(Think: three copies of C) i j = k = j i, j k = i = k j, k i = j = i k.

Transcription

1 Warm-up: The quaternion group, denoted Q 8, is the set {1, 1, i, i, j, j, k, k} with product given by 1 a = a 1 = a a Q 8, ( 1) ( 1) = 1, i 2 = j 2 = k 2 = 1, ( 1) a = a ( 1) = a a Q 8, (Think: three copies of C) i j = k = j i, j k = i = k j, k i = j = i k. (Think: cross-product with i = v 1, j = v 2, k = v 3 ) You try: 1. Write the group table (multiplication table) for Q Compute the order of each of the 8 elements of Q 8.

2 Isomorphisms Consider the subgroup of S 6 generated by ( ) and (16)(25)(34)

3 Isomorphisms Consider the subgroup of S 6 generated by r = ( ) and s = (16)(25)(34)

4 Isomorphisms Consider the subgroup of S 6 generated by r = ( ) and s = (16)(25)(34) In some sense, this subgroup is the same as D 12, but in some sense, they re not the same until I name them appropriately. Since we don t want to call them the same, we call them isomorphic.

5 Let G and H be groups. A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G.

6 Let G and H be groups. A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. Think: a linear map L : F n F m of vector spaces preserves vector space structure, i.e. for all v, v F n, c F (a field), L(cv) = cl(v) and L(v + v ) = L(v) + L(v ).

7 Let G and H be groups. A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. Think: a linear map L : F n F m of vector spaces preserves vector space structure, i.e. for all v, v F n, c F (a field), L(cv) = cl(v) and L(v + v ) = L(v) + L(v ). In a group, the corresponding structure is the binary operation.

8 Let G and H be groups. A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. Think: a linear map L : F n F m of vector spaces preserves vector space structure, i.e. for all v, v F n, c F (a field), L(cv) = cl(v) and L(v + v ) = L(v) + L(v ). In a group, the corresponding structure is the binary operation. Similarly, a homomorphism of fields would preserve both addition and multiplication.

9 Let G and H be groups. A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. Think: a linear map L : F n F m of vector spaces preserves vector space structure, i.e. for all v, v F n, c F (a field), L(cv) = cl(v) and L(v + v ) = L(v) + L(v ). In a group, the corresponding structure is the binary operation. Similarly, a homomorphism of fields would preserve both addition and multiplication. An isomorphism is a bijective homomorphism.

10 Let G and H be groups. A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. Think: a linear map L : F n F m of vector spaces preserves vector space structure, i.e. for all v, v F n, c F (a field), L(cv) = cl(v) and L(v + v ) = L(v) + L(v ). In a group, the corresponding structure is the binary operation. Similarly, a homomorphism of fields would preserve both addition and multiplication. An isomorphism is a bijective homomorphism. Two groups G and H are isomorphic, written G = H, if there exists an isomorphism between them.

11 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of isomorphisms: 1. G is always isomorphic to itself via the identity map.

12 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of isomorphisms: 1. G is always isomorphic to itself via the identity map. Note that may be other isomorphisms! An isomorphism ϕ : G G is called an automorphism.

13 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of isomorphisms: 1. G is always isomorphic to itself via the identity map. Note that may be other isomorphisms! An isomorphism ϕ : G G is called an automorphism. 2. (R, +) is isomorphic to (R >0, ) via the map ϕ : x e x.

14 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of isomorphisms: 1. G is always isomorphic to itself via the identity map. Note that may be other isomorphisms! An isomorphism ϕ : G G is called an automorphism. 2. (R, +) is isomorphic to (R >0, ) via the map ϕ : x e x. Check: ϕ is a bijection and ϕ(x + y) = e x+y = e x e y = ϕ(x)ϕ(y).

15 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of isomorphisms: 1. G is always isomorphic to itself via the identity map. Note that may be other isomorphisms! An isomorphism ϕ : G G is called an automorphism. 2. (R, +) is isomorphic to (R >0, ) via the map ϕ : x e x. Check: ϕ is a bijection and ϕ(x + y) = e x+y = e x e y = ϕ(x)ϕ(y). 3. S X is isomorphic to S X.

16 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of isomorphisms: 1. G is always isomorphic to itself via the identity map. Note that may be other isomorphisms! An isomorphism ϕ : G G is called an automorphism. 2. (R, +) is isomorphic to (R >0, ) via the map ϕ : x e x. Check: ϕ is a bijection and ϕ(x + y) = e x+y = e x e y = ϕ(x)ϕ(y). 3. S X is isomorphic to S X. 4. S 3 is isomorphic to D 6.

17 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of homomorphisms that aren t isomorphisms:

18 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of homomorphisms that aren t isomorphisms: 1. Let ϕ : Z Z/6Z be given by reducing mod 6.

19 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of homomorphisms that aren t isomorphisms: 1. Let ϕ : Z Z/6Z be given by reducing mod Let ϕ : Z R be the inclusion map.

