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1) The classification of p-groups (even p-groups of class 2) is indeed wild over $\mathbb{F}_p$ [V. Sergeichuk, The classification of metabelian p-groups (Russian), Matrix problems, Akad. Nauk Ukrain. …

One can make the argument by wildness much more concrete than in the previous answer: Sergeichuk ["Classification of metabelian p-groups", in: Matrix problems, Inst. Mat. Ukrain. Akad. Nauk, Kiev, 19 …

There has been some recent progress on algorithms for this problem. Das & Thakkar STOC '24 give the following algorithms: For groups with no abelian normal subgroups, given by a generating set of per …

Consider the coboundary matrix $C^1(G, \mathbb{Z}) \to C^2(G, \mathbb{Z})$ of the normalized bar resolution of $G$ with coefficients in the trivial $\mathbb{Z}G$-module $\mathbb{Z}$. That is, thinking …

Is there an algorithm which halts on all inputs that takes as input a finite group ($p$-group if you like) and outputs a finite presentation of the cohomology ring (with trivial coefficients $\m …

I agree w/ Holt's technical statements (not sure whether I agree about his guess on the final running time, though I agree about which groups are likely to be hardest). But I wanted to add that a lot …

If the group is given by its multiplication table then there are polynomial-time algorithms for all the tasks you mentioned (over $\mathbb{C}$; these don't handle modular representations). Babai and …

The Lazard Correspondence is often phrased (for simplicity) for $p$-groups of nilpotence class $c < p$, but it works more generally whenever every 3-generated subgroup has nilpotence class $< p$, and …