{"id":15905,"date":"2023-09-22T15:24:16","date_gmt":"2023-09-22T19:24:16","guid":{"rendered":"https:\/\/notes.math.ca\/article\/a-jargon-minimal-counting-proof-of-sylows-first-theorem\/"},"modified":"2023-10-16T11:31:16","modified_gmt":"2023-10-16T15:31:16","slug":"a-jargon-minimal-counting-proof-of-sylows-first-theorem","status":"publish","type":"article","link":"https:\/\/notes.math.ca\/fr\/article\/a-jargon-minimal-counting-proof-of-sylows-first-theorem\/","title":{"rendered":"A jargon-minimal counting proof of Sylow\u2019s first theorem"},"content":{"rendered":"

Abstract<\/strong><\/h4>\n

We present a short proof of Sylow\u2019s famous \u2018First Theorem.\u2019  Stripped to its essentials, the proof\u2014attributed to Wielandt  (1959)\u2014relies only on basic concepts (equivalence relations and divisibility). Whereas most textbook proofs invoke plenty of group-theoretic jargon (stabilizers, conjugacy  classes,  etc.),  this  one  avoids  all  that,  stands  alone,  and fits on a page.<\/p>\n

A (binary) million years ago, Helmut Wielandt published an appealing counting proof [3] of Sylow\u2019s First Theorem.  Stripped to its bare essentials, it requires surprisingly little group theory and ought to be better known. The version below lends itself to an hour or so in a classroom with second- or third-year university math enthusiasts.<\/p>\n

Theorem (\u2018Sylow\u2019s First\u2019) <\/strong>If p<\/span> is a prime, r<\/span> is an integer coprime to p<\/span>, and G<\/span> is a group <\/em>of order p^{alpha}r<\/span>, then G<\/span> contains a(t least one) subgroup of order p^{alpha}<\/span>.<\/em><\/p>\n

Proof. <\/em>Let mathcal{S}<\/span><\/em> denote the family of all subsets <\/strong><\/em>of G<\/span><\/em> of size p^{alpha}<\/span>, <\/em>and write mathcal{S}={A_1,A_2,ldots,A_n}<\/span>. <\/em>We shall argue that at least one member of mathcal{S}<\/span><\/em> <\/em>is a group\u2014indeed, is a subgroup of G<\/span>.<\/em><\/p>\n

It\u2019s convenient to know that p<\/span> <\/em>and n<\/span> <\/em>are coprime, so we first establish this fact. We have<\/p>\nn=binom{p^{alpha}r}{p^{alpha}}=frac{p^{alpha}r(p^{alpha}r-1)(p^{alpha}r-2)cdots(p^{alpha}r-(p^{alpha}-1))}{p^{alpha}cdot1cdot 2cdot~cdots~cdot(p^{alpha}-1)}=rprodlimits_{k=1}^{p^{alpha}-1}frac{p^{alpha}r-k}{k}.<\/span>\n

Consider the factors frac{p^{alpha}r-k}{k}<\/span>. If pnmid k<\/span>, then pnmid (p^{alpha}r-k)<\/span>. On the other hand, if p,|, k<\/span>, then k=p^{beta}s<\/span> for some integer beta<\/span> with 1leqbeta<alpha<\/span> and pnmid s<\/span>. So here, we have<\/p>\nfrac{p^{alpha}r-k}{k}=frac{p^{beta}(p^{alpha-beta}r-s)}{p^{beta}s}=frac{p^{alpha-beta}r-s}{s},<\/span>\n

and pnmid p^{alpha-beta}r-s<\/span> (for otherwise, p,|, s<\/span>). In either case, p<\/span> fails to divide each factor frac{p^{alpha}r-k}{k}<\/span>, and thus we see that pnmid n<\/span>.<\/p>\n

Now if xin G<\/span> and A_iinmathcal{S}<\/span>, then the set A_{i}x<\/span> also contains p^{alpha}<\/span> elements (for the map A_ito A_{i}x<\/span> given by amapsto ax<\/span> is injective). Hence, A_{i}x=A_{j}<\/span> for some jin{1,ldots,n}<\/span>. Let us define a relation sim<\/span> on mathcal{S}<\/span> by<\/p>\nA_isim A_j Longleftrightarrow A_{i}x=A_{j} text{ for some } xin G.<\/span>\n

Using the group axioms for G<\/span>, it’s easy to see that sim<\/span> is an equivalence relation. Since pnmid n<\/span>, at least one equivalence class mathcal{C}<\/span>  of sim<\/span> contains some q<\/span> sets A_i<\/span> with pnmid q<\/span>; say mathcal{C}={A_1,A_2,ldots,A_q}<\/span> (relabelling the A_i<\/span>‘s as necessary).<\/p>\n

