Two small and independent interrogative remarks:<\/p>\n
(1) Nowadays, an extension of a group G<\/em> by a group K<\/em> is a group H<\/em> fitting in a short exact sequence<\/p>\n $latex 1\\rightarrow K\\rightarrow H\\rightarrow G\\rightarrow 1$<\/p>\n in other words, and rather counterintuitively, the group G <\/em>is a quotient of the group which is the extension. When did this terminology originate? A paper of Alan Turing<\/a> (entitled, rather directly, “The extensions of a group”) defines “extension”, in the very first paragraph, exactly in the opposite (na\u00efve) way, quoting Schreier and Baer who, presumably, had the same convention.<\/p>\n (2) There’s a whole lot of discussion here and there about the mystical “field with one element”; usually, papers of Tits from around 1954 are mentioned as being the source of the whole “idea”; however, the following earlier quote from a 1951 paper of R. Steinberg (“A geometric approach to the representations of the full linear group over a Galois field”, 1951, p. 279, TAMS 71, 274–282) seems to also contain a germ of the often mentioned analogy between the formulas for the order of the Weyl groups and those of groups of Lie type over a field with q <\/em>elements, the former being obtained by specializing the latter for q=1<\/em>:<\/p>\n In closing this section, a remark on the analogy between G<\/em> and H<\/em> seems to be in order. Instead of considering G<\/em> as a group of linear transformations of a vector space, we could consider G<\/em> as a collineation group of a finite (n-1)<\/em>-dimensional geometry. If q=1<\/em>, the vector space fails to exist but the finite geometry does exist and, in fact, reduces to the n<\/em> vertices of a simplex with a collineation group isomorphic to H<\/em>. “<\/p><\/blockquote>\n In this citation, G <\/em>is GL(n,F<\/strong>q<\/sub>)<\/em>, and H<\/em> is the symmetric group on n<\/em> letters.<\/p>\n (3) Here’s a third question: when did the terminology “Galois field” become more or less obsolete within the pure mathematics community?<\/p>\n","protected":false},"excerpt":{"rendered":" Two small and independent interrogative remarks: (1) Nowadays, an extension of a group G by a group K is a group H fitting in a short exact sequence $latex 1\\rightarrow K\\rightarrow H\\rightarrow G\\rightarrow 1$ in other words, and rather counterintuitively, the group G is a quotient of the group which is the extension. When did … Continue reading Tidbits of terminology and other folklore<\/span><\/a><\/p>\n","protected":false},"author":625,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"class_list":["post-64","post","type-post","status-publish","format-standard","hentry","category-blogroll"],"_links":{"self":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/64","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/users\/625"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/comments?post=64"}],"version-history":[{"count":0,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/posts\/64\/revisions"}],"wp:attachment":[{"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/media?parent=64"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/categories?post=64"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blogs.ethz.ch\/kowalski\/wp-json\/wp\/v2\/tags?post=64"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}