The problem revolves around the quaternion group Q8={1,i,j,k,\u22121,\u2212i,\u2212j,\u2212k}Q_8 = \\{1, i, j, k, -1, -i, -j, -k\\}Q8\u200b={1,i,j,k,\u22121,\u2212i,\u2212j,\u2212k}, a group of order 8. You are tasked with investigating the structure of its inner automorphism group Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b), proving that Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b) is isomorphic to S4S_4S4\u200b, and finding specific properties related to the factor group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b).<\/p>\n\n\n\n
1. Inner Automorphism Group of Q8Q_8Q8\u200b:<\/strong><\/h3>\n\n\n\nThe inner automorphism group Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) consists of automorphisms of the form:\u03d5g(x)=gxg\u22121for some g\u2208Q8 and x\u2208Q8.\\phi_g(x) = gxg^{-1} \\quad \\text{for some } g \\in Q_8 \\text{ and } x \\in Q_8.\u03d5g\u200b(x)=gxg\u22121for some g\u2208Q8\u200b and x\u2208Q8\u200b.<\/p>\n\n\n\n
We need to identify the elements of Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b). The inner automorphisms of Q8Q_8Q8\u200b correspond to conjugation by elements of Q8Q_8Q8\u200b. Since Q8Q_8Q8\u200b has order 8, Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) is a subgroup of Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b), and the order of Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) is the index of the center of Q8Q_8Q8\u200b.<\/p>\n\n\n\n
\n- The center of Q8Q_8Q8\u200b is Z(Q8)={\u00b11}Z(Q_8) = \\{ \\pm 1 \\}Z(Q8\u200b)={\u00b11}, because 111 and \u22121-1\u22121 commute with all other elements of Q8Q_8Q8\u200b, while the other elements do not.<\/li>\n\n\n\n
- The order of Z(Q8)Z(Q_8)Z(Q8\u200b) is 2, and the index of Z(Q8)Z(Q_8)Z(Q8\u200b) in Q8Q_8Q8\u200b is 8\/2=48\/2 = 48\/2=4.<\/li>\n<\/ul>\n\n\n\n
Thus, the inner automorphism group Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) has order 4, which means it is isomorphic to Z2\u00d7Z2\\mathbb{Z}_2 \\times \\mathbb{Z}_2Z2\u200b\u00d7Z2\u200b, a group of order 4.<\/p>\n\n\n\n
2. Automorphism Group Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b):<\/strong><\/h3>\n\n\n\nThe group of automorphisms of Q8Q_8Q8\u200b, Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b), consists of all bijections of Q8Q_8Q8\u200b that preserve the group structure. It turns out that Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b) is isomorphic to S4S_4S4\u200b, the symmetric group on 4 elements.<\/p>\n\n\n\n
\n- The elements i,j,ki, j, ki,j,k in Q8Q_8Q8\u200b generate the non-commutative part of the group, and automorphisms are determined by how they map. Since conjugation by elements in Q8Q_8Q8\u200b induces specific permutations on i,j,ki, j, ki,j,k, the structure of Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b) corresponds to the possible permutations of these three elements.<\/li>\n\n\n\n
- The automorphisms of Q8Q_8Q8\u200b can be seen to map {i,j,k}\\{i, j, k\\}{i,j,k} to each other in a way that preserves the group relations. There are 24 such automorphisms, which is the order of S4S_4S4\u200b, showing that Aut(Q8)\u2245S4\\text{Aut}(Q_8) \\cong S_4Aut(Q8\u200b)\u2245S4\u200b.<\/li>\n<\/ul>\n\n\n\n
3. Function \\@\\@\\@:<\/strong><\/h3>\n\n\n\nThe function \\@:Aut(Q8)\u2192S4\\@ : \\text{Aut}(Q_8) \\to S_4\\@:Aut(Q8\u200b)\u2192S4\u200b defined by the relations in the question maps certain automorphisms to specific permutations. To explain the behavior of \\@\\@\\@:<\/p>\n\n\n\n
\n- \\@\\@\\@ sends each automorphism to a permutation in S4S_4S4\u200b, specifically mapping the automorphisms determined by the images of iii and jjj to specific elements of S4S_4S4\u200b. For example, P1(i)=jP_1(i) = jP1\u200b(i)=j implies that the automorphism P1P_1P1\u200b sends iii to jjj, and so on for other automorphisms.<\/li>\n\n\n\n
