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<h2 id="subgroup-tests">Subgroup tests</h2> <p>Suppose that G is a group, and H is a subset of G. For now, assume that the group operation of G is written multiplicatively, denoted by juxtaposition. </p> <ul><li>Then H is a subgroup of G <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> H is nonempty and <a href="/facts/Closure_(mathematics)/Ct6r2fJz">closed</a> under products and inverses. <i>Closed under products</i> means that for every a and b in H, the product ab is in H. <i>Closed under inverses</i> means that for every a in H, the inverse <i>a</i>−1 is in H. These two conditions can be combined into one, that for every a and b in H, the element <i>ab</i>−1 is in H, but it is more natural and usually just as easy to test the two closure conditions separately.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li> <li>When H is <i>finite</i>, the test can be simplified: H is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element a of H generates a finite cyclic subgroup of H, say of order n, and then the inverse of a is <i>a</i><i>n</i>−1.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li></ul> <p>If the group operation is instead denoted by addition, then <i>closed under products</i> should be replaced by <i>closed under addition</i>, which is the condition that for every a and b in H, the sum <i>a</i> + <i>b</i> is in H, and <i>closed under inverses</i> should be edited to say that for every a in H, the inverse −<i>a</i> is in H. </p> <h2 id="basic-properties-of-subgroups">Basic properties of subgroups</h2> <ul><li>The <a href="/facts/Identity_element/9nss3xHY">identity</a> of a subgroup is the identity of the group: if G is a group with identity eG, and H is a subgroup of G with identity eH, then <i>eH</i> = <i>eG</i>.</li> <li>The <a href="/facts/Inverse_element/CwMygHSj">inverse</a> of an element in a subgroup is the inverse of the element in the group: if H is a subgroup of a group G, and a and b are elements of H such that <i>ab</i> = <i>ba</i> = <i>eH</i>, then <i>ab</i> = <i>ba</i> = <i>eG</i>.</li> <li>If H is a subgroup of G, then the inclusion map <i>H</i> → <i>G</i> sending each element a of H to itself is a <a href="/facts/Homomorphism/0gyqdEse">homomorphism</a>.</li> <li>The <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> of subgroups A and B of G is again a subgroup of G.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> For example, the intersection of the x-axis and y-axis in R 2 {\displaystyle \mathbb {R} ^{2}} under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of G is a subgroup of G.</li> <li>The <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of subgroups A and B is a subgroup if and only if <i>A</i> ⊆ <i>B</i> or <i>B</i> ⊆ <i>A</i>. A non-example: 2 Z ∪ 3 Z {\displaystyle 2\mathbb {Z} \cup 3\mathbb {Z} } is not a subgroup of Z , {\displaystyle \mathbb {Z} ,} because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the x-axis and the y-axis in R 2 {\displaystyle \mathbb {R} ^{2}} is not a subgroup of R 2 . {\displaystyle \mathbb {R} ^{2}.} </li> <li>If S is a subset of G, then there exists a smallest subgroup containing S, namely the intersection of all of subgroups containing S; it is denoted by ⟨<i>S</i>⟩ and is called the <a href="/facts/Generating_set_of_a_group/B9q5lnFY">subgroup generated by S</a>. An element of G is in ⟨<i>S</i>⟩ if and only if it is a finite product of elements of S and their inverses, possibly repeated.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a></li> <li>Every element a of a group G generates a cyclic subgroup ⟨<i>a</i>⟩. If ⟨<i>a</i>⟩ is <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> to Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} } (<a href="/facts/Integers_modulo_n/0wtRa5dU">the integers mod <i>n</i></a>) for some positive integer n, then n is the smallest positive integer for which <i>an</i> = <i>e</i>, and n is called the <i>order</i> of a. If ⟨<i>a</i>⟩ is isomorphic to Z , {\displaystyle \mathbb {Z} ,} then a is said to have <i>infinite order</i>.