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Let $ G$ be a connected second countable compact Hausdorff group, and let $ H\subset G$ be a closed subgroup such that the homogeneous space $ G/H$ has continuum cardinality. For every $ x\in G/H$ let $ H_x\subset G$ be the stability subgroup at $ x$ . Question: Does there exist a toral subgroup $Read more

Consider the alternating subgroup $ A_n$ of the symmetric group $ S_n$ (or in general any Coxeter Group). Is there a simplicial complex whose $ h$ -vector $ h_i$ equals the number of elements of $ A_n$ with $ i$ descents? Note that the $ h$ -vector of the Coxeter Complex of $ S_n$ countsRead more

Let $ E$ be an elliptic curve over $ \mathbb{Q}_p$ . I was told that if $ F$ is the smallest Galois extension over $ \mathbb{Q}_p$ then the inertia subgroup $ I_{v’}$ of $ G_{F/\mathbb{Q}_p}$ is a subgroup of the automorphism group of $ \widetilde{E}$ . Here $ \widetilde{E}$ is the elliptic curve reduced modRead more

Let $ G$ be a finite group. Suppose $ K>1$ is a normal proper subgroup of $ G$ with $ K$ abelian. Then we have a new group $ \widetilde{G}=G/K\ltimes K$ . QUESTION: $ G\cong\widetilde{G}$ ? If it is wrong, then what is the relationship on these groups. I want know some information (papers, booksRead more

I’m interested in computing the cohomological dimension of the commutator subgroup $ [P_n,P_n]$ of the pure braid group $ P_n$ . I wasn’t able to find a reference in the literature. Because $ [P_n,P_n]$ has an abelian subgroup of rank $ \lfloor(n-1)/2\rfloor$ we have$ $ (n-2)/2\le\text{cd}([P_n,P_n])\le n-2.$ $ My guess is that in fact $Read more

We note that group $ A$ is a $ G$ -group (looking at $ G$ as a set), if for each $ g \in G$ and each $ a_1 \in A$ , there is $ {a_1}^g \in A$ such that for all $ a_1 , a_2 \in A$ $ (a_1 a_2)^g = {a_1}^g {a_2}^g$ .Read more

Let $ p$ be a prime. Let $ K/\mathbb{Q}_p$ be a finite extension and $ \mathcal{O}=\mathcal{O}_K$ its ring of integers. Let $ A_K$ be an abelian variety and $ \mathcal{A}_\mathcal{O}$ denote its Neron model over $ \mathcal{O}$ . We don’t assume that $ \mathcal{A}_\mathcal{O}$ has good or semistable reduction. Let $ \ell$ be a primeRead more

Let $ S_n$ denote the symmetric group on $ n$ symbols. Given a subgroup $ H\le S_n$ and a particular cycle structure, is there a relatively efficient way to compute the number of elements of $ H$ having that cycle structure?Read more

I need to find all the subgroups of order 4 in $ \mathbb{Z}_4 \oplus \mathbb{Z}_4$ . I am unsure on how to proceed with this and be thorough with it. I’m having issues getting started and knowing where to go.Read more