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diagrams</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, especially <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a> and other applications of <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a> theory, the <b>five lemma</b> is an important and widely used <a href="/wiki/Lemma_(mathematics)" title="Lemma (mathematics)">lemma</a> about <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutative diagrams</a>. The five lemma is not only valid for abelian categories but also works in the <a href="/wiki/Category_of_groups" title="Category of groups">category of groups</a>, for example. </p><p>The five lemma can be thought of as a combination of two other theorems, the <b>four lemmas</b>, which are <a href="/wiki/Duality_(category_theory)" class="mw-redirect" title="Duality (category theory)">dual</a> to each other. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Statements">Statements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Five_lemma&action=edit&section=1" title="Edit section: Statements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the following <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutative diagram</a> in any <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a> (such as the category of <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a> or the category of <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> over a given <a href="/wiki/Field_(algebra)" class="mw-redirect" title="Field (algebra)">field</a>) or in the category of <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>. </p> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:5_lemma.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/5_lemma.svg/388px-5_lemma.svg.png" decoding="async" width="388" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/5_lemma.svg/582px-5_lemma.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/5_lemma.svg/776px-5_lemma.svg.png 2x" data-file-width="388" data-file-height="131" /></a></span></dd></dl> <p>The five lemma states that, if the rows are <a href="/wiki/Exact_sequence" title="Exact sequence">exact</a>, <i>m</i> and <i>p</i> are <a href="/wiki/Isomorphism" title="Isomorphism">isomorphisms</a>, <i>l</i> is an <a href="/wiki/Epimorphism" title="Epimorphism">epimorphism</a>, and <i>q</i> is a <a href="/wiki/Monomorphism" title="Monomorphism">monomorphism</a>, then <i>n</i> is also an isomorphism. </p><p> The two four-lemmas state:</p><div><ol><li>If the rows in the commutative diagram <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:4_lemma_right.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/4_lemma_right.svg/300px-4_lemma_right.svg.png" decoding="async" width="300" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/4_lemma_right.svg/450px-4_lemma_right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/4_lemma_right.svg/600px-4_lemma_right.svg.png 2x" data-file-width="300" data-file-height="131" /></a></span></dd></dl> are exact and <i>m</i> and <i>p</i> are epimorphisms and <i>q</i> is a monomorphism, then <i>n</i> is an epimorphism.</li><li>If the rows in the commutative diagram <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:4_lemma_left.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/4_lemma_left.svg/300px-4_lemma_left.svg.png" decoding="async" width="300" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/4_lemma_left.svg/450px-4_lemma_left.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/4_lemma_left.svg/600px-4_lemma_left.svg.png 2x" data-file-width="300" data-file-height="131" /></a></span></dd></dl> are exact and <i>m</i> and <i>p</i> are monomorphisms and <i>l</i> is an epimorphism, then <i>n</i> is a monomorphism.</li></ol></div> <div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Five_lemma&action=edit&section=2" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The method of proof we shall use is commonly referred to as <a href="/wiki/Diagram_chasing" class="mw-redirect" title="Diagram chasing">diagram chasing</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> We shall prove the five lemma by individually proving each of the two four lemmas. </p><p>To perform diagram chasing, we assume that we are in a category of <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a> over some <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a>, so that we may speak of <i>elements</i> of the objects in the diagram and think of the morphisms of the diagram as <i><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a></i> (in fact, <a href="/wiki/Homomorphism" title="Homomorphism">homomorphisms</a>) acting on those elements. Then a morphism is a monomorphism <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a>, and it is an epimorphism if and only if it is <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a>. Similarly, to deal with exactness, we can think of <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernels</a> and <a href="/wiki/Image_(function)" class="mw-redirect" title="Image (function)">images</a> in a function-theoretic sense. The proof will still apply to any (small) abelian category because of <a href="/wiki/Mitchell%27s_embedding_theorem" title="Mitchell's embedding theorem">Mitchell's embedding theorem</a>, which states that any small abelian category can be represented as a category of modules over some ring. For the category of groups, just turn all additive notation below into multiplicative notation, and note that commutativity of abelian group is never used. </p><p>So, to prove (1), assume that <i>m</i> and <i>p</i> are surjective and <i>q</i> is injective. