{"id":985,"date":"2014-09-30T07:33:05","date_gmt":"2014-09-29T23:33:05","guid":{"rendered":"http:\/\/www.shuizilong.com\/house\/?p=985"},"modified":"2014-09-30T09:05:12","modified_gmt":"2014-09-30T01:05:12","slug":"hdu-4702-group","status":"publish","type":"post","link":"https:\/\/www.shuizilong.com\/house\/archives\/hdu-4702-group\/","title":{"rendered":"HDU 4702. Group"},"content":{"rendered":"<p><a href=\"http:\/\/acm.hust.edu.cn\/vjudge\/problem\/viewProblem.action?id=46279\">http:\/\/acm.hust.edu.cn\/vjudge\/problem\/viewProblem.action?id=46279<\/a><\/p>\n<h3>Brief description: <\/h3>\n<p>\u7ed9\u5b9a\u957f\u5ea6\u4e3a N \u7684\u7f6e\u6362\u96c6\u5408 S\uff0c|S| = M\uff0c\u6c42\u6ee1\u8db3\u4e00\u4e0b\u6761\u4ef6\u7684\u96c6\u5408 G \u7684\u6700\u5c0f\u7684\u9636\u3002<\/p>\n<ul>\n<li>$$S\\subset G$$<\/li>\n<li>$$\\forall \\sigma, \\tau \\in G, \\sigma \\tau^{-1} \\in G$$<\/li>\n<\/ul>\n<p><!--more--><\/p>\n<h3>Analysis: <\/h3>\n<p>&#8230;<\/p>\n<pre>\r\n\u201cA subset H of the group G is a subgroup of G if and only if it is nonempty and closed under products and inverses. (The closure conditions mean the following: whenever a and b are in H, then ab and a\u22121 are also in H. These two conditions can be combined into one equivalent condition: whenever a and b are in H, then ab\u22121 is also in H.)&quot;\r\n<\/pre>\n<p><a href=\"http:\/\/en.wikipedia.org\/wiki\/Subgroup\">http:\/\/en.wikipedia.org\/wiki\/Subgroup<\/a><\/p>\n<p>\u5305\u542b S \u7684\u6700\u5c0f\u7684\u7fa4\u5373\u4e3a S \u7684\u751f\u6210\u5b50\u7fa4\u3002\u8bb0 G = &lt;S>\uff0c\u8981\u6c42\u7684\u65e2\u662f |&lt;S>|\u3002<br \/>\n\u6211\u4eec\u7528 Schreier &#8211; Sims \u7b97\u6cd5\u6c42\u51fa &lt;S> \u7684\u57fa B\u3002\u6211\u4eec\u77e5\u9053 B \u5b9a\u4e49\u4e86\u4e00\u4e2a\u5b50\u7fa4\u94fe\u3002<\/p>\n<p>$$! G = G^{[1]} \\geq G^{[2]} \\geq \\ldots \\geq G^{[m]} \\geq G^{[m+1]} = 1 $$<\/p>\n<p>\u7531\u62c9\u683c\u6717\u65e5\u5b9a\u7406\uff0c<\/p>\n<p>$$! |G| = \\Pi_{i=1}^{m} |G^{[i]} : G^{[i+1]}| $$<\/p>\n<p>\u800c G^[i] \u4e2d G^[i+1] \u7684\u966a\u96c6\u4e2a\u6570\u5c31\u662f $$|R_i|$$\u3002<\/p>\n<h5>\u966a\u96c6\u5212\u5206: <\/h5>\n<p>\u8bbe H \u662f G \u7684\u5b50\u7fa4\uff0c\u5bf9\u4efb\u610f\u7684 a \u2208 G\uff0c\u79f0 aH = {ah | h \u2208 H} \u4e3a\u5b50\u7fa4 H \u7684\u4e00\u4e2a\u5de6\u966a\u96c6\u3002\u79f0 Ha = {ha | h \u2208 H} \u4e3a\u5b50\u7fa4 H \u7684\u4e00\u4e2a\u53f3\u966a\u96c6\u3002<br \/>\n\u8bbe G \u4e2d H \u7684\u6240\u6709\u5de6\u966a\u96c6\u5f62\u6210\u7684\u96c6\u5408\u4e3a Sl, \u6240\u6709\u53f3\u966a\u96c6\u7684\u96c6\u5408\u4e3a Sr\uff0c\u53ef\u4ee5\u8bc1\u660e\u6620\u5c04 aH -> Ha^{-1} \u662f