{"id":1280,"date":"2021-12-12T23:04:49","date_gmt":"2021-12-12T15:04:49","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1280"},"modified":"2022-10-13T15:14:16","modified_gmt":"2022-10-13T07:14:16","slug":"direct_sum_and_direct_product","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/12\/12\/direct_sum_and_direct_product\/","title":{"rendered":"\u76f4\u548c\u4e0e\u76f4\u79ef"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u5728\u770b\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u5173\u4e8e\u81ea\u7531Abel\u7fa4\u7684\u89c6\u9891\u65f6, \u591a\u6b21\u63d0\u5230\u4e86\u76f4\u548c\u4e0e\u76f4\u79ef\u7684\u6982\u5ff5, \u6bcf\u6b21\u4e00\u63d0\u5230\u8fd9\u4e9b\u6982\u5ff5\u81ea\u5df1\u90fd\u9700\u8981\u91cd\u65b0\u7ffb\u770b\u76f8\u5173\u7684\u6982\u5ff5, \u8bf4\u660e\u8fd8\u662f\u638c\u63e1\u5f97\u4e0d\u591f, \u56e0\u6b64\u8c28\u4ee5\u672c\u6587\u91cd\u65b0\u590d\u4e60\u4e00\u4e0b\u76f8\u5173\u6982\u5ff5, \u52a0\u6df1\u7406\u89e3~<\/p>\n<p><!--more--><\/p>\n<p>\u53c2\u8003\u89c6\u9891: <a href=\"https:\/\/www.bilibili.com\/video\/BV1P7411N7fW?p=53\">P53 (52)Abel\u7fa4\u3001\u81ea\u7531Abel\u7fa4<\/a><\/p>\n<p><strong>1. \u76f4\u79ef<\/strong><\/p>\n<p>\u8bbe$(G, \\cdot), (H, \\cdot)$\u4e3a\u7fa4, \u5b9a\u4e49$$G \\times H = \\{ (g, h)|g \\in G, h \\in H) \\},$$\u5b9a\u4e49\u4e8e$G \\times H$\u4e4b\u4e0a\u7684\u4e58\u6cd5\u8fd0\u7b97\u4e3a$$(g_1, h_1) \\cdot (g_2, h_2) = (g_1 g_2, h_1 h_2).$$\u8bbe$(G_\\alpha, +), \\alpha \\in I$\u4e3a\u4e00\u65cfAbel\u7fa4, \u5b9a\u4e49\u5b83\u4eec\u7684\u76f4\u79ef\u7fa4(Direct Product)$(\\prod_{\\alpha \\in I} G_\\alpha, +)$\u5982\u4e0b:$$\\prod_{\\alpha \\in I} G_\\alpha = \\{ (g_\\alpha)_{\\alpha \\in I} | g_\\alpha \\in G_\\alpha \\}.$$$\\forall (g_\\alpha), (h_\\alpha) \\in \\prod_{\\alpha \\in I} G_\\alpha, (g_\\alpha) + (h_\\alpha) := (f_\\alpha)$, \u5176\u4e2d$\\forall \\alpha \\in I$, $f_\\alpha = g_\\alpha + $$ h_\\alpha$.<br \/>\n$\\\\$ \u82e5$I$\u4e3a\u6709\u9650\u96c6, i.e. \u5bf9\u4e8e\u6709\u9650\u4e2aAbel\u7fa4$(G_1, +), \\cdots, (G_n, +)$, $(\\prod_{i = 1}^n G_i, +)$\u53c8\u8bb0\u4e3a$(G_1 \\times \\cdots \\times G_n, +)$.<\/p>\n<p><strong>2. \u5916\u76f4\u548c<\/strong><\/p>\n<p>\u8bbe$\\{ G_\\alpha \\}_{\\alpha \\in I}$\u4e3a\u4e00\u65cfAbel\u7fa4, \u5b9a\u4e49\u5b83\u4eec\u7684\u5916\u76f4\u548c(Direct Sum) $(\\oplus_{\\alpha \\in I} G_\\alpha, +)$\u5982\u4e0b:$$\\oplus_{\\alpha \\in I} G_\\alpha := \\{ (g_\\alpha)_{\\alpha \\in I} | g_\\alpha \\in G_\\alpha, \\forall \\alpha \\in I \\}$$\u4e14\u5176\u6bcf\u4e2a\u5143\u7d20\u7684\u6240\u6709$g_\\alpha$\u53ea\u6709\u6709\u9650\u4e2a\u975e\u96f6. $\\forall (g_\\alpha), (h_\\alpha) \\in \\oplus_{\\alpha \\in I} G_\\alpha$, $(g_\\alpha) + ( $$ h_\\alpha) := (f_\\alpha)$, \u5176\u4e2d$\\forall \\alpha \\in I, f_\\alpha = g_\\alpha + h_\\alpha.$<\/p>\n<p>\u6ce81: $\\oplus_{\\alpha \\in I} G_\\alpha \\subset  \\prod_{\\alpha \\in I} G_\\alpha$.