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Qayum Khan
Qayum Khan
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Let $G$ be a group and $H$ a subgroup. Consider the ascending chain $$ H \leq N_G(H) \leq N_G(N_G H) \leq \cdots \leq N^{(k)}_G(H) \leq \cdots $$ of of iterated normalizers. Does: $$ H \trianglelefteq N_G(H) \trianglelefteq N_G(N_G H) \trianglelefteq \cdots \trianglelefteq N^{(k)}_G(H) \trianglelefteq \cdots. $$ Is there exist an example of $G$ and $H$ such thatwhere all the terms of the sequence are finite subgroups, but the chain fails to stabilize?

Let $G$ be a group and $H$ a subgroup. Consider the ascending chain $$ H \leq N_G(H) \leq N_G(N_G H) \leq \cdots \leq N^{(k)}_G(H) \leq \cdots $$ of iterated normalizers. Does there exist an example of $G$ and $H$ such that all the terms of the sequence are finite subgroups, but the chain fails to stabilize?

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?

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Qayum Khan
Qayum Khan
  • 687
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  • 13

Let $G$ be a group and $H$ a subgroup. Consider the ascending chain $$ H \leq N_G(H) \leq N_G(N_G H) \leq \cdots N^{(k)}_G(H) \leq \cdots $$$$ H \leq N_G(H) \leq N_G(N_G H) \leq \cdots \leq N^{(k)}_G(H) \leq \cdots $$ of iterated normalizers. Does there exist an example of $G$ and $H$ such that all the terms of the sequence are finite subgroups, but the chain fails to stabilize?

Let $G$ be a group and $H$ a subgroup. Consider the ascending chain $$ H \leq N_G(H) \leq N_G(N_G H) \leq \cdots N^{(k)}_G(H) \leq \cdots $$ of iterated normalizers. Does there exist an example of $G$ and $H$ such that all the terms of the sequence are finite subgroups, but the chain fails to stabilize?

Let $G$ be a group and $H$ a subgroup. Consider the ascending chain $$ H \leq N_G(H) \leq N_G(N_G H) \leq \cdots \leq N^{(k)}_G(H) \leq \cdots $$ of iterated normalizers. Does there exist an example of $G$ and $H$ such that all the terms of the sequence are finite subgroups, but the chain fails to stabilize?

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Qayum Khan
Qayum Khan
  • 687
  • 5
  • 13

Ascending Chain Condition for finite normalizers

Let $G$ be a group and $H$ a subgroup. Consider the ascending chain $$ H \leq N_G(H) \leq N_G(N_G H) \leq \cdots N^{(k)}_G(H) \leq \cdots $$ of iterated normalizers. Does there exist an example of $G$ and $H$ such that all the terms of the sequence are finite subgroups, but the chain fails to stabilize?