Questions tagged [group-extensions]

This group is for questions relating to "group extensions", a general means of describing a group in terms of a particular normal subgroup and quotient group.

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29 views

Algebraic Extensions of Z_7 with polynomials?

I am studying fields and rings and I came across this statement in a textbook that I am having trouble visualising; $\mathbb{Z_7}/\left<x^2-3\right>$ is an algebraic extension of $\mathbb{Z_7}...
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3answers
76 views

Finite index subgroups of group extensions such that the quotient map is injective

Suppose we have two groups $N$ and $G$. A group extension of $G$ by $N$ is a group $E$ that fits into the short exact sequence $0 \rightarrow N \rightarrow^i E \rightarrow^s G \rightarrow 0.$ A ...
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1answer
63 views

$M<K\rtimes H$ is a semidirect product?

Let $H,K$ be two finite groups, $K$ abelian, and let $M$ be a subgroup of $K\rtimes H$. Consider the projection $\pi:K\rtimes H \rightarrow H$ on the Second factor. Let us suppose that $\pi(M)=H$. ...
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1answer
50 views

Centraliser in a group extension

Given a group $N$ with a subgroup $D \leq N$, for an automorphism $x \in {\rm Aut}(N)$ such that $x$ normalises $D$ but no power of $x$ centralises $D$, before yesterday I would have thought that if ...
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1answer
99 views

For Every Representation of a Subgroup is there a Group such that the Group Representation is Irreducible?

Let $H$ be a finite group, Let $V$ be a finite dimensional vector space over $\mathbb{C}$, and $\rho$ a representation of $H$ on $V$. Is there, for any representation $\rho$ of $H$ on $V$, another ...
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12 views

Inönü-Wigner contraction of Poincaré $\oplus$ $\mathfrak{u}$(1)

Metric = (-+++), complex $i$'s are ignored. Using the following decompositions of the Poincaré generators, I can write the Poincaré algebra as I can get the Galilei algebra using the following ...
3
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1answer
25 views

Extension-algebras of $A_4$

Consider the quiver $Q\colon 1\xrightarrow{\alpha} 2\xrightarrow{\beta} 3\xrightarrow{\gamma} 4$ and the algebra $A=k[Q]/(\gamma\beta\alpha)$. Denote the simple $A$-modules by $L(-)$ and let $M$ be ...
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3answers
116 views

The only group $G$ with one $A$ and one $B$ as composition factors is $G = A\times B$ (where $A$ and $B$ are non-abelian, finite and simple)

Is it true that if $A$ and $B$ are two non-abelian finite simple groups, then the only finite group $G$ which has one copy of $A$ and one copy of $B$ as composition factors is $G = A \times B$? If not,...
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30 views

$A_5$ or the icosahedral group $I$ is not isomorphic to any subgroup of the binary icosahedral group $2I$

I'm trying to show that the extension $1 \to \{\pm 1\} \to 2I \to I \to 1$ does not split. For that, I think it's sufficient to show that $I$ is not isomorphic to any subgroup of $2I$ (?). But I'm not ...
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28 views

Is the direct product $H \times K$ necessarily a central extension of $H$ by $K$ or $K$ by $H$?

As far as I know, a central + split extension of a group $K$ by a group $H$ is uniquely the direct product $H \times K$. But is the other direction true as well? That is, is any direct product ...
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1answer
57 views

Why does every $\varphi: K \to \mathrm{Out}(H)$ determine an unique extension of $H$ by $K$ when $Z(H) = 1$?

Every homomorphism $\varphi: K \to \mathrm{Out}(H)$ determines an unique extension of $H$ by $K$. Why is this true for groups $H$ with a trivial center? Even if we only consider split extensions, as ...
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138 views

Extensions of $A_5$ by $C_2$

I'm attempting the following exercise: Prove that there are only three finite groups whose composition factors are $A_5$ and $C_2$. The direct product $A_5 \times C_2$. The symmetric group $S_5$. A ...
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1answer
51 views

Is group extension unique?