</li> <li>The subgroups of any given group form a <a href="/facts/Complete_lattice/pQbphWeR">complete lattice</a> under inclusion, called the <a href="/facts/Lattice_of_subgroups/hdIChdWA">lattice of subgroups</a>. (While the <a href="/facts/Infimum/oXCKVopz">infimum</a> here is the usual set-theoretic intersection, the <a href="/facts/Supremum/oXCKVopz">supremum</a> of a set of subgroups is the subgroup <i>generated by</i> the set-theoretic union of the subgroups, not the set-theoretic union itself.) If e is the identity of G, then the trivial group {<i>e</i>} is the <a href="/facts/Partial_order/qb4a3FAX">minimum</a> subgroup of G, while the <a href="/facts/Partial_order/qb4a3FAX">maximum</a> subgroup is the group G itself.</li></ul> <h2 id="cosets-and-lagranges-theorem">Cosets and Lagrange's theorem</h2> <p class="note">Main articles: <a href="/facts/Coset/zFgQUTLY">Coset</a> and <a href="/facts/Lagrange%2527s_theorem_(group_theory)/VgyNjGlz">Lagrange's theorem (group theory)</a></p> <p>Given a subgroup H and some a in G, we define the left <a href="/facts/Coset/zFgQUTLY">coset</a> <i>aH</i> = {<i>ah</i> : <i>h</i> in <i>H</i>}. Because a is invertible, the map φ : <i>H</i> → <i>aH</i> given by φ(<i>h</i>) = <i>ah</i> is a <a href="/facts/Bijection/j4gTuTmW">bijection</a>. Furthermore, every element of G is contained in precisely one left coset of H; the left cosets are the equivalence classes corresponding to the <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> <i>a</i>1 ~ <i>a</i>2 <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> a 1 − 1 a 2 {\displaystyle a_{1}^{-1}a_{2}} is in H. The number of left cosets of H is called the <a href="/facts/Index_of_a_subgroup/ewY2A71Y">index</a> of H in G and is denoted by [<i>G</i> : <i>H</i>]. </p><p><a href="/facts/Lagrange%2527s_theorem_(group_theory)/VgyNjGlz">Lagrange's theorem</a> states that for a finite group G and a subgroup H, </p> [ G : H ] = | G | | H | {\displaystyle [G:H]={|G| \over |H|}} <p>where |G| and |H| denote the <a href="/facts/Order_(group_theory)/Cl3tKGFg">orders</a> of G and H, respectively. In particular, the order of every subgroup of G (and the order of every element of G) must be a <a href="/facts/Divisor/DKNa2GvC">divisor</a> of |G|.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>Right cosets are defined analogously: <i>Ha</i> = {<i>ha</i> : <i>h</i> in <i>H</i>}. They are also the equivalence classes for a suitable equivalence relation and their number is equal to [<i>G</i> : <i>H</i>]. </p><p>If <i>aH</i> = <i>Ha</i> for every a in G, then H is said to be a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a>. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if p is the lowest prime dividing the order of a finite group G, then any subgroup of index p (if such exists) is normal. </p> <h2 id="example-subgroups-of-z8">Example: Subgroups of Z8</h2> <p>Let G be the <a href="/facts/Cyclic_group/JL5zfFuL">cyclic group</a> Z8 whose elements are </p> G = { 0 , 4 , 2 , 6 , 1 , 5 , 3 , 7 } {\displaystyle G=\left\{0,4,2,6,1,5,3,7\right\}} <p>and whose group operation is <a href="/facts/Modular_arithmetic/0wtRa5dU">addition modulo 8</a>. Its <a href="/facts/Cayley_table/1L05nSE7">Cayley table</a> is </p> <table><tbody><tr><th>+</th><th>0</th><th>4</th><th>2</th><th>6</th><th>1</th><th>5</th><th>3</th><th>7</th></tr><tr><th>0</th><td>0</td><td>4</td><td>2</td><td>6</td><td>1</td><td>5</td><td>3</td><td>7</td></tr><tr><th>4</th><td>4</td><td>0</td><td>6</td><td>2</td><td>5</td><td>1</td><td>7</td><td>3</td></tr><tr><th>2</th><td>2</td><td>6</td><td>4</td><td>0</td><td>3</td><td>7</td><td>5</td><td>1</td></tr><tr><th>6</th><td>6</td><td>2</td><td>0</td><td>4</td><td>7</td><td>3</td><td>1</td><td>5</td></tr><tr><th>1</th><td>1</td><td>5</td><td>3</td><td>7</td><td>2</td><td>6</td><td>4</td><td>0</td></tr><tr><th>5</th><td>5</td><td>1</td><td>7</td><td>3</td><td>6</td><td>2</td><td>0</td><td>4</td></tr><tr><th>3</th><td>3</td><td>7</td><td>5</td><td>1</td><td>4</td><td>0</td><td>6</td><td>2</td></tr><tr><th>7</th><td>7</td><td>3</td><td>1</td><td>5</td><td>0</td><td>4</td><td>2</td><td>6</td></tr></tbody></table> <p>This