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From the Hiller-Rhea formula $$|\operatorname{Aut} H_p| = \prod_k (p^{d_k} - p^{k-1}) \prod_j (p^{e_j})^{n-d_j} \prod_i (p^{e_i-1})^{n-c_i+1},$$ given an abelian $p$-group of type $p^{e_1}\cdots p^{e_ …

If you don't specify more about the structure of the field $K$, then we can't say much about the structure of the group $E(K)$. There are special cases (described in the Wikipedia article): If $K$ i …

Given a finite group, the sums of squares of dimensions of irreducible representations add up to the order of the group, so the dimension of an irreducible representation is at most the square root of …

As I mentioned in a comment, Question 2 (in its revised form) has a negative answer, because odd natural numbers have unbounded abundancy index, while the Odd Order Theorem implies all groups of odd o …

Google gives the following paper: Greenlees, Tate cohomology in axiomatic stable homotopy theory. It gives a definition of the Tate construction using Bousfield localization and completion, and has s …

It is possible that Conway was referring to the generic construction that works for all finite groups equipped with faithful representations, given in the other answers. However, I think it is more l …

The colon means "semidirect product", but it does not specify which semidirect product. This notation is a concise shorthand that gives important structural information without necessarily uniquely s …

For any $n>1$, the lower central series for the symmetric group is $S_n > A_n\geq A_n \geq A_n \geq \cdots$, so the Lie ring formed by the sum of successive quotients is the group $\mathbb{Z}/2\mathbb …

Yes. The basic representation of $E_8$ has character $j(\tau)^{1/3} = q^{-1/3}(1+248q+4124q^2 + \cdots)$, and the 4124 decomposes as $1+248+3875$. By Frenkel-Kac-Segal, the basic representation has …

The answer to your question is Yes. Consider your covering group $C$ as a central extension: $$1 \to N \to C \to G \to 1$$ and suppose it is given by a 2-cocycle $\alpha \in H^2(G, N)$. Then for any …

You can derive bounds in a straightforward way from the complete list of finite simple groups in Wikipedia, which has references (particularly to ATLAS). If you look through all of the cases (Note: I …

Let $p$ be a prime, let $\mathbb{Z}_p$ be the ring of $p$-adic integers, and let $G$ be a cyclic group of order $p$. It is rather well-known that finite rank $\mathbb{Z}_p$-free representations of $G …

As others have mentioned, there are many CFTs, but we can narrow down our list by looking at conditions that select for interesting automorphism groups. Perhaps the easiest is to consider holomorphic …

Your observation, which was expanded into a concrete conjecture last year by Harvey and Rayhaun, is now a theorem. See M. Griffin, M. Mertens, "A proof of the Thompson Moonshine conjecture". This …

This is the unique Lemma in section 3.5 of Waterhouse's "Introduction to Affine Group Schemes". It only requires that $G$ be an affine group scheme over a field.