{"id":31015,"date":"2025-06-21T21:45:09","date_gmt":"2025-06-21T21:45:09","guid":{"rendered":"https:\/\/gaviki.com\/blog\/?p=31015"},"modified":"2025-06-21T21:45:11","modified_gmt":"2025-06-21T21:45:11","slug":"find-normal-subgroups-of-s3","status":"publish","type":"post","link":"https:\/\/gaviki.com\/blog\/find-normal-subgroups-of-s3\/","title":{"rendered":"Find normal subgroups of S3"},"content":{"rendered":"\n<p>Find normal subgroups of S3<\/p>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-vivid-cyan-blue-color\"><strong>The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n<p><strong>Correct Answer:<\/strong> The normal subgroups of S3S_3 are:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>{e}\\{ e \\} (the trivial subgroup)<\/li>\n\n\n\n<li>S3S_3 (the whole group)<\/li>\n\n\n\n<li>A3={e,(123),(132)}A_3 = \\{ e, (123), (132) \\}<\/li>\n<\/ol>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong>Explanation:<\/strong><\/p>\n\n\n\n<p>The symmetric group S3S_3 is the group of all permutations of three elements. It has six elements: S3={e,(12),(13),(23),(123),(132)}S_3 = \\{ e, (12), (13), (23), (123), (132) \\}<\/p>\n\n\n\n<p>A <strong>normal subgroup<\/strong> NN of a group GG satisfies gNg\u22121=NgNg^{-1} = N for all g\u2208Gg \\in G. This means the subgroup is invariant under conjugation by any element of the group.<\/p>\n\n\n\n<p>First, the trivial subgroup {e}\\{ e \\} and the whole group S3S_3 are always normal in any group.<\/p>\n\n\n\n<p>To find other normal subgroups, examine the structure of S3S_3. It has the following subgroups:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>{e}\\{ e \\}<\/li>\n\n\n\n<li>{e,(12)}\\{ e, (12) \\}<\/li>\n\n\n\n<li>{e,(13)}\\{ e, (13) \\}<\/li>\n\n\n\n<li>{e,(23)}\\{ e, (23) \\}<\/li>\n\n\n\n<li>A3={e,(123),(132)}A_3 = \\{ e, (123), (132) \\}<\/li>\n\n\n\n<li>S3S_3<\/li>\n<\/ul>\n\n\n\n<p>Check which of these are normal. The three subgroups of order 2 each contain one transposition. These are not normal. For example, conjugating (12)(12) by (13)(13) gives: (13)(12)(13)\u22121=(23)(13)(12)(13)^{-1} = (23)<\/p>\n\n\n\n<p>This shows that {e,(12)}\\{ e, (12) \\} is not closed under conjugation, so it is not normal.<\/p>\n\n\n\n<p>Now consider A3={e,(123),(132)}A_3 = \\{ e, (123), (132) \\}. It is of order 3. Since 3 divides 6, and the index of this subgroup is 2, which is the smallest nontrivial index, this subgroup is always normal. Any subgroup of index 2 in a group is normal because conjugation preserves the coset structure.<\/p>\n\n\n\n<p>Therefore, the only normal subgroups of S3S_3 are:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>{e}\\{ e \\}<\/li>\n\n\n\n<li>A3A_3<\/li>\n\n\n\n<li>S3S_3<\/li>\n<\/ul>\n\n\n\n<p>These subgroups are closed under conjugation and satisfy the definition of normality.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find normal subgroups of S3 The correct answer and explanation is: Correct Answer: The normal subgroups of S3S_3 are: Explanation: The symmetric group S3S_3 is the group of all permutations of three elements. It has six elements: S3={e,(12),(13),(23),(123),(132)}S_3 = \\{ e, (12), (13), (23), (123), (132) \\} A normal subgroup NN of a group GG [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-31015","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/31015","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=31015"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/31015\/revisions"}],"predecessor-version":[{"id":31017,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/31015\/revisions\/31017"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=31015"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=31015"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=31015"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}