{ localUrl: '../page/image_of_group_under_homomorphism_is_subgroup.html', arbitalUrl: 'https://arbital.com/p/image_of_group_under_homomorphism_is_subgroup', rawJsonUrl: '../raw/4b4.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'image_of_group_under_homomorphism_is_subgroup', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'The image of a group under a homomorphism is a subgroup of the codomain', clickbait: 'Group homomorphisms take groups to groups, but it is additionally guaranteed that the elements they hit form a group.', textLength: '903', alias: 'image_of_group_under_homomorphism_is_subgroup', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-06-14 19:30:27', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-06-14 19:30:27', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '0', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '29', text: 'Let $f: G \\to H$ be a [-47t], and write $f(G)$ for the set $\\{ f(g) : g \\in G \\}$.\nThen $f(G)$ is a group under the operation inherited from $H$.\n\n# Proof\n\nTo prove this, we must verify the group axioms.\nLet $f: G \\to H$ be a group homomorphism, and let $e_G, e_H$ be the identities of $G$ and of $H$ respectively.\nWrite $f(G)$ for the image of $G$.\n\nThen $f(G)$ is closed under the operation of $H$: since $f(g) f(h) = f(gh)$, so the result of $H$-multiplying two elements of $f(G)$ is also in $f(G)$.\n\n$e_H$ is the identity for $f(G)$: it is $f(e_G)$, so it does lie in the 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