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
3 answers
61 views

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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2 votes
1 answer
19 views

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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4 votes
0 answers
24 views

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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7 votes
2 answers
78 views

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 ...
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5 votes
0 answers
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 ...
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1 vote
1 answer
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
0 answers
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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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0 votes
0 answers
11 views

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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  • 2,409
3 votes
2 answers
131 views

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 $2^2$ ...
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1 vote
0 answers
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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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3 votes
1 answer
74 views

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: ...
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-1 votes
0 answers
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
1 answer
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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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  • 1,145
2 votes
2 answers
67 views

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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0 votes
0 answers
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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 ...
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1 vote
0 answers
29 views

Let $G$ be a finitely generated group, and $H \subset G$ a subgroup of finite index. Show that $H$ is finitely generated. [duplicate]

Problem: Let $G$ be a finitely generated group, and $H \subset G$ a subgroup of finite index. Show that $H$ is finitely generated. My work: I wrote this proof: Consider any finitely generated group $...
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0 votes
1 answer
41 views

Show that the set $H=\{f \in S_4 \mid f(4)=4\} \subset S_4$ is isomorphic with $S_3$. [duplicate]

Show that the set $H=\{f \in S_4 \mid f(4)=4\} \subset S_4$ is isomorphic with $S_3$. I think I need to construct a homomorphism $\varphi:S_4 \to S_3$ such that $S_4/\ker \varphi = H$? The problem I'...
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3 votes
1 answer
70 views

Groupoid embedding into a group

A necessary condition for a groupoid $(G,∗)$ to embed into a group is that for all $a,b\in G$ then $a ∗ a^{-1}=b∗b^{−1}$. Question: Is this necessary condition also sufficient? This question came to ...
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3 votes
2 answers
56 views

Let $H,K$ be subgroups of $G$. Show that if $G$ has elements $x,y$ such that $xH=yK$, then $H=K$.

Let $H,K$ be subgroups of $G$. Show that if $G$ has elements $x,y$ such that $xH=yK$, then $H=K$. Suppose that $G$ has elements $x,y$ such that $xH=yK$. Let $h \in H$, then $xh \in xH = yK$ and $$xh \...
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1 vote
0 answers
29 views

Group action on monoid with involution - Laws relating involution of element (from monoid) with inverse function (from group)

Preliminaries (Remarks on notation: To denote function application, I will use the Haskell notation $f\ x$, rather than the traditional mathematical notation $f (x)$. The expression $x^-$ denotes an ...
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-2 votes
1 answer
47 views

Abstract Algebra Normal Subgroups [closed]

I have a basic question related to abstract algebra, I read a proof in a book and partly below: $K$ is a normal subgroup of $G$ and $H_1,H_2$ are two subgroups of $G$ containing $K$. Suppose $H_1/K = ...
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-5 votes
0 answers
43 views

Jordan-Holder theorem and failure of reassembly [closed]

Why reassembling group G from its pieces from Jordan-Holder decomposition is often not possible? Thanks a lot in advance.
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-1 votes
2 answers
52 views

Outer automorphism group being a quotient group [closed]

What is the significance of ${\rm Out}(G)$ being a quotient group of ${\rm Inn}(G)$ in ${\rm Aut}(G)$? Thanks a lot in advance.
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  • 9
2 votes
2 answers
99 views

Isomorphism between two presentations of 6 order abelian group

From this link we can see the order $6$ abelian group can have two presentations: One generator: $\langle k\mid k^6\rangle $ Two generators: $\langle k,r\mid k^3, r^2, krk^{-1}r\rangle $ Both of ...
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-3 votes
1 answer
79 views

Does there exist a group homomorphism $(\mathbb{Q}_p, +) \to (\mathbb{R}, +)$?

There exist homomorphisms from the $p$-adic numbers to the multiplicative group of real numbers (additive and multiplicative characters), but is there an additive homomorphism?
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4 votes
0 answers
98 views

Is there a conceptual proof that $A_n$ is simple for $n \ge 5$, or that it is the only nontrivial normal subgroup of $S_n$?

All the proofs I’ve seen of this are very combinatorial and at least for me not very memorable. Is there a cleaner or more conceptual proof? For instance, I wondered if representation theory might ...
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4 votes
2 answers
103 views

Prove that $S$ is a coset of some subgroup of $G$ iff $S+S-S=S.$

I'm a student self-learning abstract-algebra but encountered this problem. The full problem is here: Suppose $S$ is a nonempty subset of an additive abelian group $G$. Prove that $S$ is a coset of ...
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2 votes
1 answer
53 views

Question about a proof for showing that $A_n$ has no subgroup of order $\frac{n!}{4}$ if $n>4$

For the following theorem, I don't understand how the contradiction is derived following from concluding that $H=A_n$. Theorem: If $n>4$, then $A_n$ has no subgroup of order $\frac{n!}{4}$. ...
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  • 1,635
16 votes
1 answer
245 views

Are the roots of unity the only algebraic subgroups of the multiplicative group?

$\newcommand{\G}{\mathbb{G}}$ Let $k$ a field (or maybe more generally an arbitrary ring with connected spectrum), $\G_m$ the multiplicative group over $k$. Are the $\mu_n = \{x^n = 1\}$ the only $k$-...
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  • 1,215
4 votes
1 answer
49 views

Are metacyclic $p$-groups semidirect products?