group has two nontrivial subgroups: ■ <i>J</i> = {0, 4} and ■ <i>H</i> = {0, 4, 2, 6} , where J is also a subgroup of H. The Cayley table for H is the top-left quadrant of the Cayley table for G; The Cayley table for J is the top-left quadrant of the Cayley table for H. The group G is <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a>, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="example-subgroups-of-s4">Example: Subgroups of S4</h2> <p>S4 is the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> whose elements correspond to the <a href="/facts/Permutation/JSw9ukmv">permutations</a> of 4 elements. Below are all its subgroups, ordered by cardinality. Each group (except those of cardinality 1 and 2) is represented by its <a href="/facts/Cayley_table/1L05nSE7">Cayley table</a>. </p> <h3>24 elements</h3> <p>Like each group, S4 is a subgroup of itself. </p> <table><tbody><tr><td></td><td></td></tr></tbody></table> <h3>12 elements</h3> <p>The <a href="/facts/Alternating_group/yethKwyo">alternating group</a> contains only the <a href="/facts/Parity_of_a_permutation/ludhCMJz">even permutations</a>. It is one of the two nontrivial proper <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroups</a> of S4. (The other one is its Klein subgroup.) </p> <h3>8 elements</h3> <table><tbody><tr><td></td><td> </td><td></td><td> </td><td></td></tr></tbody></table> <h3>6 elements</h3> <table><tbody><tr><td></td><td></td><td></td><td></td></tr></tbody></table> <h3>4 elements</h3> <table><tbody><tr><td></td><td></td><td></td><td></td></tr></tbody></table> <table><tbody><tr><td></td><td></td><td></td></tr></tbody></table> <h3>3 elements</h3> <table><tbody><tr><td></td><td></td><td></td><td></td></tr></tbody></table> <h3>2 elements</h3> <p>Each permutation p of order 2 generates a subgroup {1, <i>p</i>}. These are the permutations that have only 2-cycles: </p> <ul><li>There are the 6 <a href="/facts/Cyclic_permutation/UndoYWpl">transpositions</a> with one 2-cycle. (green background)</li> <li>And 3 permutations with two 2-cycles. (white background, bold numbers)</li></ul> <h3>1 element</h3> <p>The <a href="/facts/Trivial_group/XPjQkQFm">trivial subgroup</a> is the unique subgroup of order 1. </p> <h2 id="other-examples">Other examples</h2> <ul><li>The even integers form a subgroup 2 Z {\displaystyle 2\mathbb {Z} } of the <a href="/facts/Integer_ring/KEzxxMbh">integer ring</a> Z : {\displaystyle \mathbb {Z} :} the sum of two even integers is even, and the negative of an even integer is even.</li> <li>An <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> in a ring R is a subgroup of the additive group of R.</li> <li>A <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> of a <a href="/facts/Vector_space/M1pxMLx2">vector space</a> is a subgroup of the additive group of vectors.</li> <li>In an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a>, the elements of finite <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a> form a subgroup called the <a href="/facts/Torsion_subgroup/dIwLuG5j">torsion subgroup</a>.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Cartan_subgroup/4mDR1t5Y">Cartan subgroup</a></li> <li><a href="/facts/Fitting_subgroup/EIXK0ANT">Fitting subgroup</a></li> <li><a href="/facts/Fixed-point_subgroup/tekvHaSE">Fixed-point subgroup</a></li> <li><a href="/facts/Fully_normalized_subgroup/JFJUXgub">Fully normalized subgroup</a></li> <li><a href="/facts/Stable_subgroup/owD2h3d6">Stable subgroup</a></li></ul> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Nathan_Jacobson/TEDrg4Lk">Jacobson, Nathan</a> (2009), <i>Basic algebra</i>, vol. 1 (2nd ed.), Dover, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-47189-1.</li> <li><a href="/facts/Thomas_W._Hungerford/sWS5Wkkw">Hungerford, Thomas</a> (1974), <i>Algebra</i> (1st ed.), Springer-Verlag, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780387905181.