Questions tagged [subgroup-growth]

Subgroup growth is a branch of group theory, dealing with quantitative questions about subgroups of a given group.

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Computing the size of the product of multiple subgroups

Let $G$ be a group and let $S_1, \dots, S_k$ be subgroups. Consider the product of the $\{S_i\}$: $$S_1 S_2 \dots S_k := \{s_1 s_2 \dots s_k \ \mid \ s_i \in S_i \text{ for each } i\}$$ It's well ...
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Growth rate of finitely generated nilpotent groups

Let $N$ be a group and $S$ a finite, symmetric generating set with the identity. For $n \in \mathbb N$, we let $S^n = \{s_1\dots s_n\mid s_i \in S\}$ We say $N$ has polynomial growth rate if $\...
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Wreath product of subgroup with symmetric group

I am going through Generic Quantum Fourier Transforms by Moore et al. I would like to put the screenshot of the section I am confused about below. So, Why does $H$ need to be of size $poly(n)$ for ...
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Differences of Data

I have a question that might be easy...but has me curious Suppose we have this set of data: and with this we need to determine the type of growth that best describes it by finding: -1st Differences ...
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Growth rate of free nilpotent group of rank $r$ and nilpotency index $s$.

Let $F^{(r)}$ denote the free group of rank $r$ with generators $x_1, \dots, x_r$. Recall a group $G$ is nilpotent of index $s$ if $G_{s+1} = \{e\} $ and $G_s \neq \{e\}$ (where $G_i$ denotes the ...
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Growth in Groups / Helfgott paper (2013)

I'm currently working on this paper by Helfgott for a small project: https://arxiv.org/abs/1303.0239. After Lemma 3.1 (Ruzsa inequality) he says (and shows partially) that for any finite subset $A$ of ...
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Exact value of exponential growth rate depends on generating set

I am trying to solve an exercise from Clara Loh's Geometric Group Theory: An introduction. The problem uses the exponential growth rate of a finitely generated group $G$ with generating set $S$. The ...
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Element of a group, its order and inferring that it belongs to a subgroup generated by another element

As part of a larger proof, I want to make the following statement: Let G be finite abelian group. Let $g\in G$ such that the order of $g^{q}$ is $q^{e}$ where $e\geq1$ and q is prime. Let $h\in G$ ...