{"id":810,"date":"2020-08-22T18:21:26","date_gmt":"2020-08-22T18:21:26","guid":{"rendered":"http:\/\/localhost\/?page_id=810"},"modified":"2020-08-22T18:28:24","modified_gmt":"2020-08-22T18:28:24","slug":"exercicios-4","status":"publish","type":"page","link":"http:\/\/localhost\/index.php\/ensino\/grupos-e-corpos\/exercicios-4\/","title":{"rendered":"Exerc\u00edcios 4"},"content":{"rendered":"

1. Sejam $G$ um grupo, $N\\unlhd G$ e demonstre as seguintes afirma\u00e7\u00f5es.<\/p>\n

    \n
  1. Se $G$ for abeliano, ent\u00e3o $G\/N$ \u00e9 abeliano.<\/li>\n
  2. Se $G$ for c\u00edclico, ent\u00e3o $G\/N$ \u00e9 c\u00edclico.<\/li>\n
  3. Se $|g|$ for finito com $g\\in G$, ent\u00e3o $|Ng|$ divide $|g|$.<\/li>\n<\/ol>\n

    2. Seja $H$ um subgroupo de um grupo $G$. Mostre que as seguintes s\u00e3o
    \nequivalentes.<\/p>\n

      \n
    1. $H\\unlhd G$;<\/li>\n
    2. $g^{-1}hg\\in H$ para todo $g\\in G$ e $h\\in H$.<\/li>\n<\/ol>\n

      3. Seja $G$ um grupo e defina
      \n\\[
      \nZ(G) =\\{g\\in G\\mid gx=xg\\mbox{ para todo }x\\in G\\}.
      \n\\]
      \nMostre que $Z(G)\\unlhd G$. (O subgrupo $Z(G)$ \u00e9 chamado de centro de $G$}<\/em>).<\/p>\n

      4. Sejam $G$ e $H$ grupos c\u00edclicos de ordem $n$ onde $n\\in\\mathbb N\\cup\\{\\infty\\}$. Mostre que $G\\cong H$. Deduza que se $p$ \u00e9 um primo e $G$ e $H$ s\u00e3o grupos de ordem $p$, ent\u00e3o $G\\cong H$.<\/p>\n

      5. Decida com justificativa quais dos seguintes grupos s\u00e3o isomorfos.<\/p>\n

        \n
      1. $(\\mathbb Z_8^*,\\cdot)$ e $\\left<(1,2)(3,4),(1,3)(2,4)\\right>$.<\/li>\n
      2. $(\\mathbb Z_8^*,\\cdot)$ e $\\{1,-1,i,-i\\}$.<\/li>\n
      3. $S_3$ e $GL(2,2)$.<\/li>\n
      4. $S_4$ e $SL(2,3)$.<\/li>\n<\/ol>\n

        6. Demonstre que $S_4\/\\left<(1,2)(3,4),(1,3)(2,4)\\right>\\cong S_3$.<\/p>\n

        7. Demonstre que $\\mbox{Aut}(S_3)\\cong S_3$.<\/p>\n","protected":false},"excerpt":{"rendered":"

        1. Sejam $G$ um grupo, $N\\unlhd G$ e demonstre as seguintes afirma\u00e7\u00f5es. Se $G$ for abeliano, ent\u00e3o $G\/N$ \u00e9 abeliano. Se $G$ for c\u00edclico, ent\u00e3o $G\/N$ \u00e9 c\u00edclico. Se $|g|$ for finito com $g\\in G$, ent\u00e3o $|Ng|$ divide $|g|$. 2. Seja $H$ um subgroupo de um grupo $G$. Mostre que as seguintes s\u00e3o equivalentes. $H\\unlhd … Continue reading Exerc\u00edcios 4<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":684,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/810"}],"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=810"}],"version-history":[{"count":3,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/810\/revisions"}],"predecessor-version":[{"id":813,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/810\/revisions\/813"}],"up":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/pages\/684"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=810"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}