- There are 48 distinct automorphisms of Q8Q_8Q8\u200b, so the map \\@\\@\\@ can define a variety of isomorphisms from Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b) to S4S_4S4\u200b.<\/li>\n<\/ul>\n\n\n\n
4. Elements of S4S_4S4\u200b in Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b):<\/strong><\/h3>\n\n\n\nSince Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) is a subgroup of Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b), it consists of automorphisms corresponding to conjugation by elements of Q8Q_8Q8\u200b. Therefore, the elements of S4S_4S4\u200b that belong to Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) are those that correspond to the automorphisms defined by conjugation by the elements \u00b11\\pm 1\u00b11 and the non-central elements i,j,ki, j, ki,j,k.<\/p>\n\n\n\n
5. Factor Group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b):<\/strong><\/h3>\n\n\n\nFinally, the factor group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b) is isomorphic to the quotient of S4S_4S4\u200b by its subgroup isomorphic to Z2\u00d7Z2\\mathbb{Z}_2 \\times \\mathbb{Z}_2Z2\u200b\u00d7Z2\u200b. This quotient group is isomorphic to S3S_3S3\u200b, since S4\/(Z2\u00d7Z2)\u2245S3S_4 \/ (\\mathbb{Z}_2 \\times \\mathbb{Z}_2) \\cong S_3S4\u200b\/(Z2\u200b\u00d7Z2\u200b)\u2245S3\u200b. This is the group of permutations of 3 elements, corresponding to the non-trivial automorphisms of Q8Q_8Q8\u200b modulo the inner automorphisms.<\/p>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n\n- Inn(Q8)\u2245Z2\u00d7Z2\\text{Inn}(Q_8) \\cong \\mathbb{Z}_2 \\times \\mathbb{Z}_2Inn(Q8\u200b)\u2245Z2\u200b\u00d7Z2\u200b,<\/li>\n\n\n\n
- Aut(Q8)\u2245S4\\text{Aut}(Q_8) \\cong S_4Aut(Q8\u200b)\u2245S4\u200b,<\/li>\n\n\n\n
- Aut(Q8)\/Inn(Q8)\u2245S3\\text{Aut}(Q_8)\/\\text{Inn}(Q_8) \\cong S_3Aut(Q8\u200b)\/Inn(Q8\u200b)\u2245S3\u200b.<\/li>\n<\/ul>\n\n\n\n
These results provide insight into the structure of the quaternion group and its automorphism group.<\/p>\n\n\n\n![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/07\/learnexams-banner5-576.jpeg\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"Let {+1, i, j, k} be the quaternion group of order 8. Determine the structure of the inner automorphism group Inn(Q8). Show that S4 is isomorphic to Aut(Q8). Define the function @: Aut(Q8) -> S4 as follows: @: (1,2) -> P1, (1,3) -> P2, (1,4) -> P3, (2,3) -> P4, (2,4) -> P5, (3,4) -> […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-245031","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=245031"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Thus, the inner automorphism group Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) has order 4, which means it is isomorphic to Z2\u00d7Z2\\mathbb{Z}_2 \\times \\mathbb{Z}_2Z2\u200b\u00d7Z2\u200b, a group of order 4.<\/p>\n\n\n\n
2. Automorphism Group Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b):<\/strong><\/h3>\n\n\n\nThe group of automorphisms of Q8Q_8Q8\u200b, Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b), consists of all bijections of Q8Q_8Q8\u200b that preserve the group structure. It turns out that Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b) is isomorphic to S4S_4S4\u200b, the symmetric group on 4 elements.<\/p>\n\n\n\n
\n- The elements i,j,ki, j, ki,j,k in Q8Q_8Q8\u200b generate the non-commutative part of the group, and automorphisms are determined by how they map. Since conjugation by elements in Q8Q_8Q8\u200b induces specific permutations on i,j,ki, j, ki,j,k, the structure of Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b) corresponds to the possible permutations of these three elements.<\/li>\n\n\n\n
