Assignment-5

1. Let G be a group and T be a cyclic subgroup of G. If T is normal in G, then showthat every subgroup of T is normal in G.

2. If N is a normal subgroup in the finite group G such that iG(N) and o(N) arerelatively prime, show that any element x ∈ G satisfying xo(N) = e must be in N .

3. Let G be any group, g a fixed element in G. Define ϕ : G → G by ϕ(x) = gxg−1.Prove that ϕ is an automorphism of G.

4. Let G be a finite abelian group of order o(G) and suppose the integer n is relativelyprime to o(G). Prove that every g ∈ G can be written as g = xn with x ∈ G.

5. (a) Given any group G and a subset U of G, let U be the smallest subgroup of Gwhich contains U . Prove that there is such a subgroup U in G. (U is calledthe subgroup generated by U .)

(b) If gug−1 ∈ U for all g ∈ G, u ∈ U , prove that U is a normal subgroup of G.

6. For a group G, recall that the commutator subgroup G′ of G is the subgroupgenerated by all elements of the form xyx−1y−1, x, y ∈ G.

(a) Prove that G′ is normal in G.

(b) Prove that G/G′ is abelian.

(c) If G/N is abelian, prove that N ⊇ G′.

(d) Prove that if H is a subgroup of G and H ⊃ G′, then H is normal in G.

7. Prove that the center of a group is always a normal subgroup.

8. Prove that a group of order 9 is abelian.

9. Let D2n = ⟨x, y | xn = e = y2, xy = yx−1⟩ be the dihedral group of order 2n,n ≥ 3. Prove that the subgroup N = {e, x, x2, · · · , xn−1} is normal in G and thatthe quotient group G/N is isomorphic to W , where W = {1,−1} is the groupunder the multiplication of the real numbers. Find the center of D2n. Find thecommutator subgroup of D2n.

10. If G is a non-abelian group of order 6, prove that G is isomorphic to S3.

11. Let G be the group of nonzero complex numbers under multiplication and let N bethe set of complex numbers of absolute value 1 (that is, a+ bi ∈ N if a2 + b2 = 1).Show that G/N is isomorphic to the group of all positive real numbers undermultiplication.

12. Let G be the group of all nonzero complex numbers under multiplication and let G

be the group of all 2×2 matrices of the form

(a b−b a

), where not both a and b are

0, under matrix multiplication. Show that G and G are isomorphic by exhibitingan isomorphism of G onto G.

13. Let G be the group of real numbers under addition and let N be the subgroup ofG consisting of all the integers. Prove that G/N is isomorphic to the group of allcomplex numbers of absolute value 1 under multiplication.