Let H={xin Gcolon A_{1}x=A_{1}}<\/span>; one easily verifies that H<\/span> is a subgroup of G<\/span>—write h<\/span> for its order. For x,yin G<\/span>, observe that<\/p>\n A_{1}x=A_{1}y Leftrightarrow A_{1}xy^{-1}=A_{1} Leftrightarrow xy^{-1}in H Leftrightarrow Hxy^{-1}= H Leftrightarrow Hx=Hy; <\/span>\n

therefore, A_{1}xneq A_{1}y<\/span> if and only if Hxneq Hy<\/span>. It follows from this equivalence and the definition of mathcal{C}<\/span> that mathcal{C} <\/span>‘s order q<\/span> coincides with the number of distinct (right) cosets of H<\/span> in G<\/span>; i.e., q=p^{alpha}r\/h<\/span>, or h=p^{alpha}r\/q<\/span>. But since pnmid q<\/span>, this implies that q,|, r<\/span>, so that hgeq p^{alpha}<\/span>. <\/p>\n

To see the reverse inequality, consider an element ain A_1<\/span>. The definition of H<\/span> implies that the (left) coset aH<\/span> is a subset of A_1<\/span>; whence,<\/p>\n

h=|H|=|aH|leq |A_1|=p^{alpha}<\/span>,<\/p>\n

and it now follows that h=p^{alpha}<\/span>. Therefore, the subgroup H<\/span> of G<\/span> is indeed a member of mathcal{S}<\/span>. <\/p>\nBox<\/span>\n

Remark <\/strong>Other authors\u2014e.g., [1, 2]\u2014have presented Wielandt\u2019s proof; Professor Petrich (see below) distilled it particularly nicely on one chalkboard.<\/p>\n

In memoriam<\/h2>\n

Submitted in honour of Mario Petrich (1932\u20132021)<\/strong>, who showed me (and the rest of the class) this proof at Simon Fraser University in 1986.<\/p>\n

References<\/h2>\n

[1]  I.N. Herstein, Topics in Algebra<\/em>, Second edition, Xerox College Publishing, Lexington MA, 1975.<\/p>\n

[2]  I. Martin Isaacs, Algebra. A Graduate Course<\/em>, Brooks\/Cole, Pacific Grove CA, 1994.<\/p>\n

[3]  Helmut Wielandt, Ein Beweis f\u00fcr die Existenz der Sylowgruppen, Arch. Math. <\/em>(Basel), 10 <\/strong>(1959), 401\u2013402.<\/p>\n

AMS (MOS) Subject Classifications<\/strong>: 20D20, 05A10, 00A05, 11-01<\/p>\n","protected":false},"author":11,"template":"","section":[468],"keyword":[467,466,136],"class_list":["post-15905","article","type-article","status-publish","hentry","section-submissions","keyword-finite-groups","keyword-finite-groups-and-orders-of-their-subgroups","keyword-research"],"toolset-meta":{"author-4-info":{"author-4-surname":{"type":"textfield","raw":""},"author-4-given-names":{"type":"textfield","raw":""},"author-4-honorific":{"type":"textfield","raw":""},"author-4-institution":{"type":"textfield","raw":""},"author-4-email":{"type":"email","raw":""},"author-4-cms-role":{"type":"textfield","raw":""}},"author-3-info":{"author-3-surname":{"type":"textfield","raw":""},"author-3-given-names":{"type":"textfield","raw":""},"author-3-honorific":{"type":"textfield","raw":""},"author-3-institution":{"type":"textfield","raw":""},"author-3-email":{"type":"email","raw":""},"author-3-cms-role":{"type":"textfield","raw":""}},"author-2-info":{"author-2-surname":{"type":"textfield","raw":""},"author-2-given-names":{"type":"textfield","raw":""},"author-2-honorific":{"type":"textfield","raw":""},"author-2-institution":{"type":"textfield","raw":""},"author-2-email":{"type":"email","raw":""},"author-2-cms-role":{"type":"textfield","raw":""}},"author-info":{"author-surname":{"type":"textfield","raw":"Kayll"},"author-given-names":{"type":"textfield","raw":"P. Mark"},"author-honorific":{"type":"textfield","raw":""},"author-email":{"type":"email","raw":"mark.kayll@umontana.edu"},"author-institution":{"type":"textfield","raw":"University of Montana, Missoula, MT, USA"},"author-cms-role":{"type":"textfield","raw":""}},"unknown":{"downloadable-pdf":{"type":"file","raw":"https:\/\/notes.math.ca\/wp-content\/uploads\/2023\/09\/7-A-jargon-minimal-counting-proof-of-Sylows-first-theorem-\u2013-CMS-Notes-1.pdf","attachment_id":16208},"article-toc-weight":{"type":"numeric","raw":"7"},"author-surname":{"type":"textfield","raw":"Kayll"},"author-given-names":{"type":"textfield","raw":"P. Mark"}}},"_links":{"self":[{"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/article\/15905","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/article"}],"about":[{"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/types\/article"}],"author":[{"embeddable":true,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/users\/11"}],"version-history":[{"count":39,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/article\/15905\/revisions"}],"predecessor-version":[{"id":16120,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/article\/15905\/revisions\/16120"}],"wp:attachment":[{"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/media?parent=15905"}],"wp:term":[{"taxonomy":"section","embeddable":true,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/section?post=15905"},{"taxonomy":"keyword","embeddable":true,"href":"https:\/\/notes.math.ca\/fr\/wp-json\/wp\/v2\/keyword?post=15905"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}