{"id":396,"date":"2019-09-09T13:16:05","date_gmt":"2019-09-09T13:16:05","guid":{"rendered":"http:\/\/localhost\/?p=396"},"modified":"2019-09-12T11:12:26","modified_gmt":"2019-09-12T11:12:26","slug":"exercicios-3","status":"publish","type":"post","link":"http:\/\/localhost\/index.php\/2019\/09\/09\/exercicios-3\/","title":{"rendered":"Exerc\u00edcios 3"},"content":{"rendered":"

$\\newcommand{\\rot}[1]{\\mbox{Rot}(#1)}\\newcommand{\\ref}[1]{\\mbox{Ref}(#1)}$ Denote por $\\rot \\alpha$ a rota\u00e7\u00e3o do plano $\\mathbb R^2$ pela origem por \u00e2ngulo $\\alpha$\u00a0no sentido contr\u00e1rio aos ponteiros do rel\u00f3gio. Denote por $\\ref\\alpha$ a reflex\u00e3o de $\\mathbb R^2$ pelo eixo que passe pela origem e tem \u00e2ngulo $\\varphi$\u00a0com o eixo $x$.<\/p>\n

1. Ache as matrizes das transforma\u00e7\u00f5es $\\rot\\alpha$ e $\\ref\\alpha$ na base can\u00f4nica de $\\mathbb R^2$.\u00a0 (Como \u00e9 comum na teoria dos grupos, a matriz de uma transforma\u00e7\u00e3o linear $F$ \u00e9 a matriz $A$ que satisfaz $F(v)=v\\cdot A$ para todo $v\\in\\mathbb R^2$).<\/p>\n

2. Sejam $\\varphi$ e $\\psi$ \u00e2ngulos. Demonstre as seguintes igualdades:<\/p>\n

    \n
  1. $\\ref\\varphi\\ref\\psi=\\rot{2(\\psi-\\varphi)}$<\/li>\n
  2. $\\rot\\varphi\\ref\\psi=\\ref{\\psi-\\varphi\/2}$;<\/li>\n
  3. $\\ref\\varphi\\rot\\psi=\\ref{\\varphi+\\psi\/2}$.<\/li>\n<\/ol>\n

    Usando estas igualdades, determine $\\rot\\varphi^{\\ref\\psi}$, $\\ref\\varphi^{\\rot\\psi}$, $[\\ref\\varphi,\\ref\\psi]$, $[\\rot\\varphi,\\ref\\psi]$.<\/p>\n

    3.\u00a0 Denote por $O_2$ o grupo das reflex\u00f5es e rota\u00e7\u00f5es do plano $\\mathbb R^2$. Calcule a s\u00e9rie derivada, a s\u00e9rie central inferior e a s\u00e9rie central superior de $O_2$.<\/p>\n

    4. Seja $D_n$ o grupo dihedral de ordem $2n$. Calcule a s\u00e9rie derivada, a s\u00e9rie central superior, e a s\u00e9rie central inferior de $D_n$.\u00a0D\u00ea uma condi\u00e7\u00e3o suficiente e necess\u00e1ria para a exist\u00eancia de $k$ e $\\ell$ que satisfazem $\\zeta_k(D_n)=D_n$ e $\\gamma_\\ell(D_n)=1$.<\/p>\n

    5. Seja $H\\leq G$ e $N\\unlhd G$ tal que $N\\leq K$. Mostre que $[H,G]\\leq N$ se e somente se $H\/N\\leq Z(G\/N)$.<\/p>\n

    6. Demonstre as seguintes afirma\u00e7\u00f5es para $X,Y\\leq G$.<\/p>\n

      \n
    1. $[X,Y]=[Y,X]$.<\/li>\n
    2. Se $X, Y\\unlhd G$, ent\u00e3o $[X,Y]\\unlhd G$.<\/li>\n
    3. Se $X, Y$ s\u00e3o carater\u00edsticos em $G$, ent\u00e3o $[X,Y]$ \u00e9 carater\u00edstico em $G$.<\/li>\n<\/ol>\n

      7. Sejam $x,y,z\\in G$. Demonstre as seguintes igualdades:<\/p>\n

        \n
      1. $[x,y]=[y,x]^{-1}$;<\/li>\n
      2. $[x,yz]=[x,z][x,y]^{z}$;<\/li>\n
      3. $[xy,z]=[x,z]^{y}[y,z]$;<\/li>\n
      4. $[x,y^{-1}]=([x,y]^{y^{-1}})^{-1}$;<\/li>\n
      5. $[x^{-1},y]=([x,y]^{x^{-1}})^{-1}$;<\/li>\n
      6. $[x,y^{-1},z]^y[y,z^{-1},x]^z[z,x^{-1},y]^x=1$ (identidade de Hall-Witt).<\/li>\n<\/ol>\n

        8. Para $X,Y,Z\\leq G$, denote
        \n$$
        \n[X,Y,Z]=\\left<[[x,y],z]\\mid x\\in X, y\\in Y, z\\in Z\\right>.
        \n$$
        \nDemonstre as seguintes afirma\u00e7\u00f5es.<\/p>\n

          \n
        1. Se $[X,Y,Z]=[Y,Z,X]=1$, ent\u00e3o $[Z,X,Y]=1$.<\/li>\n
        2. Se $N\\unlhd G$ e $[X,Y,Z]\\leq N$ e $[Y,Z,X]\\leq N$, ent\u00e3o $[Z,X,Y]\\leq N$. (Lema dos $[Z,X,Y]\\leq [X,Y,Z][Y,Z,X]$.<\/li>\n
        3. $[\\gamma_i(G),\\gamma_j(G)]\\leq \\gamma_{i+j}(G)$ para todo $i,j\\geq 1$.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

          $\\newcommand{\\rot}[1]{\\mbox{Rot}(#1)}\\newcommand{\\ref}[1]{\\mbox{Ref}(#1)}$ Denote por $\\rot \\alpha$ a rota\u00e7\u00e3o do plano $\\mathbb R^2$ pela origem por \u00e2ngulo $\\alpha$\u00a0no sentido contr\u00e1rio aos ponteiros do rel\u00f3gio. Denote por $\\ref\\alpha$ a reflex\u00e3o de $\\mathbb R^2$ pelo eixo que passe pela origem e tem \u00e2ngulo $\\varphi$\u00a0com o eixo $x$. 1. Ache as matrizes das transforma\u00e7\u00f5es $\\rot\\alpha$ e $\\ref\\alpha$ na base can\u00f4nica … Continue reading Exerc\u00edcios 3<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/396"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/post"}],"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=396"}],"version-history":[{"count":19,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/396\/revisions"}],"predecessor-version":[{"id":416,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/396\/revisions\/416"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=396"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/categories?post=396"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/tags?post=396"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}