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1 Solutions to Homework #7 0. Prove that [S n, S n ] = A n for every n 2 (where A n is the alternating group). Solution: Since [f, g] = f 1 g 1 fg is an even permutation for all f, g S n and since A n is a subgroup, we conclude that [S n, S n ] A n. For the opposite inclusion, first note that every 3-cycle (a, b, c) lies in [S n, S n ] since (a, b, c) = (a, b)(a, c)(a, b)(a, c) = (a, b) 1 (a, c) 1 (a, b)(a, c) = [(a, b), (a, c)]. Hence [S n, S n ] contains the subgroup generated by all 3-cycles. Thus, to prove that A n [S n, S n ] it suffices to show that A n is generated by 3-cycles. Take any g A n and write it as a product of transpositions. Since g is even, the number of transpositions in this product is even, so we can write g as a product of pairs of transpositions (x, y)(z, w). It is enough to represent each such (x, y)(z, w) as a product of 3-cycles. If {x, y} = {z, w} as sets, then (x, y)(z, w) = e, and we can simply eliminate the corresponding pair of transpositions. If the sets {x, y} and {z, w} have one element in common, WOLOG we can assume that y = z (and x, y, w are distinct), in which case (x, y)(z, w) = (x, z)(z, w) = (x, z, w). Finally, if {x, y} and {z, w} are disjoint, we write (x, y)(z, w) = (x, y)(y, z)(y, z)(z, w) = (x, y, z)(y, z, w). 1. Let G = D 2n be the dihedral group of order 2n. Recall that G has a presentation G = r, s r n = 1, s 2 = 1, srs 1 = r 1. (a) Let ω C be an n th root of unity (that is, ω n = 1). Prove ( that ) G has ω 0 a representation ρ ω : G GL 2 (C) such that ρ(r) = 0 ω 1 and ( ) 0 1 ρ(s) = (b) Prove that the representation ρ ω in part (a) is irreducible if and only if ω ±1. (c) Let ω 1 and ω 2 be n th roots of unity different from ±1. Prove that ρ ω1 = ρω2 (as representations of G) if and only if ω 2 = ω 1 or ω 2 = ω1 1. (d) Since we can think of isometries of a regular 2n-gon as invertible linear operators on R 2, we get a 2-dimensional representation of G for 1

2 free (simply representing the elements of D 2n by their matrices with respect to the standard basis). If we assume that r is the counterclockwise rotation by 2π and s is the reflection with respect to the n ( ) cos 2π sin 2π x-axis, the corresponding ρ is given by ρ(r) = n n sin 2π cos 2π and ( ) n n ρ(s) =. Note that technically ρ is a real representation (a 0 1 homomorphism from G to GL 2 (R)), but since every 2 2 real matrix can be thought of as a 2 2 complex matrix, we can think of ρ as a complex representation. Prove that this ρ is equivalent to some ρ ω from part (a), explicitly find such ω as well as a matrix T GL 2 (C) such that T 1 ρ(g)t = ρ ω (g) for all g G. (e) In class we proved that G ab (which we know equals the number of onedimensional complex representations of G) is equal to 2 if n is odd and 4 if n is even. Now describe all one-dimensional complex representations of G explicitly (it is enough to specify the images of r and s under these representations). ( ) ( ) ω Solution: (a) Let A = 0 ω 1 and B =. By Theorem A, we just need to verify the relations A n = B 2 = I and BAB 1 = A 1, which is done by a (simple) direct computation. (b) We argue by contrapositive. Suppose that ω = ±1. Then by direct computation the vector v = e 1 + e 2 is an eigenvector for both ρ ω (r) and ρ ω (s), so W = Cv is both ρ ω (r)-invariant and ρ ω (s)-invariant. Let H = Stab G (W ) be the stabilizer of W in G, that is, H = {g G : W is ρ ω (g)-invariant}. It is easy to check that H is a subgroup (this is a general statement not using any specific properties of G or ρ ω ), and we just saw that H contains both r and s. Since r and s generate G, we conclude that H = G. This is equivalent to saying that W is G-invariant and hence ρ ω is not irreducible. Suppose that ω ±1. The only subspaces of C 2 different from 0 and C 2 are 1-dimensional. Thus, to prove that ρ ω is irreducible, we just need to show that there are no G-invariant 1-dimensional subspaces. Since ω ±1, we have ω 1 ω and hence the only 1-dimensional ρ ω (r)- invariant subspaces are Ce 1 and Ce 2. Neither of these subspaces is ρ ω (s)- invariant, so ρ ω is irreducible. (c) Suppose that ρ ω1 = ρω2, so there exists T GL 2 (C) such that 2

