# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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### 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 \...
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 ...