The periodic part of a locally nilpotent group.

Question

Let $G$ be a locally nilpotent group and let $T$ be the periodic subgroup of $G$ (i.e., the locally nilpotent groups have a torsion subgroup). Then $T= \prod T_p$, where $T_p=\langle \{x \in T | \exists n \in \mathbb{N}^* :|x|=p^n\}\rangle$ and $p$ is a prime.

My question is this:

Could we write the $T_p$ without $\langle \rangle$? I mean like this: $T_p =\{x \in T | \exists n \in \mathbb{N}^* :|x|=p^n\}$.

Thank you.

Answer 1

In order to close the question, here is @DerekHolt's comment answering the question.

Yes because local nilpotency implies that the product of two elements of order a power of $p$ also has order a power of $p$.