- The automorphisms of Q8Q_8Q8\u200b can be seen to map {i,j,k}\\{i, j, k\\}{i,j,k} to each other in a way that preserves the group relations. There are 24 such automorphisms, which is the order of S4S_4S4\u200b, showing that Aut(Q8)\u2245S4\\text{Aut}(Q_8) \\cong S_4Aut(Q8\u200b)\u2245S4\u200b.<\/li>\n<\/ul>\n\n\n\n
3. Function \\@\\@\\@:<\/strong><\/h3>\n\n\n\nThe function \\@:Aut(Q8)\u2192S4\\@ : \\text{Aut}(Q_8) \\to S_4\\@:Aut(Q8\u200b)\u2192S4\u200b defined by the relations in the question maps certain automorphisms to specific permutations. To explain the behavior of \\@\\@\\@:<\/p>\n\n\n\n
\n- \\@\\@\\@ sends each automorphism to a permutation in S4S_4S4\u200b, specifically mapping the automorphisms determined by the images of iii and jjj to specific elements of S4S_4S4\u200b. For example, P1(i)=jP_1(i) = jP1\u200b(i)=j implies that the automorphism P1P_1P1\u200b sends iii to jjj, and so on for other automorphisms.<\/li>\n\n\n\n
- There are 48 distinct automorphisms of Q8Q_8Q8\u200b, so the map \\@\\@\\@ can define a variety of isomorphisms from Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b) to S4S_4S4\u200b.<\/li>\n<\/ul>\n\n\n\n
4. Elements of S4S_4S4\u200b in Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b):<\/strong><\/h3>\n\n\n\nSince Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) is a subgroup of Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b), it consists of automorphisms corresponding to conjugation by elements of Q8Q_8Q8\u200b. Therefore, the elements of S4S_4S4\u200b that belong to Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) are those that correspond to the automorphisms defined by conjugation by the elements \u00b11\\pm 1\u00b11 and the non-central elements i,j,ki, j, ki,j,k.<\/p>\n\n\n\n
5. Factor Group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b):<\/strong><\/h3>\n\n\n\nFinally, the factor group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b) is isomorphic to the quotient of S4S_4S4\u200b by its subgroup isomorphic to Z2\u00d7Z2\\mathbb{Z}_2 \\times \\mathbb{Z}_2Z2\u200b\u00d7Z2\u200b. This quotient group is isomorphic to S3S_3S3\u200b, since S4\/(Z2\u00d7Z2)\u2245S3S_4 \/ (\\mathbb{Z}_2 \\times \\mathbb{Z}_2) \\cong S_3S4\u200b\/(Z2\u200b\u00d7Z2\u200b)\u2245S3\u200b. This is the group of permutations of 3 elements, corresponding to the non-trivial automorphisms of Q8Q_8Q8\u200b modulo the inner automorphisms.<\/p>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n\n- Inn(Q8)\u2245Z2\u00d7Z2\\text{Inn}(Q_8) \\cong \\mathbb{Z}_2 \\times \\mathbb{Z}_2Inn(Q8\u200b)\u2245Z2\u200b\u00d7Z2\u200b,<\/li>\n\n\n\n
- Aut(Q8)\u2245S4\\text{Aut}(Q_8) \\cong S_4Aut(Q8\u200b)\u2245S4\u200b,<\/li>\n\n\n\n
- Aut(Q8)\/Inn(Q8)\u2245S3\\text{Aut}(Q_8)\/\\text{Inn}(Q_8) \\cong S_3Aut(Q8\u200b)\/Inn(Q8\u200b)\u2245S3\u200b.<\/li>\n<\/ul>\n\n\n\n
These results provide insight into the structure of the quaternion group and its automorphism group.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"Let {+1, i, j, k} be the quaternion group of order 8. Determine the structure of the inner automorphism group Inn(Q8). Show that S4 is isomorphic to Aut(Q8). Define the function @: Aut(Q8) -> S4 as follows: @: (1,2) -> P1, (1,3) -> P2, (1,4) -> P3, (2,3) -> P4, (2,4) -> P5, (3,4) -> […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-245031","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=245031"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
3. Function \\@\\@\\@:<\/strong><\/h3>\n\n\n\nThe function \\@:Aut(Q8)\u2192S4\\@ : \\text{Aut}(Q_8) \\to S_4\\@:Aut(Q8\u200b)\u2192S4\u200b defined by the relations in the question maps certain automorphisms to specific permutations. To explain the behavior of \\@\\@\\@:<\/p>\n\n\n\n
