Quotient Groups §7.8 (Ed.2), 8.4 (Ed.2): Quotient Groups and Homomorphisms, Part I

Quotient Groups §7.8 (Ed.2), 8.4 (Ed.2): Quotient Groups and Homomorphisms, Part I

<p>Math 371 Lecture #33 <br>§7.7 (Ed.2), 8.3 (Ed.2): Quotient Groups <br>§7.8 (Ed.2), 8.4 (Ed.2): Quotient Groups and Homomorphisms, Part I </p><p>Recall that to make the set of right cosets of a subgroup K in a group G into a group under the binary operation (or product) </p><p>(Ka)(Kb) = K(ab) requires that K be a normal subgroup of G. For a normal subgroup N of a group G, we denote the set of right cosets of N in G by G/N (read G mod N). </p><p>Theorem 8.12. For a normal subgroup N of a group G, the product (Na)(Nb) = N(ab) on G/N is well-defined. We saw the proof of this before. Theorem 8.13. For a normal subgroup N of a group G, we have </p><p>(1) G/N is a group under the product (Na)(Nb) = N(ab), (2) when&nbsp;|G| &lt; ∞ that |G/N| = |G|/|N|, and (3) when&nbsp;G is abelian, so is G/N. </p><p>Proof. (1) We know that the product is well-defined. The identity element of G/N is Ne because for all a ∈ G we have </p><p>(Ne)(Na) = N(ae) = Na, (Na)(Ne) = N(ae) = Na. <br>The inverse of Na is Na<sup style="top: -0.3615em;">−1 </sup>because <br>(Na)(Na<sup style="top: -0.4113em;">−1</sup>) = N(aa<sup style="top: -0.4113em;">−1</sup>) = Ne, (Na<sup style="top: -0.4113em;">−1</sup>)(Na) = N(a<sup style="top: -0.4113em;">−1</sup>a) = Ne. <br>Associativity of the product in G/N follows from the associativity in G: <br>[(Na)(Nb)](Nc) = (N(ab))(Nc) <br>= N((ab)c) = N(a(bc)) = (N(a))(N(bc)) = (N(a))[(N(b))(N(c))]. </p><p>This establishes G/N as a group. (2) Suppose that |G| &lt; ∞. By Lagrange’s Theorem we know that |G| = |N| [G : N] where the index [G : N] is the number of distinct right cosets of N in G, which is the number of elements in the group </p><p>G/N. </p><p>Dividing by |N| ≥ 1 gives |G/N| = [G : N] = |G|/|N|. (3) Suppose that G is abelian. For any Na, Nb ∈ G/N, we have </p><p>(Na)(Nb) = N(ab) = N(ba) = (Nb)(Na). <br>Thus G/N is abelian. </p><p>ꢀ</p><p>Definition. For a normal subgroup N of a group G, the group G/N is called the quotient group or factor group of G by N. Examples. (a) The subgroup {(1), (123), (132)} of S<sub style="top: 0.1495em;">3 </sub>is normal. </p><p>What is S<sub style="top: 0.1494em;">3</sub>/N? </p><p>∼</p><p>Since |S | = 6 and |N| = 3, then |G/N| = 2, so that S /N&nbsp;Z . <br>=</p><p></p><ul style="display: flex;"><li style="flex:1">3</li><li style="flex:1">3</li><li style="flex:1">2</li></ul><p></p><p>Does S<sub style="top: 0.1494em;">3 </sub>contain a subgroup isomorphic to Z<sub style="top: 0.1494em;">2</sub>? Yes, it contains 3 subgroups isomorphic to Z<sub style="top: 0.1494em;">2</sub>, namely {(1), (12)}, {(1), (13)}, and {(1), (23)}. Is S<sub style="top: 0.1495em;">3 </sub>isomorphic to Z<sub style="top: 0.1495em;">3 </sub>× Z<sub style="top: 0.1495em;">2</sub>? No, because S<sub style="top: 0.1494em;">3 </sub>is nonabelian while Z<sub style="top: 0.1494em;">3 </sub>× Z<sub style="top: 0.1494em;">2 </sub>is abelian. (b) Let G = Z<sub style="top: 0.1494em;">4 </sub>× Z<sub style="top: 0.1494em;">4 </sub>and </p><p>N = h(3, 2)i = {(0, 0), (3, 2), (2, 0), (1, 2)}. <br>Because G is abelian, the subgroup N is normal. What is G/N? Since |G| = 16 and |N| = 4, then |G/N| = 16/4 = 4. </p><p></p><ul style="display: flex;"><li style="flex:1">∼</li><li style="flex:1">∼</li></ul><p></p><p>So either G/N Z or G/N Z × Z . </p><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1">=</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">4</li><li style="flex:1">2</li><li style="flex:1">2</li></ul><p></p><p>To decide which one we look at the four distinct cosets of N in G, which are <br>N + (0, 0) = {(0, 0), (3, 2), (2, 0), (1, 2)}, N + (1, 0) = {(1, 0), (0, 2), (3, 0), (2, 2)}, N + (1, 1) = {(1, 1), (0, 3), (3, 1), (2, 3)}, N + (2, 1) = {(2, 1), (1, 3), (0, 1), (3, 3)}. </p><p>We check the order of one of these elements of G/N: <br>(N + (1, 1)) + (N + (1, 1)) = N + (2, 2) = N + (1, 0), (N + (1, 1)) + (N + (1, 0)) = N + (2, 1), (N + (1, 1)) + (N + (2, 1)) = N + (3, 2) = N + (0, 0). </p><p>∼</p><p>This says that |N + (1, 1)| = 4, and so G/N Z . <br>=</p><p>4</p><p>Does G contain a subgroup isomorphic to Z<sub style="top: 0.1495em;">4</sub>? Yes, it does. Is G isomorphic