, 12pt amsart; v.3: further revisions after second round of refereeing
Scientific paper
We study conjugacy closed loops (CC-loops) and power-associative CC-loops (PACC-loops). If $Q$ is a PACC-loop 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 PACC-loops of order 16, three of which are not extra loops. There are eight nonassociative PACC-loops 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 CC-loop, the set of WIP and the set of extra elements are normal subloops. For each $c$ in a PACC-loop, $c^3$ is WIP, $c^6$ is extra, and $c^{12} \in N$.
Kinyon Michael K.
Kunen Kenneth
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