Omega subgroups of powerful p-groups

Details Omega subgroups of powerful p-groups Omega subgroups of powerful p-groups

2011-08-02

arxiv.org/abs/1108.0521v1

Israel Journal of Mathematics 162 (2007), 75-80

Mathematics

Group Theory

4 pages

Scientific paper

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
$p^i$.

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