Question

Let G be an infinite simple p-group. Prove that Z(G) = 1.

Answer 1

By definition of Simple group :A simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.

Lemma 1 : Z(G) is normal Group of G {Proved at last}

On contrary suppose given is not true

This implies by Simplicity and lemma 1 Z(G)=G

This implies G is abelian infinite p group

As every subgroup in Abelian group is also normal group This implies G doesnot have any subgroup

This contradicts to Lagrange theorem

As it must have some nontrivial subgroup of the order of power of prime {As it is infinite p-group}

So Z(G)=G.

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Proof of lemma 1:

This true for every So Z(G) is normal in G