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Lecture 10 More on quotient groups Last time The fibers of a homomorphism of groups can form their own group. In fact, the precise range of the homomorphism can be forgotten. We’re figuring out how to use the coset structure of those fibers to motivate forgetting the homomorphism as well (i.e. we’re looking for an internal criterion on subgroups of G for defining quotients of G). Last time The fibers of a homomorphism of groups can form their own group. In fact, the precise range of the homomorphism can be forgotten. We’re figuring out how to use the coset structure of those fibers to motivate forgetting the homomorphism as well (i.e. we’re looking for an internal criterion on subgroups of G for defining quotients of G). Let H ≤ G, and fix g ∈ G. The set gH = {gh | h ∈ H} is a left coset of H. Any element of gH is called a representative. Last time The fibers of a homomorphism of groups can form their own group. In fact, the precise range of the homomorphism can be forgotten. We’re figuring out how to use the coset structure of those fibers to motivate forgetting the homomorphism as well (i.e. we’re looking for an internal criterion on subgroups of G for defining quotients of G). Let H ≤ G, and fix g ∈ G. The set gH = {gh | h ∈ H} is a left coset of H. Any element of gH is called a representative. Some facts about cosets: 1. The cosets of H partition G into equal sized sets. 2. As a consequence, if y ∈ xH, then xH = yH. 3. Two cosets xH and yH are equal exactly when x−1 y ∈ H. 4. The fibers of any homomorphism are both left and right cosets of the kernel. The question we ended on last time: When do the set of cosets of a set form a group? i.e. when is the multiplication xH ? yH = (xy)H well-defined? The question we ended on last time: When do the set of cosets of a set form a group? i.e. when is the multiplication xH ? yH = (xy)H well-defined? Proposition Let H ≤ G. 1. The operation ? above is well defined if and only if gxg −1 ∈ H for all x ∈ H and g ∈ G. (In other words, when the normalizer NG (H) = G) 2. If ? is well-defined, then G/H = {gH | g ∈ G} forms a group with 1 = 1H and (gH)−1 = g −1 H. The question we ended on last time: When do the set of cosets of a set form a group? i.e. when is the multiplication xH ? yH = (xy)H well-defined? Proposition Let H ≤ G. 1. The operation ? above is well defined if and only if gxg −1 ∈ H for all x ∈ H and g ∈ G. (In other words, when the normalizer NG (H) = G) 2. If ? is well-defined, then G/H = {gH | g ∈ G} forms a group with 1 = 1H and (gH)−1 = g −1 H. Definition A subgroup N ≤ G is normal if gN g −1 = N for all g ∈ G, i.e. when G/N is a well-defined group. Write N E G. In summary, let N ≤ G. Then the following are equivalent: 1. N is normal in G 2. NG (N ) = G 3. gN = N g for all g ∈ G 4. the operation on left cosets given by xN ? yN = (xy)N is well-defined 5. gN g −1 ⊆ N for all g ∈ G Example: Letting D8 act on itself by conjugation (g · a = gag −1 ) yields the following table (from Lecture 6): act on ↓ act by → 1 r r2 r3 s sr sr2 sr3 1 1 1 1 1 1 1 1 1 r r r r r r3 r3 r3 r3 r2 r2 r2 r2 r2 r2 r2 r2 r2 r3 r3 r3 r3 r3 r r r r What are the normal subgroups of D8 ? s s sr2 s sr2 s sr2 s sr2 sr sr sr3 sr sr3 sr3 sr sr3 sr sr2 sr2 s sr2 s sr2 s sr2 s sr3 sr3 sr sr3 sr sr sr3 sr sr3 Example: Letting D8 act on itself by conjugation (g · a = gag −1 ) yields the following table (from Lecture 6): act on ↓ act by → 1 r r2 r3 s sr sr2 sr3 1 1 1 1 1 1 1 1 1 r r r r r r3 r3 r3 r3 r2 r2 r2 r2 r2 r2 r2 r2 r2 r3 r3 r3 r3 r3 r r r r s s sr2 s sr2 s sr2 s sr2 sr sr sr3 sr sr3 sr3 sr sr3 sr sr2 sr2 s sr2 s sr2 s sr2 s sr3 sr3 sr sr3 sr sr sr3 sr sr3 What are the normal subgroups of D8 ? Note: “subgroup” is transitive but “normal” is not!! For example, {1, s} E {1, r2 , s, sr2 } and {1, r2 , s, sr2 } E D8 , but {1, s} 5 D8 . More examples 1. G and 1 = {1G } are always normal in G, with G/1 ∼ =G and G/G ∼ = 1. More examples 1. G and 1 = {1G } are always normal in G, with G/1 ∼ =G and G/G ∼ = 1. 2. All subgroups of the center Z(G) of a group are normal in G. More examples 1. G and 1 = {1G } are always normal in G, with G/1 ∼ =G and G/G ∼ = 1. 2. All subgroups of the center Z(G) of a group are normal in G. 3. All subgroups of abelian groups are normal. More examples 1. G and 1 = {1G } are always normal in G, with G/1 ∼ =G and G/G ∼ = 1. 2. All subgroups of the center Z(G) of a group are normal in G. 3. All subgroups of abelian groups are normal. 4. Quotients of abelian groups are abelian and quotients of cyclic groups are cyclic. Back to homomorphism Let N E G. The map π : G → G/N g 7→ gN is a homomorphism, and is called the natural projection of G onto G/N . Back to homomorphism Let N E G. The map π : G → G/N g 7→ gN is a homomorphism, and is called the natural projection of G onto G/N . Proposition A subgroup N of G is normal if and only if it is the kernel of some homomorphism. Back to homomorphism Let N E G. The map π : G → G/N g 7→ gN is a homomorphism, and is called the natural projection of G onto G/N . Proposition A subgroup N of G is normal if and only if it is the kernel of some homomorphism. Theorem (First isomorphism theorem) If ϕ : G → H is a homomorphism of groups, then G/ker(ϕ) ∼ = ϕ(G). Back to homomorphism Let N E G. The map π : G → G/N g 7→ gN is a homomorphism, and is called the natural projection of G onto G/N . Proposition A subgroup N of G is normal if and only if it is the kernel of some homomorphism. Theorem (First isomorphism theorem) If ϕ : G → H is a homomorphism of groups, then G/ker(ϕ) ∼ = ϕ(G). Punchline: The study of the isomorphism classes of homomorphic images of G is “the same” as the study of the normal subgroups of G.