PROBLEM 5 Let G be an abelian group, and let n € Z+. Define G × S to be G × Sn as a set, with operation defined by + (x,σ) (y, v) = (x + y²(0), σ%), . where ɛ : S„ → {±1} = Z2 is the homomorphism from Problem 1. Show that G× S is a group.
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- Suppose G1 and G2 are groups with normal subgroups H1 and H2, respectively, and with G1/H1 isomorphic to G2/H2. Determine the possible orders of H1 and H2 under the following conditions. a. G1=24 and G2=18 b. G1=32 and G2=402. Show that is a normal subgroup of the multiplicative group of invertible matrices in .Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
- 4. Prove that the special linear group is a normal subgroup of the general linear group .Write 20 as the direct sum of two of its nontrivial subgroups.8. Let (G, *) be a group. Define the center of G by Z(G) := {x € G: x * a = a * x, Va E G}. The set Z(G) consists of all elements of G that commute with every possible element of the group. For example, one can say that the matrix 41 belongs to the center of (GL(2, R), ·) because (4I)A = A(4I) for all A in GL(2, R), since both sides are equal to 4A. (a) Show that, for every group G, the center Z(G) is a subgroup of G. (b) Find the center of (Z4, +) and (this part is moved to next homework) the center of D6, the dihedral group. (You should be able to tell from the group table.) (c) One could say that "the center Z(G) measures the abelian-ness of a group G". Please interpret this statement. 'Recall that a group (G, *) is called abelian if the operation * is commutative. Hint: What is Z(G) equal to when G is abelian?
- 1. Assume (X, o) and (Y,) are groups. Let X x Y = {(r, y)|x E X,y E Y} and define the operation * on X x Y as (11, Yı) * (#2, Y2) = (x1 0 F2, Y1 • Y2) for (r1, y1), (r2, Y2) E X x Y. Show that (X x Y, *) is a group.3. Let G, H, K be finitely generated Abelian groups. (i) Show that if G G HH then G= H. (ii) Show that if GeH GOK then H = K.Let G = S3, the symmetric group on 3 letters. Show that K(X,Y) KG - (X21, YX XY², Y³ —– 1) ' - (Hint: Write S3 (12) and Y X S3 as a group.) = {id, (12), (23), (13), (123), (132)}. Consider the map from → (123). You may assume that these two elements generate
- 3) Let G and H be two groups with e and e' as identities. Define P = G x H by P = G x H = { (g, h) | g € G,h E H } with algebraic operation (91, h4)(g2, h2) = (9192, h,h2). Prove that the set P' = {(g', h') I g' e Z(G),h' E Z(H)} is a subgroup of P. Also show that P' is the center of P. (9)5. (a) Decide whether (Z, -) forms a group where : Z x Z Z is the usual operation of subtraction, i.e. (m, n) m - n. Justify your answer fully. X1 (b) Consider R? = R x R. Elements of R? have the form in which x1, x2 E R. Define the operation : R2 x R2 R? by ) - „). y2 \Y1x2+ y2, Show that (R?, *) does not form a group. Next, find a suitable subset of R? that forms a group with the operation * as given above. Finally, decide with proof whether this group is abelian. Show (R\{-1}, o) forms a group, where for any a, be R\{-1}, (c) a ob = (a+1)(b+1) – 1. Next, find a group element g such that 2og o3 = 5. Is this element unique? Justify your answer.20. Suppose that G and G' are abelian groups such that G = H₁ H₂ and G' = H₁ H₂. If H₁ is isomorphic to H₁ and H2 is isomorphic to H₂, prove that G is isomorphic to G'.