A large part of her work is dedicated to the foundations of geometry. Her most outstanding contribution to this field is a result which adds a third important discovery to two others made previously by David Hilbert (1901 and 1930). Reversing a development going from Euclid to Descartes in which geometry is replaced by algebra as a fundamental discipline of mathematics, Hilbert had shown that a subset of his axioms for plane geometry (essentially the incidence axioms) together with the incidence theorem of Desargues permits the introduction of coordinates on a straight line which are elements of a skew field. If Desargues' theorem is replaced by that of Pappus<, the coordinates become elements of a field. Moufang (1933) showed that another incidence theorem, called the theorem of the complete quadrilateral (or of the invariance of the fourth harmonic point), allows one to introduce coordinates which are elements of an alternating division algebra. This and a subsequent paper had the effect of stimulating further research of these algebras and of other nonassociative algebraic structures (Moufang loops). Her work is based both on a powerful geometric intuition and on the development of difficult algebraic techniques. It is supplemented by a sequence of papers on continuum mechanics.The Nazis, to be precise Hitler's minister of education, refused Moufang permission to teach (because she was a woman), so from 1937 she became an industrial mathematician working on elasticity theory. In fact this gives Moufang the unique position of being the first German woman with a doctorate to be employed in industry. She may actually be the first ever such woman anywhere.
Moufang published only one paper on group theory which was published in 1937. In this paper, which is motivated by the two papers of Hilbert on geometry mentioned above (published in 1901 and 1930), she examines the group M = F/F'', the free metabelian group on two generators. She proves that the rational group algebra of this group can be embedded in an ordered division ring. As a consequence it is easy to show that M contains a copy of the free semigroup on two generators. Moufang also gives applications of the result to number theory, knot theory and the foundations of geometry.
Moufang taught at Frankfurt from 1946 where she became a professor but published nothing further. Again she holds a unique position here as the first German woman professor.