\n- \\@\\@\\@ sends each automorphism to a permutation in S4S_4S4\u200b, specifically mapping the automorphisms determined by the images of iii and jjj to specific elements of S4S_4S4\u200b. For example, P1(i)=jP_1(i) = jP1\u200b(i)=j implies that the automorphism P1P_1P1\u200b sends iii to jjj, and so on for other automorphisms.<\/li>\n\n\n\n
- There are 48 distinct automorphisms of Q8Q_8Q8\u200b, so the map \\@\\@\\@ can define a variety of isomorphisms from Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b) to S4S_4S4\u200b.<\/li>\n<\/ul>\n\n\n\n
4. Elements of S4S_4S4\u200b in Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b):<\/strong><\/h3>\n\n\n\nSince Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) is a subgroup of Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b), it consists of automorphisms corresponding to conjugation by elements of Q8Q_8Q8\u200b. Therefore, the elements of S4S_4S4\u200b that belong to Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) are those that correspond to the automorphisms defined by conjugation by the elements \u00b11\\pm 1\u00b11 and the non-central elements i,j,ki, j, ki,j,k.<\/p>\n\n\n\n
5. Factor Group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b):<\/strong><\/h3>\n\n\n\nFinally, the factor group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b) is isomorphic to the quotient of S4S_4S4\u200b by its subgroup isomorphic to Z2\u00d7Z2\\mathbb{Z}_2 \\times \\mathbb{Z}_2Z2\u200b\u00d7Z2\u200b. This quotient group is isomorphic to S3S_3S3\u200b, since S4\/(Z2\u00d7Z2)\u2245S3S_4 \/ (\\mathbb{Z}_2 \\times \\mathbb{Z}_2) \\cong S_3S4\u200b\/(Z2\u200b\u00d7Z2\u200b)\u2245S3\u200b. This is the group of permutations of 3 elements, corresponding to the non-trivial automorphisms of Q8Q_8Q8\u200b modulo the inner automorphisms.<\/p>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n\n- Inn(Q8)\u2245Z2\u00d7Z2\\text{Inn}(Q_8) \\cong \\mathbb{Z}_2 \\times \\mathbb{Z}_2Inn(Q8\u200b)\u2245Z2\u200b\u00d7Z2\u200b,<\/li>\n\n\n\n
- Aut(Q8)\u2245S4\\text{Aut}(Q_8) \\cong S_4Aut(Q8\u200b)\u2245S4\u200b,<\/li>\n\n\n\n
- Aut(Q8)\/Inn(Q8)\u2245S3\\text{Aut}(Q_8)\/\\text{Inn}(Q_8) \\cong S_3Aut(Q8\u200b)\/Inn(Q8\u200b)\u2245S3\u200b.<\/li>\n<\/ul>\n\n\n\n
These results provide insight into the structure of the quaternion group and its automorphism group.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"Let {+1, i, j, k} be the quaternion group of order 8. Determine the structure of the inner automorphism group Inn(Q8). Show that S4 is isomorphic to Aut(Q8). Define the function @: Aut(Q8) -> S4 as follows: @: (1,2) -> P1, (1,3) -> P2, (1,4) -> P3, (2,3) -> P4, (2,4) -> P5, (3,4) -> […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-245031","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=245031"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
4. Elements of S4S_4S4\u200b in Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b):<\/strong><\/h3>\n\n\n\nSince Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) is a subgroup of Aut(Q8)\\text{Aut}(Q_8)Aut(Q8\u200b), it consists of automorphisms corresponding to conjugation by elements of Q8Q_8Q8\u200b. Therefore, the elements of S4S_4S4\u200b that belong to Inn(Q8)\\text{Inn}(Q_8)Inn(Q8\u200b) are those that correspond to the automorphisms defined by conjugation by the elements \u00b11\\pm 1\u00b11 and the non-central elements i,j,ki, j, ki,j,k.<\/p>\n\n\n\n