3 ( ) T 1 ρ ω1 (g)t = ρ ω2 (g) for all g G. In particular, T 1 ω 0 ω1 1 T = ( ) ω2 0 0 ω2 1. Since eigenvalues of a matrix do not change under conjugation, we must have {ω 1, ω1 1 } = {ω 2, ω2 1 } as sets, so ω 2 = ω 1 or ω 2 = ω1 1. Now suppose that ω 2 = ω 1 or ω 2 = ω1 1. We need to find T GL 2 (C) such that T 1 ρ ω1 (g)t = ρ ω2 (g) for all g G. Since G is generated by r and s, it is enough to find T which satisfies the above equation for g = r and g = s (this is because the conjugation by a fixed element is an automorphism of a group or, in more plain terms, T -conjugate of a product is equal to the product of T -conjugates). ) If ω 2 = ω 1, we can simply set T = I, and if ω 2 = ω 1 1, then T = (d) Let ω = e 2πi n and T = ( 0 1 does the job. ( 1 1 ). By direct computation we check i i that T 1 ρ(r)t = ρ ω (r) and T 1 ρ(s)t = ρ ω (s). Arguing as in (c), we deduce that T 1 ρ(g)t = ρ ω (g) for all g G and hence ρ ω = ρ. (e) Let a and b denote the images of r and s in G ab, respectively. Then a and b commute and must also satisfy the respective relations between r and s, that is, a n = b 2 = 1 and bab 1 = a 1 or, equivalently, bab 1 a 1 = a 2. Since ab = ba, we deduce that a 2 = 1 or equivalently, a 2 = 1. Case 1: n is odd. In this case a n = 1 and a 2 = 1 force a = 1. Thus, any 1-dimensional complex representation of G ab must send a to 1 and b to ±1 (the latter holds since b 2 = 1). So we have at most two choices, and since G ab = 2, both possibilities occur. Thus, there are two 1-dimensional complex representations of G the trivial representation, which sends both r and s to 1, and the other representation which sends r to 1 and s to 1. Case 2: n is even. In this case a n = 1 follows from a 2 = 1. Since a 2 = b 2 = 1, we have (at most) 4 possibilities for 1-dimensional complex representations (send a to ±1 and b to ±1). Again since G ab = 4, all 4 possibilities occur. The corresponding representations of G have the same description with a and b replaced by r and s, respectively. 2. Given a field F, the Heisenberg group over F is the group Heis(F ) consisting of all 3 3 upper unitriangular matrices in GL 3 (F ), that is, matrices 1 a c of the form 0 1 b with a, b, c F. Denote by E ij (λ) the matrix which has 1 s on the diagonal, λ in the position (i, j) and 0 everywhere else. 3

