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Questions tagged [group-extensions]

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2answers
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The number of groups $G$ such that $G/\mathbb{Z}_3\cong D_{2n}$

I am trying to find the number of groups $G$ such that $G/\mathbb{Z}_3\cong D_{2n}$, where $\mathbb{Z}_3$ denotes the cyclic group of order $3$ and $D_{2n}$ denotes the dihedral group of order $2n$. ...
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0answers
26 views

List of extension theorems

As a post grad student, I have come across many results where a function with certain properties(eg-homomorphism) on a smaller algebraic structure is extended to a larger one. For example, extending ...
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0answers
22 views

The relation between group extension and factor set

I'm new to homological algebra. I try to state what I know, and ask the question at the end. I know that given a group extension $0\to K\to G\to Q\to 1$, it may have many liftings. Each lifting ...
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1answer
18 views

Given $\Bbb ZQ$-module $K$, how to construct $\theta:Q\to\text{Aut}(K)$?

I have a problem understanding the meaning of the "converse" part of the below proposition. Let $K,~Q$ be groups, and $K$ is abelian. If $K$ is a module over $\Bbb ZQ$, then it can induce a ...
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0answers
19 views

Automorphisms and Extensions of Cyclic Groups

I can't figure out what question III.4.2 from Hilton & Stammbach's book A Course in Homological Algebra is asking: Classify the extension classes $[E]$ given by: $$ \mathbb Z_m \...
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1answer
20 views

Extension of surface group by cyclic is residually finite

Let $G$ be the fundamental group of a surface, and consider an extension $1 \to \mathbb{Z}/p\mathbb{Z} \to E \to G \to 1$. Is $E$ residually finite? I'm interested in proving the injectivity of the ...
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1answer
30 views

Equivalent Conditions of Split Extension of Groups

Definition of split extension of $Q$ by $N$: An extension of $Q$ by $N$, $$1 \longrightarrow N \longrightarrow G \longrightarrow Q \longrightarrow 1,$$ is said to be split if it is isomorphic to ...
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1answer
42 views

Showing that any central group extension of a finite group $G$ by the trivial $G$-module $\mathbb Q$ is a semidirect product

Suppose that $G$ is a finite group and view the additive group $\mathbb{Q}$ as a trivial $G$-module. I want a concrete way of understanding why the second cohomology group $H^2(G, \mathbb{Q})$ is ...
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2answers
84 views

Is there always a way to break a group?

Let $G$ be a finite group ( some finite representation is given ) and Let $N$ be a normal subgroup of group $G$. I know that given a $N$ and $G/N$ one can't get back a $G$ ( always ). My question is ...
2
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1answer
34 views

Extension that split for all p-Sylow

In my Group Cohomology class, the professor stated the following theorem If one takes an extension $1 \rightarrow A \rightarrow E \rightarrow G \rightarrow 1 $ with $A$ abelian and finite and $G$ ...
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0answers
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Find isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent

I am asking for help on the following exercise: Find two isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent. as taken from D. Robinson, A Course in ...
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1answer
50 views

Isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent

The following exercise from D. Robinson, A Course in the Theory of Groups confuses me: Find two isomorphic extensions of $\mathbb Z_3$ by $\mathbb Z_3 \times \mathbb Z_3$ which are not equivalent. ...
0
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1answer
50 views

On the central extension of a group

Let $G$ be a central extension of a group $K$ by the perfect group $H$ ($K$ is the normal subgroup). The question is that if $(|K|,|Mult(H)|)=1$ then can we say that $G=H \times K$? I would be ...
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1answer
37 views

The extension of a connected, simply-connected Lie group by $\mathbb{Z}$ is trivial.

I am reading a paper by Calvin Moore on group extensions. On page 54, there is a statement saying that when a Lie group is connected and simply-connected, the the extension of a connected, simply-...
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0answers
20 views

Functoriality of Lyndon-Hochschild-Serre spectral sequences in coefficients.

It is a question about group cohomology. Supposing that I have a short exact sequence of $G$-modules $1\rightarrow A_1 \rightarrow A_2\rightarrow A_3\rightarrow 1$, I know that there will be a long ...
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0answers
29 views

Cup Products and Group Extensions

Let $G$ be a group and let $A$ be a $G$-module. Suppose we're given a short exact sequence: $$1 \to A \to X \to G \to 1$$ It is well known that there exists a bijection between $H^2(G, A)$ and ...
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0answers
42 views

Group actions on spectral sequences of group cohomology

Suppose I have a group extension $1 \rightarrow N \rightarrow H\rightarrow K\rightarrow 1$, and we have a group $G$ which acts on $H$, and $K$ by automorphisms and it does not have action on $N$. ...
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41 views