20 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of homomorphisms that aren t isomorphisms: 1. Let ϕ : Z Z/6Z be given by reducing mod Let ϕ : Z R be the inclusion map. 3. The determinant map det : GL n (R) R is a homomorphism.

21 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of homomorphisms that aren t isomorphisms: 1. Let ϕ : Z Z/6Z be given by reducing mod Let ϕ : Z R be the inclusion map. 3. The determinant map det : GL n (R) R is a homomorphism. Fact: G H whenever G = H is an interesting equivalence relation.

22 A homomorphism is a function ϕ : G H satisfying for all g 1, g 2 G. An isomorphism is a bijective homomorphism. Examples of homomorphisms that aren t isomorphisms: 1. Let ϕ : Z Z/6Z be given by reducing mod Let ϕ : Z R be the inclusion map. 3. The determinant map det : GL n (R) R is a homomorphism. Fact: G H whenever G = H is an interesting equivalence relation. On the other hand, G H whenever there s a homomorphism ϕ : G H is a completely uninteresting equivalence relation.

23 Representations A homomorphism from a group G to a matrix group GL n (F ) is called a group representation.

24 Representations A homomorphism from a group G to a matrix group GL n (F ) is called a group representation. (1) Denote the linear transformation in R 2 that rotates everything clockwise by φ radians by r φ and the linear transformation that flips across the y-axis by s y, i.e. ( ) cos(φ) sin(φ) r φ = sin(φ) cos(φ) and s y = ( ) Then D 2n is isomorphic to the multiplicative group of matrices generated by r 2π/n and s y.

25 Representations A homomorphism from a group G to a matrix group GL n (F ) is called a group representation. (1) Denote the linear transformation in R 2 that rotates everything clockwise by φ radians by r φ and the linear transformation that flips across the y-axis by s y, i.e. ( ) cos(φ) sin(φ) r φ = sin(φ) cos(φ) and s y = ( ) Then D 2n is isomorphic to the multiplicative group of matrices generated by r 2π/n and s y. (2) The symmetric group S n is isomorphic to the multiplicative group of n n matrices satisfying every row and column has exactly one 1 and n 1 0 s. (Since S {1,...,n} = S{v1,...,v n})

26 Properties of homomorphisms Theorem Let ϕ : G H be a homomorphism of groups. 1. ϕ(1 G ) = 1 H.

27 Properties of homomorphisms Theorem Let ϕ : G H be a homomorphism of groups. 1. ϕ(1 G ) = 1 H. 2. For any x G, ϕ(x 1 ) = ϕ(x) 1.

28 Properties of homomorphisms Theorem Let ϕ : G H be a homomorphism of groups. 1. ϕ(1 G ) = 1 H. 2. For any x G, ϕ(x 1 ) = ϕ(x) For any x G, ϕ(x) divides x.

29 Properties of homomorphisms Theorem Let ϕ : G H be a homomorphism of groups. 1. ϕ(1 G ) = 1 H. 2. For any x G, ϕ(x 1 ) = ϕ(x) For any x G, ϕ(x) divides x. 4. The image of ϕ, img(ϕ) = {h H h = ϕ(g) for some g G}, is a subgroup of H.

30 Properties of homomorphisms Theorem Let ϕ : G H be a homomorphism of groups. 1. ϕ(1 G ) = 1 H. 2. For any x G, ϕ(x 1 ) = ϕ(x) For any x G, ϕ(x) divides x. 4. The image of ϕ, img(ϕ) = {h H h = ϕ(g) for some g G}, is a subgroup of H. 5. The kernel of ϕ, ker(ϕ) = {g G ϕ(g) = 1 H } is a subgroup of G. (On homework.)

31 Recall, for all x, y G, we have xy = yx if and only if xyx 1 = 1. We call the expression xyx 1 the conjugation of y by x.

32 Recall, for all x, y G, we have xy = yx if and only if xyx 1 = 1. We call the expression xyx 1 the conjugation of y by x. For example, conjugating elements of D 6 looks like y xyx 1 1 r r 2 s sr sr r r 2 s sr sr 2 x r 1 r r 2 sr sr 2 s r 2 1 r r 2 sr 2 s sr s 1 r 2 r s sr 2 sr sr 1 r 2 r sr 2 sr s sr 2 1 r 2 r sr s sr 2

33 Recall, for all x, y G, we have xy = yx if and only if xyx 1 = 1. We call the expression xyx 1 the conjugation of y by x. For example, conjugating elements of D 6 looks like x y xyx 1 1 r r 2 s sr sr r r 2 s sr sr 2 r 1 r r 2 sr sr 2 s r 2 1 r r 2 sr 2 s sr s 1 r 2 r s sr 2 sr sr 1 r 2 r sr 2 sr s sr 2 1 r 2 r sr s sr 2 On the other hand, the subgroups of D 6 are {1}, {1, s}, {1, sr}, {1, sr 2 }, {1, r, r 2 }, and D 6.

34 More special subgroups Let A be a non-empty subset of G (not necessarily a 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).

35 More special subgroups Let A be a non-empty subset of G (not necessarily a 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). Theorem For any non-empty A G, C A (G) is a subgroup of G.