<br \/>\n$\\\\$ \u6ce82: \u5bf9\u4e8e\u6709\u9650\u4e2aAbel\u7fa4$G_1, \\cdots, G_n$, $G_1 \\times \\cdots \\times G_n = G_1 \\oplus $$ \\cdots \\oplus G_n$.<\/p>\n<p><strong>3. \u5185\u76f4\u548c<\/strong><\/p>\n<p>\u8bbe$G$\u4e3a\u4e00\u4e2aAbel\u7fa4, $\\{ G_\\alpha \\}_{\\alpha \\in I}$\u4e3a$G$\u4e2d\u7684\u4e00\u65cf\u5b50\u7fa4, \u79f0$G$\u4e3a$\\{ G_\\alpha \\}_{\\alpha \\in I}$\u7684\u5185\u76f4\u548c, \u82e5$\\forall g \\in G$, $g$\u53ef\u552f\u4e00\u5730\u8868\u793a\u4e3a$g = \\sum_\\alpha g_\\alpha$, \u5176\u4e2d$g_\\alpha \\in G_\\alpha$, $g_\\alpha$\u4e2d\u53ea\u6709\u6709\u9650\u4e2a\u975e\u96f6, \u6b64\u65f6\u8bb0$G = \\oplus_{\\alpha \\in I} G_\\alpha$.<\/p>\n<p><strong>4. \u5185\u5916\u76f4\u548c\u4e4b\u5173\u7cfb<\/strong><\/p>\n<p>\u8bbe$G$\u4e3aAbel\u7fa4, $G = \\oplus_{\\alpha \\in I} G_\\alpha$\u4e3a\u5185\u76f4\u548c\u5206\u89e3, \u5219$G$\u663e\u7136\u540c\u6784\u4e8e$G_\\alpha, $$ \\alpha \\in I$\u7684\u5916\u76f4\u548c:$$\\forall g \\in G, g = \\sum_\\alpha g_\\alpha \\mapsto (g_\\alpha)_{\\alpha \\in I}.$$\u53cd\u4e4b, \u8bbe$\\{ G_\\alpha \\}_{\\alpha \\in I}$\u4e3a\u4e00\u65cfAbel\u7fa4, $G = \\oplus_{\\alpha \\in I} G_\\alpha$\u4e3a\u5916\u76f4\u548c. \u8bb0$$i_\\alpha: G_\\alpha \\to \\oplus_{\\alpha \\in I} G_\\alpha$$\u4e3a\u5178\u5219\u5d4c\u5165(\u4e8b\u5b9e\u4e0a\u6211\u5e76\u6ca1\u6709\u67e5\u5230\u5178\u5219\u5d4c\u5165\u7684\u5b9a\u4e49\u56e7), $g \\mapsto i_\\alpha(g) = $$ (h_\\alpha)$, \u5176\u4e2d,$$h_\\beta = \\left\\{\\begin{matrix}<br \/>\n g,\\ if \\ \\beta = \\alpha \\\\<br \/>\n 0, \\ if \\ \\beta \\ne \\alpha<br \/>\n\\end{matrix}.\\right.$$\u5219$G&#8217;_\\alpha := i_\\alpha(G_\\alpha) \\subset \\oplus_{\\alpha \\in I} G_\\alpha$, \u4ece\u800c\u663e\u7136$G$\u4e3a$G&#8217;_\\alpha, \\alpha \\in I$\u7684\u5167\u76f4\u548c.<br \/>\n$\\\\$ $\\because \\forall g \\in G( = \\oplus_{\\alpha \\in I} G_\\alpha), g = (g_\\alpha)_{\\alpha \\in I} = \\sum_\\alpha i_\\alpha(g_\\alpha)$. \u63a5\u4e0b\u6765\u5b9a\u4e49\u7fa4\u540c\u6001$$\\pi_\\alpha: \\oplus_{\\alpha \\in I} G_\\alpha \\to G_\\alpha, (g_\\alpha) \\longmapsto g_\\alpha,$$\u5219$\\pi_\\alpha \\circ i_\\alpha = id_{G_\\alpha}, \\pi_\\alpha \\circ i_\\beta = 0, \\forall \\beta \\ne \\alpha$. \u4ece\u800c\u6211\u4eec\u6709$$\\pi_\\beta(g) = \\pi_\\beta(\\sum_\\alpha i_\\alpha(g_\\alpha)) = \\sum_\\alpha \\pi_\\beta \\circ i_\\alpha(g_\\alpha) \\\\ = \\pi_\\beta \\circ i_\\beta(g_\\beta) = g_\\beta, \\forall \\beta \\in I.$$$\\therefore g = \\sum_\\alpha i_\\alpha(\\pi_\\alpha(g))$. \u4e5f\u5c31\u662f\u8bf4, $g_\\alpha$\u662f\u7531$g$\u4e0e$\\pi_\\alpha$\u552f\u4e00\u786e\u5b9a\u7684, \u90a3\u4e48$$g = \\sum_\\alpha i_\\alpha(g_\\alpha)$$\u8fd9\u6837\u7684\u5206\u89e3\u81ea\u7136\u4e5f\u662f\u552f\u4e00\u7684.