Articles of group extensions

Is the embedding problem with a cyclic kernel always solvable?

This question comes from this question by user72870. I shall explain how it relates to that question at the end. Let me shortly define my question: We call an embedding problem a diagram of the form: $$\begin{matrix}&&\mathfrak G\\&&\downarrow\\G&\rightarrow&\Gamma\end{matrix},$$ where $G\rightarrow\Gamma$ is a group extension (therefore we assume that the homomorphism is surjective), and $\mathfrak G$ […]

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 property. Or one can try to argue from the exact sequence. Proof/Show: Let us call $Q=SU(N)$. If the above claim is true, namely, we cannot […]

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 contains: $$SU(2)/\mathbb{Z}_2=SO(3).$$ $$\frac{\mathbb{R}/{\mathbb{Z}}}{\mathbb{Z}_n}={\mathbb{R}}/{(n\mathbb{Z})}.$$ What else are the examples that you can provide? A systematic answer to obtain […]

Describe all extension groups of a given subgroup $H \trianglelefteq$ Aff$\mathbb{(F_q)}$ by Aff$\mathbb{(F_q)}/H$

Let $\mathbb{F_q}$ be a finite field. Consider the group Aff$\mathbb{(F_q)}$ Aff$\mathbb{(F_q)} := $ $ \ \begin{Bmatrix} \begin{pmatrix} a&b\\ 0&1\\ \end{pmatrix} \colon a, b \in \mathbb{F_q}, a \neq 0 \end{Bmatrix} $. Let $H := $ $ \ \begin{Bmatrix} \begin{pmatrix} 1&b\\ 0&1\\ \end{pmatrix} \colon b \in \mathbb{F_q} \end{Bmatrix} $ be a normal subgroup of Aff$\mathbb{(F_q)}$, which is […]

Splitting in Short exact sequence

I am trying to find whether $\{1\}\longrightarrow\mathbb{Z}\longrightarrow\mathbb{R}\longrightarrow\mathbb{R}/\mathbb{Z}\longrightarrow \{1\}$ splits. My conjecture is it is not as we cannot find a non-zero group homomorphism from $\mathbb{R}/\mathbb{Z}$ to $\mathbb{R}$. If my conjecture is correct, can we tweak the above sequence so that it splits?

Conjugacy classes in group extensions

Consider a normal subgroup $N$ of a finite group $G$. How are the conjugacy classes of $G$ related to $G/N$ and $N$?

Lie Group Decomposition as Semidirect Product of Connected and Discrete Groups

I’ve believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in this Math Overflow thread, it is mentioned that this is not true, and that there exist obstructions to this decomposition. Unfortunately, the level of complexity in […]