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Why are normal subgroups called "normal"? And who is credited with naming them this?

Mike Pierce
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2 Answers2

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The following is a quote from https://mathshistory.st-andrews.ac.uk/Miller/mathword/ where you can find some of the "Earliest Known Uses of Some of the Words of Mathematics":

NORMAL SUBGROUP. According to Kramer (p. 388), Galois used the adjective "invariant" referring to a normal subgroup.

According to The Genesis of the Abstract Group Concept (1984) by Hans Wussing, "The German Normalteiler (normal subgroup) goes back to Weber [H., Lehrbuch der Algebra, vol. 1, Braunschweig, 1895. p.511] and is possibly linked to Dedekind's term Teiler (divisor), which was employed in ideal theory" [Dirk Schlimm].

Normal subgroup is found in English in 1908 in An Introduction to the Theory of Groups of Finite Order by Harold Hilton: "Similarly, if every element of G transforms a subgroup H into itself, H is called a normal, self-conjugate, or invariant subgroup of G (or 'a subgroup normal in G')."

G. A. Miller writes in Historical Introduction to Mathematical Literature (1916), "In the newer subjects the tendency is especially strong to use different terms for the same concept. For instance, in the theory of groups the following seven terms have been used by various writers to denote a single concept: invariant subgroup, self-conjugate subgroup, normal divisor, monotypic subgroup, proper divisor, distinguished subgroup, autojug."

See here for the sources.

Suzu Hirose
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Martin Brandenburg
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    This rather answers as to "when/who" than "why". – Kan't Aug 06 '24 at 05:46
  • As mentioned in the answer, Weber's book Lehrbuch der Algebra uses the terminology "normaltheiler" for normal subgroup. This is defined in the context of Galois theory and I think the name is motivated by normal field extensions, "normalkörper" in Weber. See pp.551--553 in Weber. – testaccount Aug 07 '24 at 02:45
  • But this only moves the as to "why" from subgroups to field extensions, @testaccount. – Kan't Aug 07 '24 at 09:34
  • @Keith Conrad, in his notes “Infinite Galois Theory” (https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2020/06/CTNT-InfGaloisTheory.pdf), says that it comes from normal subgroups corresponding to normal Galois extensions. – rwilsker Feb 11 '25 at 00:48
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In Latin, "norma" means "set square" ("triangle"), whose chief purpose is to draw a right angle or, more straightly, two adiacent right angles. Such a construction has a left-right simmetry, just like the cosets of a "normal" subgroup have, indeed ("$aH=Ha,\space\forall a\in G$"). By extension, "normal" means "inducing some regularity/order" and hence "some structure": think of the group structure induced in the quotient when the subgroup is (indeed) "normal". Of course this is just my guess.