Definition: Given groups N and H, a group is said to be an extension of H by N if there exists $N_{0}\vartriangleleft G$ such that $N_{0}\cong N$ and $G/N_{0}\cong H$ We know that a normal subgroup of ...
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15 views

Explicit two-cocycle for central extension of $\mathbb{Z}_2 \hookrightarrow Spin(n) \twoheadrightarrow SO(n)$?

I was wondering if there was a simple expression for a two-cocycle representing the central extension central extension of $\mathbb{Z}_2 \hookrightarrow Spin(n) \twoheadrightarrow SO(n)$. It would ...
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13 views

Sufficiency of $c - d(f) = 0 \ \forall x + y < n$ (2-cocycles and 2-coboundaries)

In this answer, could someone please tell me how to show that proving $c - d(f) = 0 \ \forall x + y < p$, where $c: \Bbb Z/p \times \Bbb Z/p \to \Bbb Z/p$ (and $c$ is a 2-cocycle) and $d$ is a 2-...
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1answer
34 views

Relating a 2-cocycle to specific section of a central extension

I'm trying to understand from this answer how $\chi: \Bbb Z/p\Bbb Z \times \Bbb Z/p\Bbb Z \to \mathbb{Z}/p\mathbb{Z}$ (defined as $\chi(x, y) = 0 \ \forall x + y < p$ and $\chi(x, y) = 1 \ \forall ...
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1answer
33 views

Some clarifications required about the two extremes of general extensions (semi-direct products and central extensions)

This is a sequel to Why can the homomorphism $\phi$ in semi-direct products only be varied by inner automorphisms upon changing the complement group? My professor made another remark that: Let's go ...
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30 views

Why can the homomorphism $\phi$ in semi-direct products only be varied by inner automorphisms upon changing the complement group?

My professor made the following remark while teaching about group extensions: We want to classify finite groups in a manner similar to the fact that every positive integer is uniquely a product of ...
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1answer
37 views

Extensions of $A_5$ by $C_2$.

Recently I've came to result that, if $H$ is a simple group, every homomorphism $\theta :K\rightarrow \mathrm{Out}(H)$ determines an unique extension of $H$ by $K$. As an example, I tried to find all ...
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116 views

Is $Ext^1_{Lie-Gr}(\mathbb C, \mathbb C^*)=0$?

I want to know if any Lie group extension $$1\to \mathbb C^*\to A\xrightarrow{\pi} \mathbb C\to 0 \tag{1}\label{1}$$ is trivial, where the group structure is multiplicative on $\mathbb C^*$ and ...
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1answer
58 views

Schur-Zassenhaus Theorem

Im a reading Mac Lane's Book on Homology and now he wants to prove the Schur-Zassenhaus Theorem. If the integers $m$ and $n$ are relatively prime, any extension of a group of order $m$ by one of ...
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30 views

Obstruction to extensions

I am reading MacLane's Book on Homology and in page $126$ he is talking about obstruction to extensions. My problem is when he says Suppose now that just the abstract kernel $(C,G,\psi)$ is given. ...
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49 views

Are there any references on extensions $G$ of a cyclic group $C_2$ by $2$-groups $P$?

Are there any references on extensions $G$ of a cyclic group $C_2$ by $2$-groups $P$ such that $1\neq a\in C_2$ is a square element in $G$? In other words, if $G/{C_2}\cong P$, where $P$ is a 2-group,...
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28 views

Correct Way of Writing Extension of Group

Let $G=C_2\times C_m=\{(g^{t_1},h^{t_2})\mid t_1\in \{0,1\}, t_2\in \{0,1,...,m-1\}\}$. Is it correct to just say that $G$ is an extension of $C_m$ by $C_2$? Or it is more accurate to say that $G$ ...
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1answer
33 views

MacLane's Homology Book exercise doubt

I have been reading MacLane's book on Homological Algebra, and in chapter 3 section 2 exercise 1 , he asks to do the following For abelian groups, show that a normalized function on $C \times C$ ...
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1answer
60 views

Center of a split metabelian group of order $p^nq$ is direct summand.