4 Let G = Heis(Z p ) for some prime p. (a) Prove that [G, G] = {E 13 (λ) : λ Z p } and that G ab = Zp Z p. (b) Let x = E 12 ([1]), y = E 23 ([1]) and z = E 13 ([1]) where [1] is the unity element of Z p. Prove that G = x, y, z x p = 1, y p = 1, z p = 1, xz = zx, yz = zy, x 1 y 1 xy = z (c) Describe all one-dimensional complex representations of G. Solution: (a) Let Z = {E 13 (λ) : λ Z p }. By direct computation each commutator [g, h] with g, h G lies in Z. Since Z is clearly a subgroup, we conclude that [G, G] Z. On the other hand, [E 13 (λ)] = [E 12 (λ), E 23 (1)] [G, G] for every λ Z p, so Z [G, G]. Now we prove that G ab = Zp Z p. Define the map π : G Z p Z p 1 a c by π 0 1 b = (a, b). By direct computation π is a homomorphism, clearly π is surjective and Ker π = Z = [G, G]. Thus, by the first isomorphism theorem G ab = G/[G, G] = Z p Z p. (b) Let G be the group with the presentation X, Y, Z X p = 1, Y p = 1, Z p = 1, XZ = ZX, Y Z = ZY, X 1 Y 1 XY = Z. In other words, G is a group generated by abstract symbols X, Y, Z which satisfy the above six relations and where all other relations follow from those six. 1 Now define a map ϕ : G G by ϕ(w(x, Y, Z)) = w(x, y, z). In other words, we take g G, write it as w(x, Y, Z), some word in X ±1, Y ±1 and Z ±1, and map it to the same word in x ±1, y ±1, z ±1. Our goal is to prove that ϕ is an isomorphism (of groups). We proceed in 4 steps. (i) ϕ is well defined. To justify this we just need to check that the relations x p = 1, y p = 1, z p = 1, xz = zx, yz = zy, x 1 y 1 xy = z do hold in G (which is done by direct computation). Let us explain why this implies that ϕ is well defined. Suppose that w(x, Y, Z) = w (X, Y, Z) for some words w, w. This means that the equality w(x, Y, Z) = w (X, Y, Z) follows (formally) from the six 1 Here we do not address the question why such a group exists. For those of you familiar with free groups, G is the quotient F/N where F is the free group on 3 generators X, Y, Z and N is the smallest normal subgroup of F containing X p, Y p, Z p, (XZ) 1 ZX = [Z, X], (Y Z) 1 ZY = [Z, Y ] and Z 1 X 1 Y 1 XY = Z 1 [X, Y ]. 4

5 defining relations of G (and general group axioms). Since we verified that the corresponding six relations hold in G, we can conclude that w(x, y, z) = w (x, y, z) in G as well, that is, ϕ is well defined. (ii) ϕ is a homomorphism. This is pretty clear from the definition (once we know ϕ is well defined). Indeed, for any words w 1 and w 2, let w 1 w 2 be their concatenation (simply write w 2 to the right of w 1 ). We have ϕ(w 1 (X, Y, Z)w 2 (X, Y, Z)) = ϕ((w 1 w 2 )(X, Y, Z)) = (w 1 w 2 )(x, y, z) = w 1 (x, y, z)w 2 (x, y, z) = ϕ(w 1 (X, Y, Z))ϕ(w 2 (X, Y, Z)). (iii) ϕ is a surjective. From the definition of ϕ it is clear that its image Im(ϕ) is the subgroup of G generated by x, y and z. But this subgroup 1 a c is the entire G; for instance, we can write 0 1 b = x a y b z c ab for all a, b, c Z p. Thus ϕ is indeed surjective. (iv) ϕ is a injective. From (iii) we already know that G G = p 3. If we show the opposite inequality G p 3, it would follow that G = G and hence the surjective map ϕ : G G must be injective. To prove that G p 3 it suffices to show that every element of G can be written as X a Y b Z c with 0 a, b, c p 1 (since the number of distinct expressions of this form is p 3 ). The fact that every g G can be written as X a Y b Z c follows easily from the defining relations. Starting with an arbitrary word in X ±1, Y ±1 and Z ±1, we first make all the exponents positive (using X p = Y p = Z p = 1). Then using XZ = ZX and X 1 Y 1 XY = Z (or equivalently Y X = XY Z 1 = XY Z p 1 ), we move all the X s to the left and then using Y Z = ZY all the Z s to the right. We then get an element of the form X a Y b Z c, and finally using X p = Y p = Z p = 1, we move the exponents a, b, c to the interval [0, p 1]. (c) Since z = [x, y] [G, G], it must map to 1 under any 1-dimensional representation. Since x p = y p = 1, both x and y must map to p th roots of unity. This already limits the number of possibilities to p 2. Since we know that G ab = p 2, each of those possibilities must occur. It is easy to describe the 1-dimensional complex representations of G explicitly as maps from G to C (not just their images on generators). For every pair of integers 0 k, l p 1 we have a unique representation 5

