ab Define * on Q* by a * b =: Show that (Q*,*) is a group. 1.1 1.2 Let a, b be elements of a group G. Assume that a has order 5 and a'b = ba³. Prove that ab = ba. Let G be a group and Z(G) = {a e G: ax = xa for all x E G}. Show that Z(G)is 1.3 a normal subgroup of G. Let G be a group and let p:G → G be the map p(x) = x-1. (a) Prove that p is bijective. (b) Prove that p is an automorphism if G is abelian. Let a, ßE S, (Symmetric group), where a = (1,2)(4,5) and ß = (1,6,5,3,2). Verify that (aß)1 = B-1a-1 . 1.4 1.5

Exercise 2.2. Let (G, *) be a group satisfying that (a* b)² = a² * 62 for every a, b = G. Show that G is abelian. Extended quesion* (Opt): What happens if the condition is replaced by (a + b)³ = a³ * 6³?

modern algebra

Exercise 2.2.6 Consider the affine group AM9) = { (; ")-bez, a#0}. Show that Aff(3) is a group under matrix multiplication as defined in formula (1.3) of Section 1.8 and the statement that follows it. Draw a Cayley graph for this group with { (6 ):6 1)} Compare with the Cayley graph X (Da, {R, F} ) Are generating set S= these groups really the same in some sense (a sense to be known to us as isomorphic groups in Section 3.2)?

9. a) find two integers x,y such that 30x+101y=1. (hint: gcd(30,101)=1).     b) find the inverse of 30 in the group U(101).

Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x Y of X and Y, but can we also define a binary operation on X x Y such that X x Y is a group? Let's consider two elements (x1, 41) and (x2, Y2) in X x Y. Let denote a function on X x Y.We need to define so that (X x Y,•) is a group. The most natural way to proceed is to define • component-wise: (T1, Yı) • (x2, Y2) = (x1 º Y1, ¤2 * Y2) %3D Now we should show that (X × Y,•) is in fact a group.