$\\newcommand{\\sym}[1]{{\\rm Sym}\\,#1}$<\/p>\n
The problem was reported by Joris Buloron.<\/p>\n
The statement claims that $$N_G(H)\\lambda_H=C_{{\\rm Sym}[G:H]}(G\\varrho_H).$$<\/p>\n
The statement is true, but the proof is incomplete for infinite groups.<\/p>\n
The proof of the containment $N_G(H)\\lambda_H\\leq C_{\\sym[G:H]}(G\\varrho_H)$ is correct. We only show here that $C_{\\sym[G:H]}(G\\varrho_H)\\leq\u00a0N_G(H)\\lambda_H$.<\/p>\n
Let\u00a0$c\\in C_{\\sym[G:H]}(G\\varrho_H)$ and define $g_0\\in G$ by $Hc = Hg_0^{-1}$.\u00a0Note, for $g\\in G$, that\u00a0$$(Hg)c = H(g \\varrho_H)c = Hc (g \\varrho_H) = Hg_0^{-1} (g \\varrho_H) = Hg_0^{-1}g.$$\u00a0 Hence $(Hg)c = Hg_0^{-1}g$ holds for each $g\\in G$. Thus the inverse $c^{-1}$ maps $H g_0^{-1}g$ to $Hg$ for all $g$. Setting $x = g_0^{-1}g$, we obtain that $c^{-1}$ maps $Hx$ to $Hg_0 x$ for all $x \\in G$. If $x=1$, then this gives that $H c^{-1} = Hg_0$.<\/p>\n
We claim that $H^{g_0}\\leq H$. Let $h\\in H$. Then $c(h\\varrho_H)=(h\\varrho_H) c$, and so $$Hg_0^{-1}h=Hc(h\\varrho_H)=H(h\\varrho_H)c=Hhc=Hc=Hg_0^{-1}.$$ That is, $Hg_0^{-1}hg_0=H$, and hence $g_0^{-1}hg_0\\in H$ holds for all $h\\in H$.<\/p>\n
Our conclusion that $g_0 \\in N_G(H)$ is not necessarily valid when $H$ is infinite. Our proof to this point showed only that $H^{g_0}\\leq H$. We need also to show the reverse inclusion, or equivalently, we need to show that\u00a0 $H^{g_0^{-1}}\\leq H$.\u00a0<\/em>To complete the proof note that $c\\in\u00a0C_{\\sym[G:H]}(G\\varrho_H)$ implies that\u00a0 $c^{-1}\\in C_{\\sym[G:H]}(G\\varrho_H)$ and $Hc^{-1}=Hg_0$. Applying the argument above for $c^{-1}$, we obtain that $H^{g_0^{-1}} \\leq H$; that is, $H\\leq H^{g_0}$, and so $H^{g_0}= H$. Therefore $g_0\\in N_G(H)$.<\/p>\n On line 4 of page 47, we should have written $H(h \\lambda_H) = h^{-1}H$ instead of $Hh^{-1}$.<\/p>\n","protected":false},"excerpt":{"rendered":" $\\newcommand{\\sym}[1]{{\\rm Sym}\\,#1}$ Statement (i) of Lemma 3.1 The problem was reported by Joris Buloron. The statement claims that $$N_G(H)\\lambda_H=C_{{\\rm Sym}[G:H]}(G\\varrho_H).$$ The statement is true, but the proof is incomplete for infinite groups. The proof of the containment $N_G(H)\\lambda_H\\leq C_{\\sym[G:H]}(G\\varrho_H)$ is correct. We only show here that $C_{\\sym[G:H]}(G\\varrho_H)\\leq\u00a0N_G(H)\\lambda_H$. Let\u00a0$c\\in C_{\\sym[G:H]}(G\\varrho_H)$ and define $g_0\\in G$ by $Hc … Continue reading Lemma 3.1<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":74,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/118"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/comments?post=118"}],"version-history":[{"count":5,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/118\/revisions"}],"predecessor-version":[{"id":201,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/118\/revisions\/201"}],"up":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/74"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=118"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}