A group $G$ is called metacyclic if there is cyclic $N\unlhd G$ such that $G/N$ is cyclic as well. If $G$ is a metacyclic $p$-group, I know that there is a presentation $$G\cong\langle x,y\mid\, x^{p^...
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  • 176
1 vote
1 answer
57 views

Can there be a different proof be given for: If $N$ is a subgroup of a group $G$ of index $2$, then $N$ is a normal subgroup of $G$

For a proof of the following theorem, I am wondering if it can be modified in the beginning steps so that it is the same for the proof of Lagrange's theorem. Theorem: If $N$ is a subgroup of a group $...
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  • 1,635
1 vote
2 answers
38 views

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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  • 1,635
4 votes
2 answers
148 views

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 ...
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2 votes
1 answer
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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  • 3,893
-2 votes
0 answers
36 views

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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  • 447
0 votes
2 answers
51 views

First Isomorphism Theorem: Does each homomorphism has to be surjective? Is it possible to define an homomorphism $\phi:G\to H$ such that $|G|<|H|$? [closed]

I have a little bit of misunderstanding about homomorphism and the first isomorphism. Does each homomorphism has to be surjective? Is it possible to define an homomorphism $\phi:G\to H$ such that $|G|&...
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0 votes
1 answer
56 views

Let $G$ be a simple group which acts on $\Omega$. Let $\alpha \in \Omega$ such that $|O(\alpha)|=p$. Prove the order of $p$-sylow subgroup is $p$.

Let $G$ be a finite simple group which acts on $\Omega$. Let $\alpha \in \Omega$ such that $|O(\alpha)|=p$, ($O$ is the orbit of $\alpha$, $p$ is a prime number). Prove the order of $p$-sylow subgroup ...
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  • 1,799
3 votes
0 answers
46 views

Representations on $V\otimes W$ that factor as the tensor product of representations on $V$ and $W$

$G$ is a group and $V$, $W$ are finite dimensional complex vector spaces. I have a representation $\pi: G \to GL(V\otimes W)$ and I want to know if it factors via two representations $\pi_1: G \to V$ ...
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3 votes
1 answer
41 views

Let $G$ be a group with $25$ elements and $E$ a $G$-set with $32$ elements. Show that there exists $a \in E$ such that $G_a=G$.

Let $G$ be a group with $25$ elements and $E$ a $G$-set with $32$ elements. Show that there exists $a \in E$ such that $G_a=G$. So I want to show that $G_a=\{g \in G \mid ga = a\} = G$. I believe ...
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1 vote
1 answer
93 views

If $G_1\cong G_2$, $H_1\triangleleft G_1$, $H_2 \triangleleft G_2$ and $G_1/H_1\cong G_2/H_2$, then is $H_1\cong H_2$?

I am a new student in a learning group. Assume $G_1$, $G_2$ are two groups, $H_1\triangleleft G_1$, $H_2\triangleleft G_2$. If $G_1\cong G_2$, $H_1\cong H_2$, I know there is no $G_1/H_1\cong G_2/H_2$...
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  • 11
3 votes
1 answer
54 views

Group action for signal

I am working through the book Geometric Deep Learning (https://arxiv.org/abs/2104.13478) and have hit the following problem (Chapter 3.1, page 14). We have a group $\mathfrak{G}$ and a set $\Omega$ ...
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  • 2,409
0 votes
1 answer
47 views

Can we come up with a disjoint union of a subsets of the group $\Bbb{Z}$ such that they do not equal the cosets of a subgroup, yet they form a group?

If this applies to $\Bbb{Z}$ it probably will work for other groups $G$, however, for simplicity and because I'm interested in integers & their primes, let's work with $G = \Bbb{Z}$. Anyway, we ...
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3 votes
2 answers
102 views

Why is $\Bbb Z/4\Bbb Z\times \Bbb Z/12\Bbb Z\times \Bbb Z/40\Bbb Z$ not isomorphic to $\Bbb Z/8\Bbb Z\times \Bbb Z/10\Bbb Z\times \Bbb Z/24\Bbb Z$?

Why is $\Bbb Z/4\Bbb Z\times \Bbb Z/12\Bbb Z\times \Bbb Z/40\Bbb Z$ not isomorphic to $\Bbb Z/8\Bbb Z\times \Bbb Z/10\Bbb Z\times \Bbb Z/24\Bbb Z$? I was thinking to show that they have distinct ...
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  • 117
-3 votes
0 answers
36 views

Does $H\cap K= e$ imply $HK \cap K=K$? [closed]

Given two subgroups $H$ and $K$ in a group $G$. Does $H\cap K= \{e\}$ imply that $HK \cap K=K$?
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2 votes
1 answer
31 views

Is there a way to define other homorphisms, different from the conjugation mapping, in the definition of outer semidirect-products?

I've found the following definition of the outer semidirect product on wikipedia Let us now consider the outer semidirect product. Given any two groups $N$ and $H$ and a group homomorphism $φ: H \to \...
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  • 495
4 votes
0 answers
58 views

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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  • 655
8 votes
1 answer
138 views

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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  • 501
1 vote
1 answer
49 views

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 ...
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  • 1,977
-4 votes
0 answers
29 views

What is a subgroup in a finite field [closed]

If I draw a elliptic curve on this website with a = 0 and b=7 and r= 223 and click on a random point, I read below that that point belongs to a subgroup 42. What is meant by a subgroup? And how can I ...
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