Prove that any noncommutative roup contains nontrivial subgr

Prove that any noncommutative roup contains nontrivial subgroups.

Solution

SOLUTION:

Let G be a non-commutative group.

Suppose that G has no nontrivial proper sub group.

Let x be a non identity elwment in G(i.e xe).

Consider the cyclic subgroup <x> of G generated by x.

i.e, <x>={e,x,x²,x³,....}. This is a nontrivial proper subgroup of G.

By assumption G has no nontrivial proper subgroup,and hence <x> must be G.

Thus G becomes cyclic and hence commutative - a contradiction that G is non commutative.

Therefore G must have a non trivial proper sub group.