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Four_lemma_epic_and_zero.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Four_lemma_epic_and_zero.gif/300px-Four_lemma_epic_and_zero.gif" decoding="async" width="300" height="147" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Four_lemma_epic_and_zero.gif/450px-Four_lemma_epic_and_zero.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Four_lemma_epic_and_zero.gif/600px-Four_lemma_epic_and_zero.gif 2x" data-file-width="1953" data-file-height="960" /></a><figcaption>A proof of (1) in the case where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(c')=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t(c')=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a49a11020e551e28faea375e56b9dcaf6a80309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.601ex; height:3.009ex;" alt="{\displaystyle t(c')=0}" /></span></figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:4lemma-epic-nonzero.gif" class="mw-file-description"><img alt="An animation showing a diagram chase to prove (1) of the 4 lemma. This is the case where we assume c' gets sent to a nonzero element and want to show the map from B to B' is epic." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/4lemma-epic-nonzero.gif/302px-4lemma-epic-nonzero.gif" decoding="async" width="302" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/4lemma-epic-nonzero.gif/453px-4lemma-epic-nonzero.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/4lemma-epic-nonzero.gif/604px-4lemma-epic-nonzero.gif 2x" data-file-width="1255" data-file-height="607" /></a><figcaption>A proof of (1) in the case where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(c')\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">(</mo> <msup> <mi>c</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t(c')\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bc3d5ba4514d12e07edfb03dd8f4f7b0dc9c07b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.601ex; height:3.009ex;" alt="{\displaystyle t(c')\neq 0}" /></span> is nonzero</figcaption></figure> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:4_lemma_right.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/4_lemma_right.svg/300px-4_lemma_right.svg.png" decoding="async" width="300" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/4_lemma_right.svg/450px-4_lemma_right.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/4_lemma_right.svg/600px-4_lemma_right.svg.png 2x" data-file-width="300" data-file-height="131" /></a></span></dd></dl> <ul><li>Let <i>c′</i> be an element of <i>C′</i>.</li> <li>Since <i>p</i> is surjective, there exists an element <i>d</i> in <i>D</i> with <i>p</i>(<i>d</i>) = <i>t</i>(<i>c′</i>).</li> <li>By commutativity of the diagram, <i>u</i>(<i>p</i>(<i>d</i>)) = <i>q</i>(<i>j</i>(<i>d</i>)).</li> <li>Since im <i>t</i> = ker <i>u</i> by exactness, 0 = <i>u</i>(<i>t</i>(<i>c′</i>)) = <i>u</i>(<i>p</i>(<i>d</i>)) = <i>q</i>(<i>j</i>(<i>d</i>)).</li> <li>Since <i>q</i> is injective, <i>j</i>(<i>d</i>) = 0, so <i>d</i> is in ker <i>j</i> = im <i>h</i>.</li> <li>Therefore, there exists <i>c</i> in <i>C</i> with <i>h</i>(<i>c</i>) = <i>d</i>.</li> <li>Then <i>t</i>(<i>n</i>(<i>c</i>)) = <i>p</i>(<i>h</i>(<i>c</i>)) = <i>t</i>(<i>c′</i>). Since <i>t</i> is a homomorphism, it follows that <i>t</i>(<i>c′</i> − <i>n</i>(<i>c</i>)) = 0.</li> <li>By exactness, <i>c′</i> − <i>n</i>(<i>c</i>) is in the image of <i>s</i>, so there exists <i>b′</i> in <i>B′</i> with <i>s</i>(<i>b′</i>) = <i>c′</i> − <i>n</i>(<i>c</i>).</li> <li>Since <i>m</i> is surjective, we can find <i>b</i> in <i>B</i> such that <i>b′</i> = <i>m</i>(<i>b</i>).</li> <li>By commutativity, <i>n</i>(<i>g</i>(<i>b</i>)) = <i>s</i>(<i>m</i>(<i>b</i>)) = <i>c′</i> − <i>n</i>(<i>c</i>).</li> <li>Since <i>n</i> is a homomorphism, <i>n</i>(<i>g</i>(<i>b</i>) + <i>c</i>) = <i>n</i>(<i>g</i>(<i>b</i>)) + <i>n</i>(<i>c</i>) = <i>c′</i> − <i>n</i>(<i>c</i>) + <i>n</i>(<i>c</i>) = <i>c′</i>.</li> <li>Therefore, <i>n</i> is surjective.</li></ul> <p>Then, to prove (2), assume that <i>m</i> and <i>p</i> are injective and <i>l</i> is surjective. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Four_lemma_monic_case.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Four_lemma_monic_case.gif/300px-Four_lemma_monic_case.gif" decoding="async" width="300" height="179" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Four_lemma_monic_case.gif/450px-Four_lemma_monic_case.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Four_lemma_monic_case.gif/600px-Four_lemma_monic_case.gif 2x" data-file-width="1947" data-file-height="1163" /></a><figcaption>A proof of (2)</figcaption></figure> <dl><dd><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:4_lemma_left.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/4_lemma_left.svg/300px-4_lemma_left.svg.png" decoding="async" width="300" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/4_lemma_left.svg/450px-4_lemma_left.