Lecture 6 Special Subgroups

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1 Lecture 6 Special Subgroups

2 Review: Recall, for any homomorphism ϕ : G H, the kernel of ϕ is and the image of ϕ is ker(ϕ) = {g G ϕ(g) = e H } G img(ϕ) = {h H ϕ(g) = h for some g G} H. (They are subgroups of G and H respectively). Subgroup criterion: A non-empty subset H G is a subgroups if and only if xy 1 H for all x, y H.

3 From last time: A group action of a group G on a set A is a map from G A (g, a) A g a which satisfies g (h a) = (gh) a and 1 a = a for all g, h G, a A. We say G acts on A.

4 From last time: A group action of a group G on a set A is a map from G A (g, a) A g a which satisfies g (h a) = (gh) a and 1 a = a for all g, h G, a A. We say G acts on A. Any group action is equivalent to a homomorphism ρ : G S A g σ g defined by ρ(g)(a) = σ g (a) = g a.

5 Group action: g (h a) = (gh) a and 1 a = a. Homomorphism: ρ : G S A, where ρ(g)(a) = σ g (a) = g a. Example: D 8 (1) on single vertices, and (2) on unordered pairs of opposite vertices.

6 Group action: g (h a) = (gh) a and 1 a = a. Homomorphism: ρ : G S A, where ρ(g)(a) = σ g (a) = g a. Definition Let a be a fixed element of A. The stabilizer of a in G (with respect to a given action) is G a = {g G g a = a} G. Example: D 8 (1) on single vertices, and (2) on unordered pairs of opposite vertices.

7 Group action: g (h a) = (gh) a and 1 a = a. Homomorphism: ρ : G S A, where ρ(g)(a) = σ g (a) = g a. Definition Let a be a fixed element of A. The stabilizer of a in G (with respect to a given action) is If S A, then G a = {g G g a = a} G. G S = {g G g s = s for all s S} G. The kernel of the group action is G A. Example: D 8 (1) on single vertices, and (2) on unordered pairs of opposite vertices.

8 Group action: g (h a) = (gh) a and 1 a = a. Homomorphism: ρ : G S A, where ρ(g)(a) = σ g (a) = g a. Definition Let a be a fixed element of A. The stabilizer of a in G (with respect to a given action) is If S A, then G a = {g G g a = a} G. G S = {g G g s = s for all s S} G. The kernel of the group action is G A. Theorem For any non-empty S A, G S is a subgroup of G. Example: D 8 (1) on single vertices, and (2) on unordered pairs of opposite vertices.

9 A group acts on itself in several ways (A = G). Two important ways are 1. by left multiplication: g a = ga, and 2. by conjugation: g a = gag 1.

10 A group acts on itself in several ways (A = G). Two important ways are 1. by left multiplication: g a = ga, and 2. by conjugation: g a = gag 1. Example: Let D 8 act on itself by conjugation (g a = gag 1 ). Fill out the following table: act by 1 r r 2 r 3 s sr sr 2 sr 3 act on 1 r r 2 r 3 s sr sr 2 sr 3 What subset of G fixes r? fixes s? fixes both s and r? fixes everything?

11 A group acts on itself in several ways (A = G). Two important ways are 1. by left multiplication: g a = ga, and 2. by conjugation: g a = gag 1. Example: Let D 8 act on itself by conjugation (g a = gag 1 ). Fill out the following table: act by act on 1 r r 2 r 3 s sr sr 2 sr r r 2 r 3 s sr sr 2 sr 3 r 1 r r 2 r 3 sr 2 sr 3 s sr r 2 1 r r 2 r 3 s sr sr 2 sr 3 r 3 1 r r 2 r 3 sr 2 sr 3 s sr s 1 r 3 r 2 r s sr 3 sr 2 sr sr 1 r 3 r 2 r sr 2 sr s sr 3 sr 2 1 r 3 r 2 r s sr 3 sr 2 sr sr 3 1 r 3 r 2 r sr 2 sr s sr 3 What subset of G fixes r? fixes s? fixes both s and r? fixes everything?

12 More special subgroups

13 More special subgroups Definition Let A be a non-empty subset of G (not nec. subgroup). The centralizer of A in G is Since C G (A) = {g G gag 1 = a for all a A}. gag 1 = a ga = ag this is the set of elements which commute with all a in A. If A = {a}, we write C G ({a}) = C G (a).

14 More on the centeralizers C G (A) = {g G gag 1 = a for all a A}.

15 More on the centeralizers C G (A) = {g G gag 1 = a for all a A}. Theorem For any non-empty A G, C A (G) is a subgroup of G.

16 More on the centeralizers C G (A) = {g G gag 1 = a for all a A}. Theorem For any non-empty A G, C A (G) is a subgroup of G. Definition The center of a group G, denoted Z(G), is the set of elements which commute with everything in G, i.e. Z(G) = C G (G).

17 More on the centeralizers C G (A) = {g G gag 1 = a for all a A}. Theorem For any non-empty A G, C A (G) is a subgroup of G. Definition The center of a group G, denoted Z(G), is the set of elements which commute with everything in G, i.e. Z(G) = C G (G). Corollary The center Z(G) is a subgroup of G of C G (A) for all A G.

18 And one more... Again, let A G be a subset of G, and fix an element g G. Let gag 1 = {h G h = gag 1 for some a A} G be the set of all elements one can arrive at by conjugating elements of A by g.

19 And one more... Again, let A G be a subset of G, and fix an element g G. Let gag 1 = {h G h = gag 1 for some a A} G be the set of all elements one can arrive at by conjugating elements of A by g. Definition The normalizer of A in G is the set N G (A) = {g G gag 1 = A} G of all the elements of G which setwise fix A (individual elements don t have to be fixed!)

20 And one more... Again, let A G be a subset of G, and fix an element g G. Let gag 1 = {h G h = gag 1 for some a A} G be the set of all elements one can arrive at by conjugating elements of A by g. Definition The normalizer of A in G is the set N G (A) = {g G gag 1 = A} G of all the elements of G which setwise fix A (individual elements don t have to be fixed!) Theorem For any A G, the normalizer N G (A) is a subgroup of G. Moreover, Z(G) C G (A) N G (A) G.

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