Question

this is under the study of group symmetry

So assume an element x in the group G has the order of infinity i.e. |x|=∞

find all generators of <x⁵>

Answer 1

There are only two generators, ie x^5 and x^(-5) the inverse of x^5

Reason In an infinite cyclic group only two generators are there.

Proof

Let <a> be an infinite cyclic group Then |a| is infinity. Suppose b is an another generator then

<a> = <b> . Since b belongs to <a> we have b = a^n for some n

Then Since a belongs to <b> , a = b^m. This implies a = (a^n)^m = a^mn

ie, a = a^mn . Then mn =1 so there are two possibility

m =1 and n=1 or m=-1 and n=-1

Then b = a or b= a^-1

So there are only two generators . Here |x| is infinity . so |x^5| is also infinity  

so <x^5> is an infinite cyclic group . So the generators are x^5 and x^(-5)