$\\newcommand{\\F}{\\mathbb F}$$G$-m\u00f3dulos, representa\u00e7\u00f5es, subm\u00f3dulos, m\u00f3dulos simples (irredut\u00edveis), redut\u00edveis, completamente redut\u00edveis. A \u00e1lgebra do grupo, seus subm\u00f3dulos. M\u00f3dulo quociente. O Teorema de Maschke.<\/p>\n
Teorema (Maschke<\/a>).\u00a0<\/strong>Seja $G$ um grupo finito e $\\F$ um corpo. As seguintes s\u00e3o equivalentes:<\/p>\n Demonstra\u00e7\u00e3o.\u00a0<\/strong>1.$\\Rightarrow$ 2. Considere $\\F G$ como um $G$-m\u00f3dulo. Seja $a=\\sum_{g\\in G}g\\in \\F G$ e note que $U=\\left<a\\right>$ \u00e9 um $G$-subm\u00f3dulo de $\\F G$ de dimens\u00e3o um. Assuma que existe um $G$-subm\u00f3dulo $W\\leq \\F G$ tal que $U\\oplus W=\\F G$.\u00a0 Comparando dimens\u00f5es, temos que $\\dim W=|G|-1$.<\/p>\n Seja Ora, se $\\mbox{char}\\,\\F\\mid |G|$, ent\u00e3o $U\\leq I$, ent\u00e3o $U\\oplus W$ \u00e9 imposs\u00edvel.<\/p>\n 2. $\\Rightarrow$ 1.<\/p>\n","protected":false},"excerpt":{"rendered":" $\\newcommand{\\F}{\\mathbb F}$$G$-m\u00f3dulos, representa\u00e7\u00f5es, subm\u00f3dulos, m\u00f3dulos simples (irredut\u00edveis), redut\u00edveis, completamente redut\u00edveis. A \u00e1lgebra do grupo, seus subm\u00f3dulos. M\u00f3dulo quociente. O Teorema de Maschke. Teorema (Maschke).\u00a0Seja $G$ um grupo finito e $\\F$ um corpo. As seguintes s\u00e3o equivalentes: Todo $G$-m\u00f3dulo de dimens\u00e3o finita \u00e9 completamente redut\u00edvel. $\\mbox{char}\\,\\F\\nmid |G|$. Demonstra\u00e7\u00e3o.\u00a01.$\\Rightarrow$ 2. Considere $\\F G$ como um $G$-m\u00f3dulo. Seja … Continue reading G-m\u00f3dulos e representa\u00e7\u00f5es<\/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\/542"}],"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=542"}],"version-history":[{"count":4,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/542\/revisions"}],"predecessor-version":[{"id":548,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/542\/revisions\/548"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=542"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/categories?post=542"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/tags?post=542"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
\n\\[
\nI=\\left\\{\\sum_{g\\in G} \\alpha_g g\\mid \\sum \\alpha_g=0\\right\\}.
\n\\]
\nClaramente, $I$ \u00e9 um $G$-subm\u00f3dulo de dimens\u00e3o $|G|-1$. N\u00f3s afirmamos que $W=I$. Como $\\dim W=\\dim I$, \u00e9 suficiente provar que $W\\leq I$. Seja $x=\\sum \\beta_g g\\in W$. Como $W$ \u00e9 um $G$-subm\u00f3dulo,
\n\\begin{align*}
\nW\\ni&\\sum_{h\\in G} xh=\\sum_{h\\in G}\\left(\\sum_{g\\in G}\\beta_g g\\right)h=\\sum_{g\\in G}\\sum_{h\\in G}\\beta_g (gh)=\\\\&\\sum_{g\\in G}\\beta_g \\sum_{h\\in G} gh=
\n\\sum_{g\\in G}\\beta_g \\sum_{y\\in G}y=\\left(\\sum_{g\\in G}\\beta_g\\right) a\\in U.
\n\\end{align*}
\nComo $U\\cap W=0$, temos que $\\sum_{g\\in G}\\beta_g=0$; ou seja $x\\in I$. Portanto $W=I$ como for afirmado.<\/p>\n