Extension of a cyclic group by an abelian group

What is meant by "an extension of a cyclic group of prime order $p$ by an abelian group of odd order relatively prime to $p$"? Can it be a semidirect product of a cyclic group of order $p^2$ by an ...
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0answers
60 views

A question related to group cohomology and spectral sequences

It is actually a follow-up question of a mathoverflow question. I don't quite understand the answer there. I tried to compute the group cohomology of $H^n(\mathbb{Z}_4,\mathbb{Z})$ via the Lyndon-...
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0answers
55 views

Split extensions of $G/U$ by elementary abelian $p$-group and $H^1(U, \mathbb{Z}/p\mathbb{Z})$

This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 \to P \to E \to W \to 1$ a finite group extension and $f\colon G \to W$ a continuous ...
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0answers
35 views

Upper central series, semidirect products and a lifting property (argument verification)

Let $G$ be a profinite group, and define the lifting property: $(*_p)$ For every extension $1 \to P \to E \to W \to 1$ where $E$ is finite and $P$ is a $p$-group and for every surjective morphism $f\...
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2answers
19 views

Can someone check If this extension is correct?

I had to find $a$ such that $\mathbb Q(2^{1/4}):\mathbb Q$ is equal to $\mathbb Q(\sqrt 2)(a)$ But i found that $a=2^{1/4}$, is that right? For it i used that $\mathbb Q(\sqrt 2) =$ { $a+b\sqrt 2$} ...
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1answer
15 views

Candidate for showing extension is simple

I have $\mathbb Q (i,\sqrt 5)$ and i need to find $a \in \mathbb Q (i,\sqrt 5)$ that $\mathbb Q (i,\sqrt 5) = \mathbb Q (a)$, i have been playing with $\sqrt 5 + i$ but got nowhere, can anyone give a ...
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2answers
29 views

Finding whether an extension is normal

I have $\mathbb Q(i,\sqrt5)$ and i think i found the base for the extension on $\mathbb Q$ as ${ a+b\sqrt5 +ci + di\sqrt5 ; a,b,c,d in \mathbb Q}$ But i don't know at which polynom to look in order to ...
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0answers
79 views

How to compute sheaf cohomology of a classifying space?

David Wigner showed that the group cohomology invented by Calvin Moore can be related to sheaf cohomology when the coefficient is discrete by constructing a locally constant sheaf on the classifying ...
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1answer
54 views

Algorithms to determine the explicit forms of possible group extensions

I learned from link 1, link 2 link3 that sometimes it is possible to write down all possible group extension. I also know that the group extension is classified by group cohomology. My questions are: ...
3
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1answer
129 views

Baer sum of two extensions

As we all know, all extensions of $Z/p$ by $Z/p$ are split extensions and following extensions:$$\epsilon_{i}:0\rightarrow Z/p\rightarrow Z/p^{2} \xrightarrow{i} Z/p \rightarrow0$$. Now the question ...
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0answers
19 views

Determine possible $p$-groups from center and quotient

Consider the following situation: I have given a finite $p$-group $P$ (in the case I am interested in $p = 2$) with cyclic center $Z(P)$ and I also know the structure of the quotient $P/Z(P)$ (which ...
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0answers
37 views

Generated a non-Abelian group from a Lie group and nontrivial $\mathbb{Z}_2$ action

Let a rank-3 matrix representation of a group $G$ generated by two group elements. Say one is a group as special unitary SU(2), $$ A=\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\frac{\theta}{2}...
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Computing extension group of finite commutative group schemes over a field

Let $k$ be a perfect field with $\text{char} k=p>0$. Then the category of finite commutative group schemes over $k$ (denoted by $FC/k$) is abelian. It seems that $FC/k$ does not have enough ...
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0answers
83 views

Proof of the Schur–Zassenhaus theorem without using homological algebra?

I am currently working on a masters project in which I prove the Schur-Zassenhaus theorem using the classification of group extensions. This classification relies on homological algebra and to my ...
2
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1answer
120 views

Non-split central extension of Z by a finite simple non-abelian group

Is there a non-split central extension of an infinite cyclic group by a finite simple non-abelian group? I have tried, very naïvely, I know, to just take a presentation of $A_5$, say $\langle x,y|x^2=...
2
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1answer
69 views

Extensions of finite groups by compact Lie groups

Let $K$ and $Q$ be discrete groups, with $K$ abelian. Central extensions of $Q$ by $K$, i.e. short exact sequences $1 \to K \to G \to Q \to 1$ such that $K$ is lies in the center of $G$, are ...
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0answers
49 views

