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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31 views

Are all intermediate growth branch groups just-infinite?

Are all branch groups of intermediate growth just-infinite? I can't seem to find an answer to this one way or another; the question is motivated by the fact all examples of intermediate growth branch ...
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3answers
85 views

Will the intersection of a subgroup and a normal subgroup create a normal subgroup for the main group

If $G$ is a group and $H \leqslant G$ and $K \triangleleft G$, is $H \cap K \triangleleft G$? I think the intersection of $H$ and $K$ is a subgroup, and if $K \triangleleft G$ is $H\cap K \...
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60 views

Are there examples of subgroups of $\Bbb Z[\frac1n]/q\Bbb Z$ not totally ordered by inclusion?

I have the theorem that the Prufer P-groups (of which $\Bbb Z[\frac12]/\Bbb Z$ is one example) are the only infinite groups whose subgroups are ordered by inclusion. That this property holds for $\...
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Sum of subsets is the whole set?

Let $A$ be subset of $\mathbb{Z}_n$. Define $A \oplus A = {\{a \oplus a’ : a \in A, a’ \in A}\}$, where $a \oplus a’ = a+a \mod n$. If $A$ is not contained in a coset of a proper additive ...
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1answer
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A useful invariant representing the “size” of a multiplicative subgroup of $\Bbb Q^+$

For any rational $r=n/d$, define $$h_s(r) = (nd)^s$$ where $s > 0$ is a free parameter. The intent is for this to be a representation of how "simple" each rational is; simpler rationals are ...
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0answers
29 views

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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47 views

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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41 views

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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1answer
66 views

Bound on the Number of Normal Subgroups of Index $n$

I'm reading Tamas Szamuely's "Galois Groups and Fundamental Groups" and have a question about an argument used in lemma 3.4.11 on page 83: Here $\hat{F}(X)$ is a free profinite group of finite rank $...
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1answer
51 views

Does a group have polynomial growth of the same degree under all generating sets?

(a) Let $G$ have polynomial growth of degree d. Let the polynomial growth function of G under the generating set S given by $\gamma_S(n)\leq c_1n^d$. Does this imply that under any other generating ...
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How many integer r-tuples are there such that sum of the absolute values of their entries is less than or equal to n.

How many r-tuples are there such that sum of absolute values of entries is less than or equal to $n$? That is, what is the cardinality of the set $ \{(x_1,...,x_r): x_i \in Z \text{ and }\mid x_1\mid+...
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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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1answer
26 views

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 $h\in G$ ...
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50 views

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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From an anual growth rate to a monthly compunded

Let's say I have an amount that grew in 1 year of 5%. From 100, to 105. How do I calulcate the average monthly growth? Using the CAGR, I could write: $(105/100)^{1/12}-1 = 0.41\%$ Is this formula ...
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1answer
146 views

Setting up a differential equation where population growth rate $\propto$ population size. [closed]

I'm having a hard time with this one problem: A populations growth rate is proportional to the population size, when the population is below a certain threshold. The proportionality factor is $0....
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1answer
89 views

Subgroup of $\mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/18\mathbb{Z}$.

I want to check my solution of this (simple) problem: find all subgroups $H$ of $\mathbb{Z}/6\mathbb{Z}\times \mathbb{Z}/18\mathbb{Z}$, such that $|H|=36$. My attempt: $|(\mathbb{Z}/6\mathbb{Z}\times ...
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1answer
326 views

All subgroups of a group with order the square of a prime

So I have a group of order $p^2$ (where $p$ is a prime number) and I'm wondering how many subgroups it can have. By Lagrange's theorem I know that if a subgroup exists its order has to divide the ...
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165 views

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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2answers
89 views

Getting 25% growth over last year sale

I want to calculate 25% growth over prior year's sale (by month). The way I am solving this currently is by multiplying (sale)*1.25; this works fine for sale values that are positive but for those ...
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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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1answer
137 views

Is $SL_1(D)$ topologically finitely generated, for $D$ a division algebra over a local field?

I've been struggling with this one all day, and I was wondering if someone can give me a hand with the proof. I'm not even sure if the group in question is finitely generated, so I would appreciate if ...