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edited Jan 30, 2014 at 1:34

(It is possible that an answer to this question can be found in the literature, but I couldn't find anything after searching for about an hour.)

Let $G$ be a compact, totally disconnected, second countable group. (Equivalently, a profinite group: it is the inverse limit of finite groups.) It makes sense to talk about normal series: $\{e\} \lhd \cdots\cdots G_2\lhd G_1\lhd G$ where (as the notation indicates)$G_n$ is a closed subrgoup of $G$, $G_n$ is normal in $G_{n-1}$ and $G_{n-1}/G_n$ is finite and $\bigcap_{n\in\mathbb{N}}G_n=\{e\}$. This is the same as having an inverse limit expresion $$G\to \cdots \to H_2\to H_1\to \{e\}$$ where each $H_n$ is finite.

Call the normal series a composition series if each $G_{n-1}/G_n$ is simple. (In the inverse limit picture, this is the same as the kernel of each map $H_n\to H_{n-1}$ being simple.)

Are the simple factors appearing in a composition series unique up to permutation? The result is false without compactness, but I do not know if it is true with it.

(It is possible that an answer to this question can be found in the literature, but I couldn't find anything after searching for about an hour.)

Let $G$ be a compact, totally disconnected, second countable group. (Equivalently, a profinite group: it is the inverse limit of finite groups.) It makes sense to talk about normal series: $\{e\} \lhd \cdots\cdots G_2\lhd G_1\lhd G$ where (as the notation indicates) $G_n$ is normal in $G_{n-1}$ and $G_{n-1}/G_n$ is finite and $\bigcap_{n\in\mathbb{N}}G_n=\{e\}$. This is the same as having an inverse limit expresion $$G\to \cdots \to H_2\to H_1\to \{e\}$$ where each $H_n$ is finite.

Call the normal series a composition series if each $G_{n-1}/G_n$ is simple. (In the inverse limit picture, this is the same as the kernel of each map $H_n\to H_{n-1}$ being simple.)

Are the simple factors appearing in a composition series unique up to permutation? The result is false without compactness, but I do not know if it is true with it.

(It is possible that an answer to this question can be found in the literature, but I couldn't find anything after searching for about an hour.)

Let $G$ be a compact, totally disconnected, second countable group. (Equivalently, a profinite group: it is the inverse limit of finite groups.) It makes sense to talk about normal series: $\{e\} \lhd \cdots\cdots G_2\lhd G_1\lhd G$ where $G_n$ is a closed subrgoup of $G$, $G_n$ is normal in $G_{n-1}$ and $G_{n-1}/G_n$ is finite and $\bigcap_{n\in\mathbb{N}}G_n=\{e\}$. This is the same as having an inverse limit expresion $$G\to \cdots \to H_2\to H_1\to \{e\}$$ where each $H_n$ is finite.

Call the normal series a composition series if each $G_{n-1}/G_n$ is simple. (In the inverse limit picture, this is the same as the kernel of each map $H_n\to H_{n-1}$ being simple.)

Are the simple factors appearing in a composition series unique up to permutation? The result is false without compactness, but I do not know if it is true with it.

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asked Jan 29, 2014 at 20:37

Eusebio Gardella

Uniqueness of composition series for profinite groups

(It is possible that an answer to this question can be found in the literature, but I couldn't find anything after searching for about an hour.)

Let $G$ be a compact, totally disconnected, second countable group. (Equivalently, a profinite group: it is the inverse limit of finite groups.) It makes sense to talk about normal series: $\{e\} \lhd \cdots\cdots G_2\lhd G_1\lhd G$ where (as the notation indicates) $G_n$ is normal in $G_{n-1}$ and $G_{n-1}/G_n$ is finite and $\bigcap_{n\in\mathbb{N}}G_n=\{e\}$. This is the same as having an inverse limit expresion $$G\to \cdots \to H_2\to H_1\to \{e\}$$ where each $H_n$ is finite.

Call the normal series a composition series if each $G_{n-1}/G_n$ is simple. (In the inverse limit picture, this is the same as the kernel of each map $H_n\to H_{n-1}$ being simple.)

Are the simple factors appearing in a composition series unique up to permutation? The result is false without compactness, but I do not know if it is true with it.

gr.group-theory