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/4_lemma_left.svg/600px-4_lemma_left.svg.png 2x" data-file-width="300" data-file-height="131" /></a></span></dd></dl> <ul><li>Let <i>c</i> in <i>C</i> be such that <i>n</i>(<i>c</i>) = 0.</li> <li><i>t</i>(<i>n</i>(<i>c</i>)) is then 0.</li> <li>By commutativity, <i>p</i>(<i>h</i>(<i>c</i>)) = 0.</li> <li>Since <i>p</i> is injective, <i>h</i>(<i>c</i>) = 0.</li> <li>By exactness, there is an element <i>b</i> of <i>B</i> such that <i>g</i>(<i>b</i>) = <i>c</i>.</li> <li>By commutativity, <i>s</i>(<i>m</i>(<i>b</i>)) = <i>n</i>(<i>g</i>(<i>b</i>)) = <i>n</i>(<i>c</i>) = 0.</li> <li>By exactness, there is then an element <i>a′</i> of <i>A′</i> such that <i>r</i>(<i>a′</i>) = <i>m</i>(<i>b</i>).</li> <li>Since <i>l</i> is surjective, there is <i>a</i> in <i>A</i> such that <span class="nowrap"><i>l</i>(<i>a</i>) = <i>a′</i></span>.</li> <li>By commutativity, <span class="nowrap"><i>m</i>(<i>f</i>(<i>a</i>)) = <i>r</i>(<i>l</i>(<i>a</i>)) = <i>m</i>(<i>b</i>)</span>.</li> <li>Since <i>m</i> is injective, <i>f</i>(<i>a</i>) = <i>b</i>.</li> <li>So <i>c</i> = <i>g</i>(<i>f</i>(<i>a</i>)).</li> <li>Since the composition of <i>g</i> and <i>f</i> is trivial, <i>c</i> = 0.</li> <li>Therefore, <i>n</i> is injective.</li></ul> <p>Combining the two four lemmas now proves the entire five lemma. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Five_lemma&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The five lemma is often applied to <a href="/wiki/Long_exact_sequence" class="mw-redirect" title="Long exact sequence">long exact sequences</a>: when computing <a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">homology</a> or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object. This alone is often not sufficient to determine the unknown homology groups, but if one can compare the original object and sub object to well-understood ones via morphisms, then a morphism between the respective long exact sequences is induced, and the five lemma can then be used to determine the unknown homology groups. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Five_lemma&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Short_five_lemma" title="Short five lemma">Short five lemma</a>, a special case of the five lemma for <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequences</a></li> <li><a href="/wiki/Snake_lemma" title="Snake lemma">Snake lemma</a>, another lemma proved by diagram chasing</li> <li><a href="/wiki/Nine_lemma" title="Nine lemma">Nine lemma</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Five_lemma&action=edit&section=5" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMassey1991" class="citation book cs1">Massey (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=svbU9nxi2xQC&pg=PA184&dq=%22diagram+chasing%22"><i>A basic course in algebraic topology</i></a>. p. 184.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+basic+course+in+algebraic+topology&rft.pages=184&rft.date=1991&rft.au=Massey&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsvbU9nxi2xQC%26pg%3DPA184%26dq%3D%2522diagram%2Bchasing%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFive+lemma" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Five_lemma&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFScott1987" class="citation book cs1">Scott, W.R. (1987) [1964]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fAPLAgAAQBAJ&pg=PP11"><i>Group Theory</i></a>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65377-8" title="Special:BookSources/978-0-486-65377-8"><bdi>978-0-486-65377-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+Theory&rft.pub=Dover&rft.date=1987&rft.isbn=978-0-486-65377-8&rft.aulast=Scott&rft.aufirst=W.R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DfAPLAgAAQBAJ%26pg%3DPP11&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFive+lemma" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMassey1991" class="citation cs2"><a href="/wiki/William_S._Massey" title="William S. Massey">Massey, William S.</a> (1991), <i>A basic course in algebraic topology</i>, Graduate texts in mathematics, vol. 127 (3rd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97430-9" title="Special:BookSources/978-0-387-97430-9"><bdi>978-0-387-97430-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+basic+course+in+algebraic+topology&rft.series=Graduate+texts+in+mathematics&rft.edition=3rd&rft.pub=Springer&rft.date=1991&rft.isbn=978-0-387-97430-9&rft.aulast=Massey&rft.aufirst=William+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFive+lemma" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Five_lemma&oldid=1207268442">https://en.wikipedia.org/w/index.php?title=Five_lemma&oldid=1207268442</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Homological_algebra" title="Category:Homological algebra">Homological algebra</a></li><li><a href="/wiki/Category:Lemmas_in_category_theory" title="Category:Lemmas in category theory">Lemmas in category theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Articles_containing_proofs" title="Category:Articles containing proofs">Articles containing proofs</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 14 February 2024, at 11:06<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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