EX1 A group of order 22 contains elements x and y where x no

EX1. A group of order 22 contains elements x and y, where x not equal to e and y is not a power of x. Show that the subgroup generated by X and y is actually the whole group G.

Solution

G, the group has 22 elements.

x and y are elements such that y is not a power of x

If a is the generator of G, then

G = {e, a, a²,...., a²¹)

Then x = a^p and y = a^q

and p does not divide q.

If we consider the elements of powers of x and y

then x^l , y^m will be elements in the subgroup generated by x and y, where l and m are natural numbers

Obviously e belongs to the subgroup.

We have to prove that there is no element in the group G which cannot be expressed in powers of x or y

If possible let z be in G such z cannot be expressed in powers of x or y.

Then z = a^r where r is not multiple of p or q

This implies all p,q,r should divide 22

But 22 has factors as 1,2, 11

Hence e has order 1, if x has order 2, the y has order 11 in that case there cannot be a z such that z^r = a where q and p do not divide r.

It follows that subgroup generated by x and y is actually the whole group G