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        Questions about the branch of abstract algebra that deals with groups.

        3
        votes
        0answers
        49 views

        It there a nice way to describe the structure of Malcev-complete groups?

        Let $\mathbb k$ be a field of characteristic zero. The grouplike functor $\mathbb G$ from complete Hopf algebras to groups is a faithful functor. Its image is the category of Malcev-complete groups ...
        3
        votes
        1answer
        132 views

        Finite dimensional compact abelian group that is not a product of connected and a totally disconnected

        Let $G$ be a compact abelian group. A compact abelian group is said to have dimension $n$ if $\dim_\mathbb{Q} \mathbb{Q}\otimes \hat G = n$. Equivalently one can show that this holds if $G$ is ...
        11
        votes
        3answers
        430 views

        Which partitions realise group algebras of finite groups?

        Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$). Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...
        3
        votes
        0answers
        97 views

        Induced Homomorphism on Cohomology of Symmetric Group 3

        For the symmetric group $S_3$, there is an inclusion $i:\mathbb{Z}/3\mathbb{Z}\hookrightarrow S_3$. How can I assert that the induced homomorphism $$i^{\ast}:H^{n}(S_3,\mathbb{Z})\rightarrow H^{n}(\...
        1
        vote
        1answer
        173 views

        A generalization of Landau's function

        For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$ the least common multiple of all ...
        2
        votes
        1answer
        69 views

        Why Triangle of Mahonian numbers T(n,k) forms the rank of the vector space?

        I am looking for an explanation of why Triangle of Mahonian numbers T(n,k) form the rank of the vector space $H^k(GL_n/B)$? With respect to the property of Kendall-Mann numbers where the statement ...
        16
        votes
        1answer
        243 views

        Gluing hexagons to get a locally CAT(0) space

        I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space: The first two give the torus and the Klein bottle, respectively. What are the ...
        10
        votes
        1answer
        341 views

        Is there a name of semidirect product of a group with its automorphism group?

        Consider the construction $G \rtimes \text{Aut}(G)$. Here $ G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action. 1) Is there any name ...
        2
        votes
        0answers
        96 views

        How large can a symmetric generating set of a finite group be?

        Let $G$ be a finite group of order $n$ and let $\Delta$ be its generating set. I'll say that $\Delta$ generates $G$ symmetrically if for every permutation $\pi$ of $\Delta$ there exists $f:G\...
        6
        votes
        1answer
        230 views

        Understanding the functoriality of group homology

        EDIT: I've decided to rephrase my question in order for it to be more concise and to the point. Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of ...
        12
        votes
        1answer
        289 views

        Is $\text{PSL}_2(\mathbb{F}_{p^m})$ known to be a Galois group over $\mathbb{Q}$ for $m>1$?

        Let $\mathbb{F}$ be a finite field of characteristic $p$, is it known that $\text{PSL}_2(\mathbb{F})$ can be realized as a Galois extension of $\mathbb{Q}$ for any/all cases when $\mathbb{F}$ is not $\...
        5
        votes
        1answer
        272 views

        Can a Shelah semigroup be commutative?

        A semigroup $S$ is called $\bullet$ $n$-Shelah for a positive integer $n$ if $S=A^n$ for any subset $A\subset S$ of cardinality $|A|=|S|$; $\bullet$ Shelah if $S$ is $n$-Shelah for some $n\in\...
        0
        votes
        1answer
        50 views

        About generator of minimal length coset representatives

        Let $(W, S)$ be a Coxeter system. Let $J \subseteq S$ and recall that $W_J = \langle s: s \in J\rangle \subseteq W$. Define $W^J = \{w \in W : \ell(ws) > \ell(w)\ \text{for all } s \in J\}$. ...
        6
        votes
        1answer
        213 views

        Subsets of a group with special property

        Let $G$ be a finite group. We say a subset $A$ of $G$, $|A|=m$, is $(m,i)$-good, $m\geq 1$ and $0\leq i\leq m$, if there exist $g_A\in G$ such that we have $|gA\cap A|=m-i$. I need some groups such ...
        9
        votes
        0answers
        79 views

        Can every non-inner automorphism of a group with residually finite outer automorphisms be realised as an non-inner automorphism of a finite quotient?

        First some motivation: most proofs that show that the group of outer automorphisms is residually finite do not only show that the subgroup of inner automorphisms is closed in the profinite topology, ...

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