Does anyone know why "normal subgroups" are "normal"? Is it because they are the kernel of a homomorphism, and that's the way it's supposed to be? Also, does the use of "normal" for a field come from the relationship to a normal subgroup, in Galois Theory? Thanks, Bill Cordwell
On Tue, 14 Mar 2006, Cordwell, William R wrote:
Does anyone know why "normal subgroups" are "normal"? Is it because they are the kernel of a homomorphism, and that's the way it's supposed to be?
Also, does the use of "normal" for a field come from the relationship to a normal subgroup, in Galois Theory?
It doesn't answer your question, but here's what is said about Normal Subgroup at the website: Earliest Known Uses of Some of the Words of Mathematics http://hometown.aol.com/jeff570/mathword.html ---------------------------------------------------------------------------- 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." ---------------------------------------------------------------------------
One way and another, this terminology is rather overworked. I'm currently trying to recast elementary geometry in terms of Clifford algebra, so frequently trying to avoid referring simultaneously to normal subgroups, vectors normal to a surface, and (the only remotely justifiable use of the word) normalised coordinates. Considering their classical education, 19-th century mathematicians were often remarkably cavalier in their choice of nomenclature. Still, I must ruefully admit that, in spite of endlessly agonising over such questions, I usually manage to get it wrong as well. I always felt that John Conway has a very good ear for neologism. Fred Lunnon On 3/14/06, Edwin Clark <eclark@math.usf.edu> wrote:
On Tue, 14 Mar 2006, Cordwell, William R wrote:
Does anyone know why "normal subgroups" are "normal"? Is it because they are the kernel of a homomorphism, and that's the way it's supposed to be?
Also, does the use of "normal" for a field come from the relationship to a normal subgroup, in Galois Theory?
It doesn't answer your question, but here's what is said about Normal Subgroup at the website: Earliest Known Uses of Some of the Words of Mathematics http://hometown.aol.com/jeff570/mathword.html
---------------------------------------------------------------------------- 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." ---------------------------------------------------------------------------
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participants (3)
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Cordwell, William R -
Edwin Clark -
Fred lunnon