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Just so we agree on the setup, you have an exact sequence $$ 1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1, $$ where $K\leq Z(G)$, and you assume that the extension is non-split. You are askin …

You can make $[N:H]$ as big as you want: start with an arbitrary group $E$ and non-normal subgroup $H$ (e.g. $E=C_p\rtimes C_2$) dihedral of order twice a prime $p$, and $H=C_2$; for every $x\not\in H …

Note that $\text{Inn}(X)$ is isomorphic to $X/Z(X)$. So your requirement is equivalent to $X$ being simple modulo the centre. For example $SL_n(\mathbb F_{p^m})$ satisfies this, since the centre is gi …

This should have been a comment, but got too long. Even if we assume that everything is semi-simple and that the field is algebraically closed, as you seem to be doing, the scalars you are talking ab …

The answer is "no", since for every sufficiently large prime $p$ there are simple non-trivial $\mathbb{F}_p[{\rm PSL}_2(\mathbb{F}_{2^f})]$-modules. You can take $N$ to be any such module and form the …

See Gorenstein, Finite Groups, Chapter 5, Theorem 4.10. Such groups are as follows: if $p$ is odd, then $G$ is cyclic; if $p=2$, then $G$ is either cyclic, or generalised quaternion of order $2^l$ f …

I think, the article by Vahid Dabbaghian-Abdoly, Journal of Symbolic Computation Volume 39, Issue 6, June 2005, Pages 671-688, entitled "An algorithm for constructing representations of finite groups" …

To answer the question about natural examples/classification, Tim Dokchitser and I have completely classified all such sets in the following sense. If $\tilde{G}\leq G$, and $M$, $N$ are two $\tilde{ …

Here is the MAGMA code to generate your group: G:=sub<Sym(12)|(1,3,5,7,9,11)*(2,4,6,8,10,12), (2,12)*(3,11)*(4,10)*(5,9)*(6,8), (2,3)*(5,6)*(8,9)*(11,12)*(4,10)>; I have a small function written by …

If I understood Joel David Hamkins's explanation correctly, then the problem of classifying finite ($p$-)groups is smooth for silly reasons. Attach to each group the following datum: randomly number t …

For any field $K$, finite-dimensional simple $K$-algebras (simple in the sense of having no proper non-zero two sided ideals) are famously classified by a theorem of Wedderburn. More generally, simple …

In the infinite family $G_p=(\mathbb{Z}/p\mathbb{Z})\rtimes (\mathbb{Z}/p\mathbb{Z})^{\times}$, as $p$ runs over prime numbers, the bound is tight up to $O(1)$, and the representation in question is f …

The answer is no. Counterexamples include the Weyl groups of types E6, E7, and E8. For a proof that the representations of these Weyl groups are indeed all realisable over $\mathbb{Q}$ (equivalently o …

$\DeclareMathOperator{\SL}{SL}\DeclareMathOperator{\GL}{GL}$To determine the real representations of a finite group, it suffices to determine the complex irreducible representations and their Schur in …

This is the second follow-up to this question on square roots of elements in symmetric groups and is concerned with generalisations to $n$-th roots. Let $G$ be a finite group and let $r_n(g)$ be the n …