</li> <li><a href="/facts/Michael_Artin/Tfk0f6rK">Artin, Michael</a> (2011), <i>Algebra</i> (2nd ed.), Prentice Hall, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780132413770.</li> <li>Dummit, David S.; Foote, Richard M. (2004). <i>Abstract algebra</i> (3rd ed.). Hoboken, NJ: Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9780471452348. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/248917264">248917264</a>.</li> <li><a href="/facts/Joseph_Gallian/NMXD7GYd">Gallian, Joseph A.</a> (2013). <a href="https://www.worldcat.org/oclc/807255720"><i>Contemporary abstract algebra</i></a> (8th ed.). Boston, MA: Brooks/Cole Cengage Learning. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-133-59970-8. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/807255720">807255720</a>.</li> <li>Kurzweil, Hans; Stellmacher, Bernd (1998). <a href="https://dx.doi.org/10.1007/978-3-642-58816-7"><i>Theorie der endlichen Gruppen</i></a>. Springer-Lehrbuch. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-642-58816-7">10.1007/978-3-642-58816-7</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-60331-3.</li> <li>Ash, Robert B. (2002). <a href="https://faculty.math.illinois.edu/~r-ash/Algebra.html"><i>Abstract Algebra: The Basic Graduate Year</i></a>. Department of Mathematics University of Illinois.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Gallian 2013, p. 61. - Gallian, Joseph A. (2013). Contemporary abstract algebra (8th ed.). Boston, MA: Brooks/Cole Cengage Learning. ISBN 978-1-133-59970-8. OCLC 807255720. <a href="https://www.worldcat.org/oclc/807255720" target="_blank">https://www.worldcat.org/oclc/807255720</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Hungerford 1974, p. 32. - Hungerford, Thomas (1974), Algebra (1st ed.), Springer-Verlag, ISBN 9780387905181 <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Artin 2011, p. 43. - Artin, Michael (2011), Algebra (2nd ed.), Prentice Hall, ISBN 9780132413770 <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Kurzweil & Stellmacher 1998, p. 4. - Kurzweil, Hans; Stellmacher, Bernd (1998). Theorie der endlichen Gruppen. Springer-Lehrbuch. doi:10.1007/978-3-642-58816-7. ISBN 978-3-540-60331-3. <a href="https://dx.doi.org/10.1007/978-3-642-58816-7" target="_blank">https://dx.doi.org/10.1007/978-3-642-58816-7</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Kurzweil & Stellmacher 1998, p. 4. - Kurzweil, Hans; Stellmacher, Bernd (1998). Theorie der endlichen Gruppen. Springer-Lehrbuch. doi:10.1007/978-3-642-58816-7. ISBN 978-3-540-60331-3. <a href="https://dx.doi.org/10.1007/978-3-642-58816-7" target="_blank">https://dx.doi.org/10.1007/978-3-642-58816-7</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Jacobson 2009, p. 41. - Jacobson, Nathan (2009), Basic algebra, vol. 1 (2nd ed.), Dover, ISBN 978-0-486-47189-1 <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Ash 2002. - Ash, Robert B. (2002). Abstract Algebra: The Basic Graduate Year. Department of Mathematics University of Illinois. <a href="https://faculty.math.illinois.edu/~r-ash/Algebra.html" target="_blank">https://faculty.math.illinois.edu/~r-ash/Algebra.html</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>See a didactic proof in this video. <a href="https://www.youtube.com/watch?v=TCcSZEL_3CQ" target="_blank">https://www.youtube.com/watch?v=TCcSZEL_3CQ</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Dummit & Foote 2004, p. 90. - Dummit, David S.; Foote, Richard M. (2004). Abstract algebra (3rd ed.). Hoboken, NJ: Wiley. ISBN 9780471452348. OCLC 248917264. <a href="https://search.worldcat.org/oclc/248917264" target="_blank">https://search.worldcat.org/oclc/248917264</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Gallian 2013, p. 81. - Gallian, Joseph A. (2013). Contemporary abstract algebra (8th ed.). Boston, MA: Brooks/Cole Cengage Learning. ISBN 978-1-133-59970-8. OCLC 807255720. <a href="https://www.worldcat.org/oclc/807255720" target="_blank">https://www.worldcat.org/oclc/807255720</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> </ol>