Let $p$ and $q$ be odd primes such that such that $q \ | \ p-1.$ Suppose we have a group $G$ which is a split extension of the cyclic group $B$ of order $q$ by a finite abelian $p$-group $A.$ Can we ...
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84 views

Isomorphism of $\mathbb{Z}\ltimes_A \mathbb{Z}^m$ and $\mathbb{Z}\ltimes_B \mathbb{Z}^m$

Let $A$ and $B$ be matrices of finite order with integer coefficients. Let $n\in\mathbb{N}$ and let $G_A=\mathbb{Z}\ltimes_A \mathbb{Z}^n$ be the semidirect product, where the action is $\varphi(n)\...
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35 views

Concise notation for the central extension defined by a cocycle

Any extension of a group $G$ by the Abelian group $A$ is determined (up to isomorphism) by $\varphi$ and $f$, where $\varphi:G\times A\to A$ is a group action of $G$ on $A$ $f:G\times G\to A $ is a ...
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1answer
50 views

Calculate $H^{2}(\mathbb{Z}_{2} ; Z(T))$

I would like to find how many group extensions of $T$ by $\mathbb{Z}_{2}$, where $T$ is the tetrahedral group (in the sense of the definition in https://en.wikipedia.org/wiki/Group_extension). I was ...
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1answer
63 views

Understanding group extensions

This question is about understand what is the intuition behind the following Definition: An extension of a group $G$ by the group $A$ is given by an exact sequence of group homomorphisms $$1\...
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1answer
88 views

Why are group extensions and split extensions defined the way they are?

I would have expected definitions of "group G extends group F" to capture the idea that elements of $G$ can be described by elements of $F$ plus some remainder. For example Foo-extensions A group $G$ ...
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1answer
42 views

Number of embeddings $F_q^n \rightarrow F_q^{n+m}$

Let $F=F_q$ be a finite field where, $q$ is a power of prime. I wish to compute the number of extnesions $$ 0 \rightarrow F^m \rightarrow F^{n+m} \rightarrow F^m \rightarrow 0 $$ This is in ...
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1answer
39 views

Proving this function is an isomorphism

Let $1\longrightarrow H\longrightarrow G\longrightarrow F\longrightarrow 1$ be a short exact sequence with $H$ abelian. Let $T=\{t_{\sigma}\}$ be a transversal of $H$ in $G$. Let also be $Z^{1}(F,H)$ ...
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2answers
54 views

Extension of a group $G$ by a commutative group $F$

Let $F,G$ be two groups. An extension of $G$ by $F$ is a triple $\mathscr{E}=(E,i,p)$, where $E$ is a group, $i:F\rightarrow E$ is an injective homomorphism, and $p:E\rightarrow G$ is a surjective ...
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1answer
36 views

Constructing a group action from a group extension + a section

Let $F,G$ be two groups. An extension of $G$ by $F$ is a triple $\mathscr{E}=(E,i,p)$, where $E$ is a group, $i:F\rightarrow E$ is an injective homomorphism, and $p:E\rightarrow G$ is a surjective ...
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1answer
39 views

A morphism between two extension of a group $G$ by a group $F$ is an isomorphism. (Not sure if the terminology is standard.)