to N × G/N? Yes it is. We are seeing in these examples that there is some kind of relationship between G, a normal subgroup N of G, and the quotient group G/N. Theorem 8.14. Let N be a normal subgroup of a group G. Then&nbsp;G/N is abelian if and only if aba<sup style="top: -0.3615em;">−1</sup>b<sup style="top: -0.3615em;">−1 </sup>∈ N for all a, b ∈ G. </p><p>Proof. The quotient group G/N is a abelian if and only if Nab = Nba for all a, b ∈ G. We have Nab = Nba for all a, b ∈ G if and only if (ab)(ba)<sup style="top: -0.3615em;">−1 </sup>∈ N for all a, b ∈ G. Since (ab)(ba)<sup style="top: -0.3615em;">−1 </sup>= aba<sup style="top: -0.3615em;">−1</sup>b<sup style="top: -0.3615em;">−1</sup>, then G/N is abelian if and only if aba<sup style="top: -0.3615em;">−1</sup>b<sup style="top: -0.3615em;">−1 </sup>∈ N for all </p><p>a, b ∈ G. </p><p>ꢀ</p><p>Recall that Z(G) = {c ∈ G : cg = gc for all g ∈ G} is a normal subgroup of G. Theorem 8.15. For a group G, we have G/Z(G) is cyclic if and only if G is abelian. </p><p>Proof. If G is abelian, then Z(G) = G so that G/Z(G) = G/G consists of only one coset, namely Ge which is trivially cyclic, i.e., G/G = hGei. </p><p>Now suppose that G/Z(G) is cyclic. Then there is a coset Z(G)d for some d ∈ G which generators G/Z(G), so that every element of G/Z(G) is of the form (Z(G)d)<sup style="top: -0.3615em;">k </sup>= Z(G)d<sup style="top: -0.3615em;">k </sup>for some k ∈ Z. </p><p>For a, b ∈ G we want to show that ab = ba. Since a = ea is in the coset Z(G)a and since Z(G)a = Z(G)d<sup style="top: -0.3616em;">i </sup>for some i ∈ Z, we have that a = c<sub style="top: 0.1494em;">1</sub>d<sup style="top: -0.3616em;">i </sup>for some c ∈ Z(G). Similarly we have b = c<sub style="top: 0.1495em;">2</sub>d<sup style="top: -0.3615em;">j </sup>for some c<sub style="top: 0.1495em;">2 </sub>∈ Z(G) and j ∈ Z. </p><p>Since c<sub style="top: 0.1495em;">1</sub>, c<sub style="top: 0.1495em;">2 </sub>commute with every element of G, we have </p><p>ab = c<sub style="top: 0.1495em;">1</sub>d<sup style="top: -0.4113em;">i</sup>c<sub style="top: 0.1495em;">2</sub>d<sup style="top: -0.4113em;">j </sup>= c<sub style="top: 0.1494em;">1</sub>c<sub style="top: 0.1494em;">2</sub>d<sup style="top: -0.4113em;">i</sup>d<sup style="top: -0.4113em;">j </sup>= c<sub style="top: 0.1494em;">2</sub>c<sub style="top: 0.1494em;">1</sub>d<sup style="top: -0.4113em;">j</sup>d<sup style="top: -0.4113em;">i </sup>= c<sub style="top: 0.1494em;">2</sub>d<sup style="top: -0.4113em;">j</sup>c<sub style="top: 0.1494em;">1</sub>d<sup style="top: -0.4113em;">i </sup>= ba. </p><p>This holds for all a, b ∈ G, so that G is abelian. </p><p>ꢀ</p><p>Another normal subgroup that plays a role in the classification of groups and in the theory of group homomorphisms is kernel of a homomorphism f : G → H, which is K = {a ∈ G : f(a) = e<sub style="top: 0.1494em;">H</sub>} (see Theorem 8.16 Ed.3). Theorem 8.17. Let f : G → H be a homomorphism of groups with kernel K. Then f is injective if and only if K = {e<sub style="top: 0.1495em;">G</sub>}. </p><p>Proof. If K = {e<sub style="top: 0.1494em;">G</sub>} then for f(a) = f(b) we have f(ab<sup style="top: -0.3615em;">−1</sup>) = f(a)f(b<sup style="top: -0.3615em;">−1</sup>) = f(a)[f(b)]<sup style="top: -0.3615em;">−1 </sup><br>=f(a)[f(a)]<sup style="top: -0.3615em;">−1 </sup>= e<sub style="top: 0.1494em;">H</sub>, so that ab<sup style="top: -0.3615em;">−1 </sup>∈ K, implying that ab<sup style="top: -0.3615em;">−1 </sup>= e<sub style="top: 0.1494em;">G </sub>or a = b. </p><p>If f is injective, then for any c ∈ K we have f(c) = e<sub style="top: 0.1494em;">H </sub>which together with f(e<sub style="top: 0.1494em;">G</sub>) = e<sub style="top: 0.1494em;">H </sub>implies that c = e<sub style="top: 0.1494em;">G</sub>. </p><p>ꢀ</p><p>Theorem 8.18. Let N be a normal subgroup of a group G. Then&nbsp;the map π : G → G/N given by a → Na is a surjective homomorphism with kernel N. </p><p>Proof. The map π is surjective because for any element Na of G/N we have π(a) = Na. Since Ne is the identity element of G/N, the kernel of π is </p><p>{a ∈ G : π(a) = Ne} = {a ∈ G : Na = Ne} = {a ∈ G : ae<sup style="top: -0.4113em;">−1 </sup>∈ N}, </p><p>which says that the kernel of π is N. </p><p>ꢀ</p>

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