5. Factor Group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b):<\/strong><\/h3>\n\n\n\nFinally, the factor group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b) is isomorphic to the quotient of S4S_4S4\u200b by its subgroup isomorphic to Z2\u00d7Z2\\mathbb{Z}_2 \\times \\mathbb{Z}_2Z2\u200b\u00d7Z2\u200b. This quotient group is isomorphic to S3S_3S3\u200b, since S4\/(Z2\u00d7Z2)\u2245S3S_4 \/ (\\mathbb{Z}_2 \\times \\mathbb{Z}_2) \\cong S_3S4\u200b\/(Z2\u200b\u00d7Z2\u200b)\u2245S3\u200b. This is the group of permutations of 3 elements, corresponding to the non-trivial automorphisms of Q8Q_8Q8\u200b modulo the inner automorphisms.<\/p>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n\n- Inn(Q8)\u2245Z2\u00d7Z2\\text{Inn}(Q_8) \\cong \\mathbb{Z}_2 \\times \\mathbb{Z}_2Inn(Q8\u200b)\u2245Z2\u200b\u00d7Z2\u200b,<\/li>\n\n\n\n
- Aut(Q8)\u2245S4\\text{Aut}(Q_8) \\cong S_4Aut(Q8\u200b)\u2245S4\u200b,<\/li>\n\n\n\n
- Aut(Q8)\/Inn(Q8)\u2245S3\\text{Aut}(Q_8)\/\\text{Inn}(Q_8) \\cong S_3Aut(Q8\u200b)\/Inn(Q8\u200b)\u2245S3\u200b.<\/li>\n<\/ul>\n\n\n\n
These results provide insight into the structure of the quaternion group and its automorphism group.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"Let {+1, i, j, k} be the quaternion group of order 8. Determine the structure of the inner automorphism group Inn(Q8). Show that S4 is isomorphic to Aut(Q8). Define the function @: Aut(Q8) -> S4 as follows: @: (1,2) -> P1, (1,3) -> P2, (1,4) -> P3, (2,3) -> P4, (2,4) -> P5, (3,4) -> […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-245031","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=245031"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Finally, the factor group Aut(Q8)\/Inn(Q8)\\text{Aut}(Q_8)\/\\text{Inn}(Q_8)Aut(Q8\u200b)\/Inn(Q8\u200b) is isomorphic to the quotient of S4S_4S4\u200b by its subgroup isomorphic to Z2\u00d7Z2\\mathbb{Z}_2 \\times \\mathbb{Z}_2Z2\u200b\u00d7Z2\u200b. This quotient group is isomorphic to S3S_3S3\u200b, since S4\/(Z2\u00d7Z2)\u2245S3S_4 \/ (\\mathbb{Z}_2 \\times \\mathbb{Z}_2) \\cong S_3S4\u200b\/(Z2\u200b\u00d7Z2\u200b)\u2245S3\u200b. This is the group of permutations of 3 elements, corresponding to the non-trivial automorphisms of Q8Q_8Q8\u200b modulo the inner automorphisms.<\/p>\n\n\n\n
Conclusion:<\/h3>\n\n\n\n\n- Inn(Q8)\u2245Z2\u00d7Z2\\text{Inn}(Q_8) \\cong \\mathbb{Z}_2 \\times \\mathbb{Z}_2Inn(Q8\u200b)\u2245Z2\u200b\u00d7Z2\u200b,<\/li>\n\n\n\n
- Aut(Q8)\u2245S4\\text{Aut}(Q_8) \\cong S_4Aut(Q8\u200b)\u2245S4\u200b,<\/li>\n\n\n\n
- Aut(Q8)\/Inn(Q8)\u2245S3\\text{Aut}(Q_8)\/\\text{Inn}(Q_8) \\cong S_3Aut(Q8\u200b)\/Inn(Q8\u200b)\u2245S3\u200b.<\/li>\n<\/ul>\n\n\n\n
These results provide insight into the structure of the quaternion group and its automorphism group.<\/p>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"Let {+1, i, j, k} be the quaternion group of order 8. Determine the structure of the inner automorphism group Inn(Q8). Show that S4 is isomorphic to Aut(Q8). Define the function @: Aut(Q8) -> S4 as follows: @: (1,2) -> P1, (1,3) -> P2, (1,4) -> P3, (2,3) -> P4, (2,4) -> P5, (3,4) -> […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-245031","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=245031"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
These results provide insight into the structure of the quaternion group and its automorphism group.<\/p>\n\n\n\n Let {+1, i, j, k} be the quaternion group of order 8. Determine the structure of the inner automorphism group Inn(Q8). Show that S4 is isomorphic to Aut(Q8). Define the function @: Aut(Q8) -> S4 as follows: @: (1,2) -> P1, (1,3) -> P2, (1,4) -> P3, (2,3) -> P4, (2,4) -> P5, (3,4) -> […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-245031","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=245031"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/245031\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=245031"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=245031"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=245031"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}<\/figure>\n","protected":false},"excerpt":{"rendered":"