Omega subgroups of powerful p-groups

TL;DR

This paper provides elementary proofs for properties of omega subgroups in powerful finite p-groups, relating their exponents and element orders to subgroup indices, for all primes p.

Contribution

It offers a simplified, elementary proof of key properties of omega subgroups in powerful p-groups, extending understanding of their structure.

Findings

Exponent bounds for omega subgroups are established.

The subgroup index equals the count of elements with bounded order.

Results hold uniformly for all primes p.

Abstract

Let G be a powerful finite p-group. In this note, we give a short elementary proof of the following facts for all $i\ge 0$: (i) $\exp \Omega_-i(G)\le p^i$ for odd p, and $\exp \Omega_-i(G)\le 2^{i+1}$ for p = 2; (ii) the index $|G:G^{p^i}|$ coincides with the number of elements of G of order at most $p^i$.