<br \/>\n$\\\\$ \u7531\u4e0a\u8ff0\u8ba8\u8bba\u4ea6\u53ef\u77e5, \u5185\u76f4\u548c\u4e0e\u5916\u76f4\u548c\u603b\u662f\u76f8\u751f\u76f8\u4f34\u7684, \u6211\u4eec\u53ef\u4ee5\u7531\u5185\u76f4\u548c\u5f97\u5230\u5916\u76f4\u548c, \u4e5f\u53ef\u7531\u5916\u76f4\u548c\u5f97\u5230\u5185\u76f4\u548c, \u6545\u7ecf\u5e38\u4e0d\u52a0\u533a\u5206\u5730\u8bb0\u5185\u76f4\u548c\u4e0e\u5916\u76f4\u548c.<\/p>\n<p><strong>5. \u76f4\u548c, \u76f4\u79ef\u4e0e$Hom$\u4e4b\u5173\u7cfb<\/strong><\/p>\n<p>\u4e0d\u59a8\u5148\u5b9a\u4e49\u51fd\u5b50$Hom$, \u8bbe$G, H$\u4e3a\u4e24\u4e2aAbel\u7fa4, \u5b9a\u4e49$$Hom(G, H) = \\{ f:G \\to H | f \\ is \\ a \\ group \\ homomorphism. \\}$$\u5176\u4e0a\u7684\u52a0\u6cd5\u8fd0\u7b97\u4e3a$$+: Hom(G, H) \\times Hom(G, H) \\to Hom(G, H), \\\\ (\\varphi, \\psi) \\longmapsto \\varphi + \\psi,$$\u5176\u4e2d, $(\\varphi + \\psi)(g) := \\varphi(g) + \\psi(g)$.<br \/>\n$\\\\$ \u8bbe$(G_\\alpha, \\alpha \\in I)$\u4e3a\u4e00\u65cfAbel\u7fa4, $H$\u4e3a\u53e6\u4e00\u4e2aAbel\u7fa4, \u5219\u6709$$\\varphi: Hom(\\oplus_{\\alpha \\in I} G_\\alpha, H) \\to \\prod_{\\alpha \\in I} Hom(G_\\alpha, H), \\\\ f \\longmapsto \\varphi(f) := (f_\\alpha)_{\\alpha \\in I},$$\u5176\u4e2d$f_\\alpha = f \\circ i_\\alpha$($i_\\alpha : G_\\alpha \\to \\oplus_{\\alpha \\in I} G_\\alpha$\u4e3a\u5178\u5219\u5d4c\u5165). \u63a5\u4e0b\u6765\u8bc1$\\varphi$\u662f\u4e00\u4e2a\u7fa4\u540c\u6784.<br \/>\n$\\\\$ $\\varphi$\u663e\u7136\u4e3a\u5355\u5c04. \u518d\u8bc1$\\varphi$\u4e3a\u6ee1\u5c04: $\\forall (f_\\alpha)_{\\alpha \\in I} \\in \\prod_{\\alpha \\in I} Hom(G_\\alpha,H)$, \u5b9a\u4e49$$f: \\oplus_{\\alpha \\in I} G_\\alpha \\to H, \\\\ \\sum_{\\alpha \\in I} g_\\alpha \\longmapsto \\sum_{\\alpha \\in I} f_\\alpha(g_\\alpha).$$\u663e\u7136$f \\in Hom(\\oplus_{\\alpha \\in I} G_\\alpha, H)$, \u4e14$\\varphi(f) = (f_\\alpha)_{\\alpha \\in I}$.<br \/>\n$\\\\$ $\\therefore Hom(\\oplus_{\\alpha \\in I} G_\\alpha, H) \\cong \\prod_{\\alpha \\in I} Hom(G_\\alpha, H)$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5728\u770b\u5e84\u6653\u6ce2\u8001\u5e08\u7684\u5173\u4e8e\u81ea\u7531Abel\u7fa4\u7684\u89c6\u9891\u65f6, \u591a\u6b21\u63d0\u5230\u4e86\u76f4\u548c\u4e0e\u76f4\u79ef\u7684\u6982\u5ff5, \u6bcf\u6b21\u4e00\u63d0\u5230\u8fd9\u4e9b\u6982\u5ff5\u81ea\u5df1\u90fd\u9700\u8981\u91cd\u65b0\u7ffb &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/12\/12\/direct_sum_and_direct_product\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u76f4\u548c\u4e0e\u76f4\u79ef<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1280"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=1280"}],"version-history":[{"count":22,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1280\/revisions"}],"predecessor-version":[{"id":2493,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1280\/revisions\/2493"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1280"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1280"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1280"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}