# Questions tagged [group-theory]

For questions about the study of algebraic structures consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility.

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### Finding automorphism group of $\langle a,b\mid a^5=b^{11}=1,\ aba^{-1}=b^3\rangle.$

Consider the group $$G = \langle a,b\mid a^5=b^{11}=1,\ aba^{-1}=b^3\rangle.$$ I wish to understand $\operatorname{Aut}(G)$, the automorphism group of this presentation. I could not find a systematic ...
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1 vote
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### Let $A,B$ be matrices under certain conditions, then $AB=B$

Let $k, n$be positive integers. Let $G= \{ A_1, A_2, \cdots, A_k\}$ be a set of real $n\times n$ matrices such that $G$ is a group under the usual matrix multiplication. Let $B=A_1 + A_2 + \cdots +A_k$...
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### Let $H$ be an infinite hyperbolic group. Can $H \rtimes \mathbb Z$ ever be hyperbolic?

It is well known that hyperbolic groups cannot contain a copy of $\mathbb Z^2$. It follows that if $H$ is infinite and hyperbolic, then $G = H \times \mathbb Z$ cannot be hyperbolic, as $H$ will ...
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### Under what condition on $G$, every descending sequence of retracts of $M$ stops?

Let $G$ be a finitely generated group and $M$ be a finitely generated $\mathbb{Z}G$-module. My question: Under what condition on $G$, every descending sequence of retracts of $M$ stops? What I've ...
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### Parallel transport on the Group manifold

We think of the group as a manifold G (called a Group manifold), whose points are the elements of our Lie group. More generally, we could think of any manifold H on which the elements act as smooth ...
118 views

### Prereqs to learn p-adic numbers

I am a junior (high school) and have been trying to self teach my way through math. Recently I have been curious about p-adic numbers and was wondering how to go down that path. I already have some ...
1 vote
58 views

### Are matrices in $\text{PSL}(2, \mathbb{Z})$ conjugate to their inverses?

As I understand it, this comes down to calculating the slope of the expanding eigenvector of each matrix... but I am having trouble with the details. I feel that the fact that we have identified every ...
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1 vote
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### Let $G$ be a group and $S \subset G$. How can I show that the normal closure of $S$ in $G$ is equal to $\langle gsg^{-1}\mid g\in G,s\in S\rangle$? [duplicate]

Let $G$ be a group and $S \subset G$. How can I show that the normal closure of $S$ in $G$ is equal to $\langle gsg^{-1} \mid g \in G, s \in S \rangle$? It is worth remembering that the normal closure ...
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### Distance between domains based on invertible mapping

In the book Geometric Deep Learning (https://arxiv.org/abs/2104.13478), I have encountered a description, which is not clear to me (Chapter 3.3, page 22): If $\mathcal{D}$ denotes the space of all ...
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### About the first Sylow theorem

The first Sylow theorem states the existence of $p$-subgroups. I was wondering if there can be other subgroups of the same order and different structure. Let's take a big group which has a $2^2$ ...
1 vote
31 views

### Reference request-dimension of projective representations of nilpotent groups.

There is folklore theorem concerning dimensions of complex projective representations of nilpotent groups that I want a reference for. I have searched Karpilovsky monographs but no success. The ...
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### About subgroups of the order of the normalizer of a p-subgroup

I recently came across a doubt I've already partially exposed in this post without getting a solution. So I'd like to isolate my final concern from the textbook I mentioned there. The real doubt is: ...
50 views

### Is there any algorithm known that finds more than one complement of a normal subgroup, assuming they exist? [closed]

Let $N$ is of prime order minimal normal subgroup of a nilpotent group $G$. Is there any algorithm known in the literature that finds more than one complement of $N$ in $G$, assuming they exists ? If ...
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1 vote
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### split maximal torus construction

In PGL(n,q) there is a split maximal torus T of order $(q-1)^{n-1}$. How to construct this in Magma? Let's use the example of $PGL(4,11)$. I took a detour to construct it: ...
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### Show that the group contains an element of order 4

Given that a finite group $G$ has a commutator subgroup with order $2$, show that the index of the commutator subgroup is even. The hint for this problem told me to show that the commutator subgroup ...
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### About the number of normalizers of Sylow p-subgroups [duplicate]

A textbook I'm reading claims: if $G$ is a finite group and if $p$ is prime diving $|G|$, then the normalizers of the Sylow p-subgroups of $G$ have order $|G|/s_p$ (where $s_p$ is the number of Sylow ...
1 vote
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1 vote
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### Quick question about a proof of the theorem: If $N$ is a subgroup of a group $G$ of index $2$, then $N$ is a normal subgroup of $G$

I have a minor question about a proof of the following standard theorem in group theory Theorem: If $N$ is a subgroup of a group $G$ of index $2$, then $N$ is a normal subgroup of $G$. It is the ...
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### Application of nonfamous finite groups in computer science [closed]

I have searched a lot about applications of finite groups in computer science. Most of the results include: Finite fields or groups of numbers coprime to $n$ which are widely used in cryptography and ...
61 views

### Determination of order of cosets in a factor group of a finite abelian group.

Consider the group $G=\frac{\mathbb{Z}_{3^{10}}\times \mathbb{Z}_{3^7} }{\langle (3^2, 3^3)\rangle}$. Let $a=(1,0)+\langle (3^2, 3^3)\rangle$ and $b=(0,1)+\langle (3^2, 3^3)\rangle$. Since the order ...
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### Is the free group $F_2$=$F_3$? [duplicate]

Fix $r\in N$, and let $F_r=\langle g_1,g_2,...,g_r\rangle$ be the rank-r free group. So $F_2$ is a subgroup of $F_3$, while $F_3$ is also a subgroup of $F_2$. Then can we get $F_2=F_3$?
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### Does there exist a finitely generated, torsion group $G$ with a residually finite ascending HNN extension?

Let $G$ be a group with an injective endomorphism $\phi$, then the HNN-extension $$G_\phi = \left<G,t \mid t^{-1} gt= \phi(g) \right>$$ is called the ascending HNN-extension of $G$ determined ...
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### Is the symplectic group over the rationals $\text{Sp}(2n,\mathbb Q)$ dense on the symplectic group $\text{Sp}(2n,\mathbb R)$ over the reals?

The symplectic group is defined as $$\text{Sp}(2n,F)=\{M\in M_{2n\times 2n}(F) : M^T\Omega M=\Omega\},$$ where $$\Omega =\left( \begin{matrix}0&I_n\\-I_n&0\end{matrix}\right).$$ Is the ...
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### Does a free group $F$ of finite rank $n$ have finitely many retracts (as a subgroup)?
A subgroup $H$ of a group $G$ is called a retract of $G$ if there exists an epimorphism $r:G\to H$ such that $r(h)=h$ for all $h\in H$. Does a free group $F$ of finite rank $n$ have finitely many ...