Editing Évariste Galois (section)

=== Algebra ===
While many mathematicians before Galois gave consideration to what are now known as [[group (algebra)|groups]], he was the first one to use the word ''group'' (in French ''groupe'') in a sense close to the technical sense that is understood today, making him among the founders of the branch of algebra known as [[group theory]]. He called the decomposition of a group into its left and right [[coset]]s a ''proper decomposition'' if the left and right cosets coincide, which leads to the notion of what today are known as [[normal subgroup]]s.<ref name="chevalier-letter" /> He also introduced the concept of a [[finite field]] (also known as a [[Galois field]] in his honor) in essentially the same form as it is understood today.<ref name="numtheory" />

In his last letter to Chevalier<ref name="chevalier-letter" /> and attached manuscripts, the second of three, he made basic studies of linear groups over finite fields:
*He constructed the [[General linear group#Over finite fields|general linear group over a prime field]], GL(''ν'', ''p'') and computed its order, in studying the Galois group of the general equation of degree ''p<sup>ν</sup>''.<ref>Letter, p. 410</ref>
*He constructed the [[projective special linear group]] PSL(2,''p''). Galois constructed them as fractional linear transforms, and observed that they were simple except if ''p'' was 2 or 3.<ref>Letter, p. 411</ref> These were the second family of finite [[simple group]]s, after the [[alternating group]]s.<ref name="raw">{{cite book | last1=Wilson | first1=Robert A. | author-link = Robert Arnott Wilson | title = The finite simple groups | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=[[Graduate Texts in Mathematics]]|volume= 251 | isbn=978-1-84800-987-5 | doi=10.1007/978-1-84800-988-2 | zbl=1203.20012 | year=2009 |chapter = Chapter 1: Introduction |chapter-url=http://www.maths.qmul.ac.uk/~raw/fsgs_files/intro.ps}}</ref>
*He noted the [[exceptional object|exceptional fact]] that PSL(2,''p'') is simple and [[Projective linear group#Action on p points|acts on ''p'' points]] if and only if ''p'' is 5, 7, or 11.<ref>Letter, pp. 411–412</ref><ref>{{cite web| url = http://people.math.umass.edu/~tevelev/475_2016/galois_lc.pdf| title = Galois's last letter, translated}}</ref>