TITLE: Ascending Chain Condition for finite normalizers
QUESTION [7 upvotes]: Let $G$ be a group and $H$ a subgroup.  Consider the ascending chain of iterated normalizers:
$$
H \trianglelefteq N_G(H) \trianglelefteq N_G(N_G H) \trianglelefteq \cdots \trianglelefteq N^{(k)}_G(H) \trianglelefteq \cdots.
$$
Is there an example where all the terms are finite, but the chain fails to stabilize?

REPLY [7 votes]: Can't we just take $G$ to be the union of the chain $H_1<H_2<\cdots$, where $H_1$ is a Klein 4-group and $H_i$ is the dihedral group of order $2^{i+1}$? Then $N_{G}(H_i) = H_{i+1}$.
So
$G = \langle x_i (i \ge 0), y \mid x_1^2=1, x_{i+1}^2=x_i (i\ge 1), y^2=1, (yx_i)^2 = 1 (i \ge 1) \rangle$,
where $H_i$ is the subgroup $\langle x_i, y \rangle$.