Émile Mathieu is remembered especially for his discovery (in 1860 and 1873) of five sporadic simple groups named after him. These were studied in his thesis on transitive functions.
Mathieu was brought up in Metz, and he attended school in that town. He excelled at school, first in classical studies showing remarkable abilities in Latin and Greek. However, once he had met mathematics when he was in his teenage years, it became the only subject which he wanted to pursue. Entering the École Polytechnique in Paris his progress was almost unbelievable, even given the remarkable achievements of the brilliant mathematicians in this archive who also attended this institution. It took Mathieu only eighteen months to complete the whole course and he continued to study for a doctorate. By 1859 he had been awarded his Docteur ès Sciences for a thesis on transitive functions, the work which led to his initial discovery of sporadic simple groups.
Progress as remarkable as that achieved by Mathieu would seem to put him in the ideal position to obtain a university appointment but this was not forthcoming. He took on work as a private tutor of mathematics and he continued in this role for ten years. He did suffer a rather serious illness in 1866 and this seems to have stopped him taking over Lamé's courses at the Sorbonne in that year. He was appointed professor of mathematics at Besançon in 1869 and, after five years teaching at Besançon, Mathieu moved to Nancy to take up the chair of mathematics there.
Mathieu's main work, after his initial interest in pure mathematics, was in mathematical physics although he did do some important work on the hypergeometric function. As Grattan-Guinness writes in :-
Although Mathieu showed great promise in his early years, he never received such normal signs of approbation as a Paris chair or election to the Académie des Sciences. From his late twenties his main efforts were devoted to the then unfashionable continuation of the great French tradition of mathematical physics, and he extended in sophistication the formation and solution of partial differential equations for a wide range of physical problems.
Perhaps had Mathieu continued to follow up his remarkable discoveries in group theory, he might have achieved more fame and better posts in his lifetime.
Some of his earliest work in mathematical physics was related to his study of light and he looked at the surfaces of vibrations arising from Fresnel waves. He also worked on the polarisation of light where he highlighted some weaknesses in Cauchy's results on the topic.
He worked on potential theory applied to elasticity, heat diffusion, and the vibration of bells, a very hard problem. Mathieu studied fluids, in particular examining capillary forces. He also studied magnetic induction and the three body problem where he applied his work to the perturbations of Jupiter and Saturn.
In addition to being remembered for the Mathieu groups, he is also remembered for the Mathieu functions. He discovered these functions, which are special cases of hypergeometric functions, while solving the wave equation for an elliptical membrane moving through a fluid. The Mathieu functions are solutions of the Mathieu equation which is
d2u/dz2= (a + 16b cos 2z)u = 0.
In  Grattan-Guinness describes Mathieu's nature as:-
... shy and retiring [which] may have accounted to some extent for the lack of worldly success in his life and career; but among his colleagues he won only friendship and respect.
Article by: J J O'Connor and E F Robertson