Question

In: Advanced Math

Let G be a group and a be an element of G. Let φ:Z→G be a...

Let G be a group and a be an element of G. Let φ:Z→G be a map defined by φ(n) =a^{n} for all n∈Z. a)Show that φ is a group homomorphism. b) Find the image ofφ, i.e.φ(Z), and prove that it is a subgroup ofG.

Solutions

Expert Solution



Related Solutions

Let G be a group, and let a ∈ G be a fixed element. Define a...
Let G be a group, and let a ∈ G be a fixed element. Define a function Φ : G → G by Φ(x) = ax−1a−1. Prove that Φ is an isomorphism is and only if the group G is abelian.
Let a be an element of a finite group G. The order of a is the...
Let a be an element of a finite group G. The order of a is the least power k such that ak = e. Find the orders of following elements in S5 a. (1 2 3 ) b. (1 3 2 4) c. (2 3) (1 4) d. (1 2) (3 5 4)
Let G be a cyclic group generated by an element a. a) Prove that if an...
Let G be a cyclic group generated by an element a. a) Prove that if an = e for some n ∈ Z, then G is finite. b) Prove that if G is an infinite cyclic group then it contains no nontrivial finite subgroups. (Hint: use part (a))
Let G be a group of order 40. Can G have an element of order 3?
Let G be a group of order 40. Can G have an element of order 3?
Let G be a finite group and H a subgroup of G. Let a be an...
Let G be a finite group and H a subgroup of G. Let a be an element of G and aH = {ah : h is an element of H} be a left coset of H. If b is an element of G as well and the intersection of aH bH is non-empty then aH and bH contain the same number of elements in G. Thus conclude that the number of elements in H, o(H), divides the number of elements...
Let G be a group acting on a set S, and let H be a group...
Let G be a group acting on a set S, and let H be a group acting on a set T. The product group G × H acts on the disjoint union S ∪ T as follows. For all g ∈ G, h ∈ H, s ∈ S and t ∈ T, (g, h) · s = g · s, (g, h) · t = h · t. (a) Consider the groups G = C4, H = C5, each acting...
Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂...
Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂ G be a subgroup of G such that K ⊂ H Suppose that H is also a normal subgroup of G. (a) Show that H/K ⊂ G/K is a normal subgroup. (b) Show that G/H is isomorphic to (G/K)/(H/K).
Let G be a group and let N ≤ G be a normal subgroup. (i) Define...
Let G be a group and let N ≤ G be a normal subgroup. (i) Define the factor group G/N and show that G/N is a group. (ii) Let G = S4, N = K4 = h(1, 2)(3, 4),(1, 3)(2, 4)i ≤ S4. Show that N is a normal subgroup of G and write out the set of cosets G/N.
Let G be a group and let C={g∈G|xg=gx for all x∈G} be the center of G....
Let G be a group and let C={g∈G|xg=gx for all x∈G} be the center of G. Prove that for any a ∈ G, aC = Ca.
Let f be a group homomorphism from a group G to a group H If the...
Let f be a group homomorphism from a group G to a group H If the order of g equals the order of f(g) for every g in G must f be one to one.
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT