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A theorem of Borovik, Pyber and Shalev (Corollary 1.6) shows that the number of subgroups of a group $G$ of order $n=\lvert G\rvert$ is bounded by $n^{(\frac{1}{4}+o(1)) \log_2(n)}$. This is essential …

Jesper Grodal and I once looked at this, cf. this answer. In particular Eilenberg and MacLane constructs a universal obstruction in $H^3(\text{Out}(G);Z(G))$ for the extension $1\rightarrow G\rightarr …

The answer to Q1 is yes, the order of $a$ might be smaller than the gcd: Let $G=\langle x,y\mid x^8=y^2=1,x^y=x^3\rangle$ be the semidihedral group of order $16$. Let $a=[x]\in G_{\text{ab}}=C_2\times …

The following is only an answer to the first question: Consider the subgroups $G_1=\langle (12)(34),(13)(24)\rangle$ and $G_2=\langle (12)(34),(34)(56)\rangle$ of $\Sigma_6$ which are both isomorphic …

I'm not a history expert, but according to Miller, Blichfeldt and Dickson: "Theory and applications of finite groups" (1916), footnote p. 46: "The concept of holomorph was used by many early writers, …

As Derek Holt points out, the existence of such an extension is equivalent to the universal cohomology class in $H^3(\text{Out}(G);Z(G))$ being zero. The Eilenberg-MacLane paper is the original refere …

For $G=A_5=\text{PSL}_2(\mathbb{F}_5)$, the minimal dimension of a flat manifold with holonomy group $G$ is 12 15 according to Theorem (V.1) in W. Plesken:``Minimal dimensions for flat manifolds with …

I think the following is an example of $Ш(G,M(G,H,\Bbb{F}_2))\neq 0$: Take $G=A_4$ and $H$ of order $2$. Then $M$ has dimension $5$ and a (computer) calculation shows that $Ш(G,M(G,H,\Bbb{F}_2))$ has …

While the following might not be conceptual, but at least it's simple if one knows group cohomology. The obstruction to splitting lives in $H^2(W;T)$, and we have $H^n(W;T)=0$ for all $n\geq 0$! One w …

The name metacyclic is normally used for a group which is cyclic-by-cyclic (ie. a group $G$ with a cyclic normal subgroup $N$ such that $G/N$ is also cyclic). I will therefore refer to a finite group …

I dont know a proof (other than by inspecting the list of degrees) in the irreducible case. However in the reducible case, a finite real reflection group is not determined up to isomorphism by its deg …

(Too long for a comment.) A Magma computation shows that for $k=\mathbf{F}_p$ with $p$ prime the group $H^1(\operatorname{PSL}_2(k);k^3)$ equals $0$ for $p=3$ and $7\le p\le 17$ while the cohomology g …