Zaku, Garba Gambo
(2020)
Moufang Loops, Magmas And The
Moufang Identities.
PhD thesis, Universiti Sains Malaysia.
Abstract
Loop theory is a generalization of group theory; Moufang loops are a variety of
loops. Four equivalent (Moufang) identities axiomatize these loops. Moufang loops
also share many similar properties as groups though generally they are not associative;
Moufang’s Theorem is pivotal in establishing this close relationship. The existing
proof of the equivalence of the Moufang identities involves the notion of "autotopism",
a completely difficult concept in itself, whereas there is no known complete proof of the
Moufang’s Theorem (though several reasonably acceptable proofs exist). This thesis
provides a simple, basic and complete proof of both. Moreover, the equivalence of
the localized versions of the four identities is studied under the generalized setting of
magmas and proven under necessary and sufficient conditions. Finally, this research
gives a (partial) resolution of Moufang loops of odd order p2q4.
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