Answer to Question #179481 in Algebra for swarnajeet kumar

Question

hich of the following statements are true? Give reasons for your answers. i) If a group G is isomorphic to one of its proper subgroups, then G = Z. ii) If x and y are elements of a non-abelian group (G, ∗) such that x ∗ y = y ∗ x, then x = e or y = e, where e is the identity of G with respect to . ∗ iii) There exists a unique non-abelian group of prime order. iv) If (a, b)∈A× A, where A is a group, then o((a, b)) = o(a)o(b). v) If H and K are normal subgroups of a group G, then hk = kh ∀ h ∈H, k ∈K.

Answer 1

(i)True- Take the set of polynomials in a variable x

x with integer coefficients, it is an infinitely generated group on the generating set 

{$1,x,x^2,x^3,…$ }. Take the proper subgroup of polynomials involving only the even powers: It is a proper subgroup isomorphic to the whole group (it is infact isomorphic even as a ring).

(ii) True- since x and y follows asccociative law and also holds the same results for multiplication with identity element.

(iii) False- It is not necessary that there always exists a unique non-abelian group of prime order.

(iv) False- As the elemnts a and b are in the cartesian product form so $o((a,b))\neq o(a)o(b)$ .

(v) True- As H and K are the normal subgroups of a group G, So due to the reversible properties of Normal subgroups we will easily get $hk = kh ,∀ h ∈H, k ∈K$ .