Comments on: The Normalizer of a Proper Subgroup of a Nilpotent Group is Strictly Bigger https://yutsumura.com/the-normalizer-of-a-proper-subgroup-of-a-nilpotent-group-is-strictly-bigger/ Mon, 20 Jan 2020 23:39:22 +0000 hourly 1 https://wordpress.org/?v=5.3.4 By: Yu https://yutsumura.com/the-normalizer-of-a-proper-subgroup-of-a-nilpotent-group-is-strictly-bigger/#comment-81069 Mon, 20 Jan 2020 23:39:22 +0000 https://yutsumura.com/?p=4223#comment-81069 Dear Alka,

The statement
\[G^{k+1} \subset H \text{ but } G^{k} \nsubseteq H\]
does not say $H$ is placed between two consecutive elements in its lower central series. Consider the lower central series from the identity element $G^n$. Of course, the identity element is contained in $H$. Is the next component $G^{n-1}$ contained in $H$? If so, how about next? Eventually, there is some $k$ such that $G^k$ is not contained in $H$ as $H$ is a proper subgroup.

I hope this helps.

]]> By: Alka https://yutsumura.com/the-normalizer-of-a-proper-subgroup-of-a-nilpotent-group-is-strictly-bigger/#comment-76493 Sun, 03 Nov 2019 15:03:46 +0000 https://yutsumura.com/?p=4223#comment-76493 In problem 523,I am having a doubt regarding the inclusion of G”k+1 in H.Is it confirmed that any subgroup of G can be placed between two consecutive elements in its lower central series?thankyou.

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