Tagged Questions

Questions on group theory which concern finite groups.

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Generalization of a theorem of Øystein Ore in group theory

Theorem (Øystein Ore, 1938): A finite group $G$ is cyclic iff its lattice of subgroups $\mathcal{L}(G)$ is distributive. Proof: see below. Let $(H \subset G)$ be an inclusion of finite groups and ...
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Products of maximal inclusions of finite groups with a non-obvious intermediate

Let $(H_1 \subset G_1)$ and $(H_2 \subset G_2)$ be core-free maximal inclusions of finite groups. Their product, the inclusion $(H_1 \times H_2 \subset G_1 \times G_2)$, admits four obvious ...
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The usual second isomorphism theorem for groups is: let $G$ be a group, $S$ and $N$ subgroups with $N$ normal, then $SN$ is a subgroup of $G$, $S\cap N$ is a normal subgroup of $S$ and $SN/N \simeq ... 14answers 5k views Fantastic properties of Z/2Z Recently I gave a lecture to master's students about some nice properties of the group with two elements$\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ... 7answers 5k views The finite subgroups of SU(n) This question is inspired by the recent question "The finite subgroups of SL(2,C)". While reading the answers there I remembered reading once that identifying the finite subgroups of SU(3) is still an ... 4answers 2k views For which$n$is there only one group of order$n$? Let$f(n)$denote the number of (isomorphism classes of) groups of order$n$. A couple easy facts: If$n$is not squarefree, then there are multiple abelian groups of order$n$. If$n \geq 4$is ... 9answers 3k views How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.? Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings, fields, graphs, partial orders, etc. ... 6answers 2k views Measures of non-abelian-ness Let$G$be a finite non-abelian group of$n$elements. I would like a measure that intuitively captures the extent to which$G$is non-commutative. One easy measure is a count of the non-commutative ... 2answers 2k views Order of products of elements in symmetric groups Let$n \in \mathbb{N}$. Is it true that for any$a, b, c \in \mathbb{N}$satisfying$1 < a, b, c \leq n-2$the symmetric group${\rm S}_n$has elements of order$a$and$b$whose product has order ... 2answers 1k views Definition of “finite group of Lie type”? The list of finite simple groups of Lie type has been understood for half a century, modulo some differences in notation (and identifications between some of the very small groups coming from ... 2answers 1k views (co)homology of symmetric groups Let$S_n=\{\text{bijections }[n]\to[n]\}$be the n-th symmetric group. Its (co)homology will be understood with trivial action. What are the$\mathbb{Z}$-modules$H_k(S_n;\mathbb{Z})$? Using GAP, we ... 3answers 3k views Feit-Thompson Theorem: The Odd Order Paper For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of ... 2answers 609 views A group action of the Heisenberg group with special symmetries Suppose we look at the Heisenberg group$H_{d}$as a matrix group of upper triangular matrices over the ring$\mathbb{Z}/d\mathbb{Z}$. You can even choose$d$to be prime if you want. A natural ... 0answers 783 views Given a lattice L with n elements, are there finite groups H < G such that L$\cong$the lattice of subgroups between H and G? If there is no restriction on$n$, this is a famous open problem. I'm wondering if any recent work has been done for small$n>6$. I believe the question is answered (positively) for$n=6$by ... 2answers 338 views If all real conjugacy classes are strongly real, then all real irreps are “strongly real”(symmetric), true ? Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real"(symmetric) ? And vice verse ? ... 1answer 213 views permutation action on cohomology of Stiefel manifolds Let$V_k(\mathbb{R}^n)$be Stiefel manifolds. In the paper The cohomology rings of real Stiefel manifolds with integer coefficients, Martin Čadek, Mamoru Mimura, and Jiří Vanžura, J. Math. Kyoto ... 4answers 880 views Structure of the adjoint representation of a (finite) group (Hopf algebra) ? Every group acts on itself by conjugation$h \mapsto g h g^{-1}$. Respectively considering functions on a group we obtain a linear representation. Question 1: what is known about this representation ... 3answers 353 views A problem with pointwise stabilizer subgroups of fixed-point subspaces II Definitions: Let$W$be a representation of a group$G$,$K$a subgroup of$G$, and$X$a subspace of$W$. Let the fixed-point subspace$W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$. Let ... 