6 ϕ k,l : G C given by ϕ k,l 1 a c 0 1 b = e 2π(ak+bl)i n. 3. A representation (ρ, V ) of a group G is called cyclic if there exists v V such that the smallest G-invariant subspace of V containing v is V itself. (a) Prove that any irreducible representation is cyclic. (b) Give an example of a cyclic representation (of some group) which is not irreducible. Solution: (a) Let (ρ, V ) be irreducible. By definition V 0, so we can find a nonzero v V. The smallest G-invariant subspace of V containing v cannot be 0, so by irreducibility it must equal the entire V. Thus, (ρ, V ) is cyclic. (b) Let G = a be a cyclic group of ( order ) 2 and ρ : G GL 2 (C) the unique representation such that ρ(a) =. Then Ce is G-invariant (so the representation is not irreducible), but the smallest G-invariant subspace containing e 1 + e 2 is the entire C 2, so the representation is cyclic. 4. A representation (ρ, V ) of a group G is called faithful if ρ : G GL(V ) is injective. Prove that if G is a finite abelian group which is NOT cyclic, then G does not have any faithful irreducible complex representations. Solution: We will prove the following general result: Theorem: Let G be a finite group. If ρ : G C is any homomorphism, then ρ(g) is cyclic. Let us first explain why this Theorem implies the result of Problem 4. Indeed, Theorem implies that any finite non-cyclic group does not have any faithful 1-dimensional complex representations (since if ρ is faithful, then ρ(g) = G). We also know that if G is abelian, then any irreducible complex representation of G is 1-dimensional. Combining these two facts, we conclude that if G is finite abelian and not cyclic, it does not have any complex representations which are both irreducible and faithful. Let us now prove the Theorem. Let n = G. Then g n = 1 for all g G and hence ρ(g) n = ρ(g n ) = 1. In other words, ρ(g) is contained in µ n (C), the set of n th roots of unity. But µ n (C) is a subgroup of C which is clearly 6

7 cyclic generated by e 2πi n. Thus, ρ(g) is a subgroup of a cyclic group and hence itself cyclic. 5. Let (ρ 1, V 1 ) and (ρ 2, V 2 ) be equivalent representations of a group G, and let (ρ, V 1 V 2 ) be their direct sum. Let T : V 1 V 2 be an isomorphism of representations and let W = {(v, T (v)) : v V 1 } V 1 V 2. Prove that W is a G-invariant subspace and that (W, ρ W ) is isomorphic to (ρ 1, V 1 ) as a G-representation. Note: We will use the result of this problem in the course of the proof of orthogonality relations between characters. Solution: We are given that T ρ 1 (g) = ρ 2 (g)t for all g G. Hence for all v V 1 and g G we have ρ(g)(v, T (v)) = (ρ 1 (g)v, ρ 2 (g)t v) = (ρ 1 (g)v, T (ρ 1 (g)v)) W. Thus, W is G-invariant. Define S : V 1 W by S(v) = (v, T (v)). It is clear that S is an isomorphism of vector spaces. Finally, for every g G and v V we have Sρ 1 (g)v = (ρ 1 (g)v, T (ρ 1 (g)v)) = (ρ 1 (g)v, ρ 2 (g)t v) = ρ(g)(v, T v) = ρ W (g)s(v). (Note that this calculation is essentially equivalent to the one we did to prove G-invariance of W ). Thus, Sρ 1 (g) = ρ W (g)s, so S is an isomorphism of representations. 7

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