Fundamental group of $\mathbb{R}P^2\sharp \mathbb{R}P^2$

Note that $ \langle a,b|a^2b^{-1}a^{-2} b \rangle =\pi_1 ( X):=G$ where $X=\mathbb{R}P^2\sharp \mathbb{R}P^2$. So $a^2,\ b$ commute in $G$ so that $G$ contains $H:=\mathbb{Z}^2$. In further note ...
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1answer
175 views

Largest abelian subgroup

Consider a group $G$ which is generated by three element $a,\ b,\ c$, i.e. $G= (a)\ast (b)\ast (c)$ and $(a)=(b)=(c)=\mathbb{Z}$. Assume that $N$ is a ${\bf smallest\ normal\ subgroup}$ containing $c^...
3
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2answers
173 views

central extension of $\mathbb Z_3$ by $\mathbb Z_3$

Suppose we are interested in all possible central extensions of $\mathbb Z_3$ by $\mathbb Z_3$, i.e. in all possible groups $E$ satisfying a short exact sequence $1 \to \mathbb Z_3 \to E \to \mathbb ...
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1answer
111 views

Extensions to a homomorphism

This is the problem I've been working on. I have some ideas on parts (a) and (b), but I'm not sure if that's correct or not. Please double check my work. Also, any hints on part (c) will be ...
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1answer
283 views

Central extensions versus semidirect products

Consider an extension $E$ of a group $G$ by an abelian group $A$. $$1 \to A \overset{\iota}{\to} E \overset{\pi}{\to} G \to 1$$ Two special kinds of extensions are: Central Extensions: $A$ is ...
2
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1answer
159 views

group extension with finite abelian group

I am a student majoring physics. The question below may be simple but confuse me. Thanks for any suggestion or detailed answer. Given two finite abelian group $N$ and $A$, the question is how many ...
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0answers
32 views

Group extensions rational matrices by infinite cyclic squared

Are there any nontrivial group extensions : $0 \rightarrow \mathbb{Z}^2 \rightarrow G \rightarrow Gl_2(\mathbb{Q}) \rightarrow 0 $ ? Any referall to literature is greatly welcomed. Thank you for all ...
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1answer
125 views

Example of disconnected Lie groups with finite extensions

I am trying to understand better the connectedness, simply-connectedness, and disconnectedness of Lie groups $L$ have anything to do with giving a nontrivial finite extension such that $G/N=L$ with $G$...
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0answers
70 views

Are there any non-trivial group extensions of $SU(N)$?

Are there any non-trivial group extensions of $SU(N)$? If not, can one show/prove there are no non-trivial group extensions of $SU(N)$? It is possibly partial related to the homotopy group ...
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2answers
106 views

Examples of group extension $G/N=Q$ with continuous $G$ and $Q$, but finite $N$

Can we have some (new) examples of group extensions $G/N=Q$ with continuous (e.g. Lie groups) $G$ and $Q$, but a finite discrete $N$? Note that $1 \to N \to G \to Q \to 1$. What I know already ...
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2answers
258 views

Central extension and direct product

Let $G,H,K$ be finite groups and suppose $G$ is a central extension of $H$ by $K$. The question is that under which condition on this extension we will have $G \cong K \times H$.
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57 views

“Higher-order” group extensions

Let $A$, $B$, and $C$ be Abelian groups (more generally, $R$-modules. Even more generally, objects of some Abelian category. I will suppress any decorations on my $\text{Ext}$ groups.) I'm interested ...
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0answers
83 views

Group with unique Involution is determined by a certain Quotient

Let $G$ be a finite group with a unique involution $u$ and let $Q = G/\langle u \rangle$. I am looking for sufficient conditions on $Q$ under which the isomorphism type of $G$ is uniquely determined ...
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139 views

Residually finite extension of a finite group

Do you know how to prove that the extension of a finte group by a residually finite group is residually finite? I know that there is a result of Mal'cev proving something stronger, but I cannot find ...
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1answer
86 views

Meaning of $\phi_h$ in outer semidirect product

In the definition of the outer semidirect product it is usually said that there is a group homomorphism $\phi: H \to Aut(N)$ from the elements of $H$ to the automorphism group of $N$. I get that. I ...
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2answers
84 views

Are extensions quasi-isometric

Let $0 \rightarrow K \rightarrow G_1 \rightarrow L \rightarrow 0 $ $$$$ $0 \rightarrow K \rightarrow G_2 \rightarrow L \rightarrow 0 $ $$$$ be two group extensions with $G_1,G_2$ finitely generated. ...
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0answers
59 views

Isomorphic Group Extensions

I need to make a statement which I think should be an elementary group theory fact. Here is the setup. I have two exact sequences of groups: $$ 1\longrightarrow \ker\varphi_1\longrightarrow G\...