Let $F,G$ be two groups. An extension of $G$ by $F$ is a triple $\mathscr{E}=(E,i,p)$, where $E$ is a group, $i:F\rightarrow E$ is an injective homomorphism, and $p:E\rightarrow G$ is a surjective ...
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1answer
55 views

Galois Groups vs $\mathbb{Q}(\alpha_i)$

Suppose I have $f(x) = (x^2-3)(x^2-2)$. The roots are $\pm\sqrt{3},\pm\sqrt{2}$. So the splitting field of $f$ over $\mathbb{Q}$, which is a Galois extension, is the smallest subfield of $\mathbb{C}$ ...
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72 views

Generalization of Symmetric Group: $S_n$ to ${\bf S'}_n$

We know the symmetric group is defined over any set is the group whose elements are all the bijections from the set to itself. S$_n$ is defined over a finite set of $n$ symbols consists of the ...
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37 views

Torsion of simple abelian group extension

I'm curious if there is a general way to determine the torsion of a group extension. I'm most interested in the simple example where we have a central extension $$ 1 \to \mathbb Z \xrightarrow{f} G \...
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1answer
38 views

splitting of a central extension

I want to prove the following version of the splitting lemma. Let $G,K$ be topological groups. Let $1\to G\xrightarrow{\phi}H\xrightarrow{\eta}K\to1$ be a central extension of $K$ by $G$. Then $\phi$ ...
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0answers
45 views

Isomorphism between free abelian groups that fit into a short exact sequence

Let $A$ and $B$ be free abelian groups and $0\to A\to B\to\Bbb{Z}/p\to 0$ a short exact sequence. Are $A$ and $B$ necessarily isomorphic? The only examples I can come up with are of the form $...
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0answers
44 views

General formula for group extensions by an Abelian group

In an older question I wrote this: It's a well-known fact that if $A$ an Abelian group and $G$ is a group, then all group extension of $G$ by $A$ is isomorphic with the group ($A\times G,\,\...
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0answers
21 views

Lifting outer automorphisms of Lie groups to representations

Let $G$ be a semisimple complex Lie group, $\mathrm{Aut}(G)$ its group of automorphisms, $\mathrm{Inn}(G)$ - group of inner automorphisms (isomorphic to $G$ modulo center), $\mathrm{Out}(G) = \frac{\...
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1answer
47 views

Does every $G$-by-$C_2$ extension split?

Given a group $G$ and a short exact sequence $$1 \longrightarrow G \longrightarrow E \longrightarrow \mathbb{Z}/2\mathbb{Z} \longrightarrow 1$$ does the extension always split? That is, is it always ...
4
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1answer
110 views

Free normal subgroup of an HNN-extension

Suppose $F$ is a finitely generated free group and $a,b$ are not in $F'$ but $b^{-1}a \in F'$. By taking the HNN extension $G=\langle F,t | t^{-1}atb^{-1}\rangle$, is there a way to find a normal free ...
4
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1answer
118 views

Constructing group extensions in GAP.

I have the following general question: Given two finite groups $N$ and $H$, how can we find, using GAP, all the groups $G$ (up to an isomorphism, of course) such that $$1 \rightarrow N \rightarrow G \...
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62 views

Proving associativity of addition with weird carry operation

There is a somewhat famous example of group cohomology witnessing $\mathbb{Z}/100$ as an extension of $\mathbb{Z}/10$ by $\mathbb{Z}/10$, with the standard carry function $c$ as a 2-cocycle (cf. this ...
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1answer
42 views

Splitting field of degree $p(p+1)$ contains a Galois subextension of degree $p$.

I've been studying for an algebra qualifying exam. Any help with the following result would be appreciated. Suppose $E$ is a splitting over $\mathbb{Q}$ of an irreducible polynomial $f(x)\in\mathbb{Q}...
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0answers
111 views

Cocycles and group extensions

I'm trying to understand how elements in the second cohomology group with coefficients in some other group correspond to group extensions. This is what I understand: Suppose we have two (countable) ...
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33 views

Is every non-simple finite group constructible from simple groups of lesser order by extensions?

The question is easy to formulate: Let be $G$ a finite group with some normal subgroup, let's say $N \triangleleft G$ where $N\neq \{1\}$ and $N\neq G$. The question is, there exists an group ...