Sl \u5230 Sr \u7684\u4e00\u4e00\u6620\u5c04\u3002<\/p>\n<p>\u5b9a\u7406 1\uff1a\u8bbe H \u4e3a G \u7684\u5b50\u7fa4\uff0c\u4efb\u7ed9 H \u7684\u53f3\u966a\u96c6 Ha, Hb\u3002\u8981\u4e48 Ha = Hb\uff0c\u8981\u4e48 Ha \u2229 Hb = \u2205\u3002<br \/>\n\u8fd9\u4e2a\u5b9a\u7406\u8bf4\u660e\u4e86\u7fa4 G \u53ef\u4ee5\u5212\u5206\u6210\u4e24\u4e24\u4e0d\u76f8\u4ea4\u7684\u53f3\u966a\u96c6\u7684\u5e76\uff0c\u7ed9\u6211\u4eec\u4e00\u79cd\u5206\u6790\u548c\u7b80\u5316\u7fa4\u7ed3\u6784\u7684\u601d\u8def\u3002<\/p>\n<h5>\u62c9\u683c\u6717\u65e5\u5b9a\u7406<\/h5>\n<p>\u8bb0 [G: H] \u8868\u793a G \u4e2d\u5b50\u7fa4 H \u7684\u4e0d\u540c\u53f3\u966a\u96c6\u7684\u4e2a\u6570\u3002\u7528 1 \u8868\u793a\u53ea\u542b\u5355\u4f4d\u5143\u7684\u5b50\u7fa4\uff0c[G: 1] \u8868\u793a G \u7684\u9636\u3002\u6211\u4eec\u53ef\u4ee5\u7acb\u5373\u5f97\u5230\uff1a[G : 1] = [G : H][H : 1]\u3002<\/p>\n<p><a href=\"http:\/\/zh.wikipedia.org\/wiki\/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%AE%9A%E7%90%86_(%E7%BE%A4%E8%AB%96)\">http:\/\/zh.wikipedia.org\/wiki\/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%AE%9A%E7%90%86_(%E7%BE%A4%E8%AB%96)<\/a><\/p>\n<h3>References: <\/h3>\n<p>\u5bf9\u6700\u6362\u7fa4\u6709\u5173\u7b97\u6cd5\u7684\u521d\u6b65\u7814\u7a76 by crx<br \/>\nGroup \u89e3\u9898\u62a5\u544a by ftiasch<\/p>\n","protected":false},"excerpt":{"rendered":"<p>http:\/\/acm.hust.edu.cn\/vjudge\/problem\/viewProblem.action?id=46279 Brief description: \u7ed9\u5b9a\u957f\u5ea6\u4e3a N \u7684\u7f6e\u6362\u96c6\u5408 S\uff0c|S| = M\uff0c\u6c42\u6ee1\u8db3\u4e00\u4e0b\u6761\u4ef6\u7684\u96c6\u5408 G \u7684\u6700\u5c0f\u7684\u9636\u3002 $$S\\subset G$$ $$\\forall \\sigma, \\tau \\in G, \\sigma \\tau^{-1} \\in G$$<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","enabled":false}}},"categories":[1],"tags":[135],"class_list":["post-985","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-135"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p2tdP7-fT","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/985","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/comments?post=985"}],"version-history":[{"count":1,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/985\/revisions"}],"predecessor-version":[{"id":986,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/posts\/985\/revisions\/986"}],"wp:attachment":[{"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/media?parent=985"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/categories?post=985"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.shuizilong.com\/house\/wp-json\/wp\/v2\/tags?post=985"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}