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.
279
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Understanding how group cohomology classifies extensions using the derived functor point of view
I am rereading some material about group extensions, in particular because I needed to recall the formula
$$H^2(G;A)\cong \mathcal{E}(G;A).$$
We have that $G$ is some group acting on an abelian group $...
- 1,495
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Vanishing of Ext involving unipotent group schemes and $\mathbb{G}_m$
I am currently reading Serre's Algebraic Groups and Class Fields. In chapter 8.1.6, Prop. 7 says that if $A$ and $B$ are linear algebraic groups then $H^2_{reg}(A,B)_s=Ext(A,B)$. The definition of $H^...
- 111
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Sum of nth power of some of the roots of irreducible polynomial over $ \mathbb{Q}$ is in $ \mathbb{Q}$
So i know that for a splitting field K over $ \mathbb{Q}$ of the polynomial f(x), where a,b,c,d are the roots of f(x). Taking the following sum $ a^n +b^n+c^n +d^n $ is in the FixGal(K,$ \mathbb{Q}$) ....
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Computing action on cohomology group induced by conjugation
Consider the extension $1 \to C_3 \to S_3 \to C_2 \to 1$. I am trying to see intuitively why the action induced by the conjugation action $C_2 \curvearrowright C_3$ on $H^{\bullet}(C_3, \mathbb{Z})$ ...
- 412
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45
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Is the group of 4 by 4 nilpotent triangular matrices a semidirect product?
Let $N$ be the Lie group of 4 by 4 nilpotent triangular matrices with 1 on the diagonal. Let us denote by $E_{ij}$ the square matrix with entry 1 where the $i$th row and $j$th column meet, all the ...
- 123
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Does the extension of L3(5) in the Lyons group split
According to the ATLAS of Finite groups, the sporadic Lyons group has a subgroup 53.L3(5), and the notation specifies that this extension does not split. However, according to the Wikipedia page on ...
- 171
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Is the involution centralizer in the Held group a split extension
According to the ATLAS of Finite groups, the centralizer of a 2B element in the sporadic Held group is 21+6.L3(2). According to the introductory section "How to read this ATLAS: Information about ...
- 171
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Extensions of metacyclic groups
A group $G$ is metacyclic if has the following exact sequence
$$1 \to N \to G \to K \to 1$$
where $N$ and $K$ are cyclic groups. In the split extensions, the wikipedia says that direct and semidirect ...
- 422
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Extensions of $G$-modules parametrized by $H^1$
Let $G$ be a finitely generated group and let $V$, $W$ be one-dimensional representations of $G$ over $\mathbb{F}_q$. (I guess we can think of $V$ and $W$ simply as $G$-modules, which are isomorphic ...
- 3,270
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How to show that 1-dimensional affine group over F is a split extension of subgroup of translations by an abelian subgroup.
I am working on this problem where first it was asked to show that the set of translations $t_{1\beta}, \beta \in F$ forms a transitive normal abelian subgroup $T$ of $AGL_1(F)$. This was trivial, I ...
- 125
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If the group extension is torsion-free, what about the quotient group?
Suppose that
$$1 \to H \to G \to K \to 1$$
is a short exact sequence (which means that $G$ is an extension of $K$ by $H$). Assume that $G$ is torsion-free. Is $K$ torsion-free? It seems that is not ...
- 422
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Group Structures of a Group Extension
Let $A,G$ be groups with $A$ abelian, and $0 \to A \xrightarrow i E \xrightarrow \pi G \to 0$ a group extension. We denote the induced action of $G$ on $A$ as $g \cdot a$. A paper that I am reading ...
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Group extensions of a nontrivial group [closed]
A group $G$ is called an extension of a group $Q$ by $N$ if we have the following short exact sequence: $1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$. If there is a homomorphism $f: G \...
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Group extension analogous to the symmetric group.
Recently, My Professor taught us about group extension. It is the following: A group $G$ is an extension of $Q$ by $N$ if we have the following short exact sequence:
$1 \rightarrow N \rightarrow G \...
- 149
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Equivalent Definition of Solvable Groups in Serre’s Book
In Serre's Linear Representations of Finite Groups, Section 8.2, the following claim is given, as equivalent to a definition of solvable groups.
Solvable groups. One says that G is solvable if there ...
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