2answers 402 views Decomposing the conjugacy representation of Sym$(n)$for small$n$I am trying to compute the decomposition of the conjugacy representation of some small symmetric groups. Perhaps someone has undertaken a similar calculation. My own calculations are quite slow, ... 2answers 637 views Decomposing representations of finite groups Let$G$be a finite group,$p$a prime number. We denote by$\mathbb{F}_p$the field of cardinality$p$. Let$V$be an infinite dimensional representation of$G$over$\mathbb{F}_p$. Must there be ... 1answer 212 views An upper bound for the maximal subgroups at fixed index? Let us call a subgroup an injective homomorphism between groups. I warn the reader that a subgroup designates here an inclusion$(H \subset G)$, not$H$alone. A subgroup$H \subset G$is ... 1answer 177 views relatively free groups in$Var(S_3)$Suppose$S_3$is the symmetric group of order 6. Which elements of the variety$Var(S_3)$are relatively free? This question is related to my previous question Relatively free algebras in a variety ... 1answer 299 views Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs? There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ... 0answers 190 views $\mathcal{L}(H_i \subset G_i)$distributive$\Rightarrow\mathcal{L}(H_1 \times H_2 \subset G_1 \times G_2)$modular? Let$\mathcal{L}( G)$be the lattice of subgroups of$G$and$\mathcal{L}(H \subset G)$the lattice of intermediate subgroups. Definitions: A lattice$(L, \wedge, \vee)$is distributive if, ... 4answers 2k views Is there a 0-1 law for the theory of groups? Several months ago, Dominik asked the question Is there a 0-1 law for the theory of groups? on mathstackexchange, but although his question received attention there is still no answer. By asking the ... 1answer 4k views Why can't a nonabelian group be 75% abelian? This question asks for intuition, not a proof. An earlier question, Measures of non-abelian-ness was thoroughly answered by Arturo Magidin. A paper by Gustafson1 proves that, for a nonabelian group, ... 1answer 994 views Are there$n$groups of order$n$for some$n>1$? Given a positive integer$n$, let$N(n)$denote the number of groups of order$n$, up to isomorphism. Question: Does$N(n)=n$hold for some$n>1$? I checked the OEIS-sequence ... 3answers 1k views Being a subgroup: proof by character theory Let me first cite a theorem due to Frobenius: Let$G$be a finite group, with$H$a proper subgroup ($H\ne (1)$and$G$). Suppose that for every$g\not\in H$, we have$H\cap gHg^{-1}=(1)$. Then ... 1answer 2k views Do all exact 1 -> A -> AxB -> B -> 1 split for finite groups? Let$A$,$B$be finite groups. Is it true that all short exact sequences$1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$split on the right? In other words, do there exist ... 3answers 831 views Do finite groups acting on a ball have a fixed point? Suppose that$G$is a finite group, acting via homeomorphisms on$B^n$, the closed$n$-dimensional ball. Does$G$have a fixed point? A fixed point for$G$is a point$p \in B^n$where for all ... 2answers 1k views learning Deligne-Lusztig theory Can someone give me a roadmap for learning Deligne-Lusztig theory? (Except for the original article by Deligne and Lusztig) Edit: You may assume knowledge of representation theory of finite groups ... 3answers 1k views How much of the ATLAS of finite groups is independently checked and/or computer verified? In a recent talk Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he said that a proof that relied on the ... 2answers 1k views A group-theoretic perspective on Frankl's union closed problem Here is a group theoretic phrasing of a special case of the union closed conjecture: Question: Given a finite group$G$, is there an element of prime power order which is contained in at most half ... 2answers 802 views Nilpotency of a group by looking at orders of elements For any finite group$G$, let $$\theta(G) := \sum_{g \in G} \frac{o(g)}{\phi(o(g))},$$ where$o(g)$denotes the order of the element$g$in$G$, and where$\phi$is the Euler totient function. It is ... 4answers 3k views Classification of finite groups of isometries Consider the problem of classifying the finite groups of isometries of R^n. --For n=2 it is cyclic and dihedral groups. --For n=3 they are well known, probably from Kepler and are related to ... 3answers 2k views Smallest n for which G embeds in$S_n$? Question: Given a finite group$G$, how do I find the smallest$n$for which$G$embeds in$S_n$? Equivalently, what is the smallest set$X$on which$G$acts faithfully by permutations? This ... 3answers 2k views What is this subgroup of$\mathfrak S_{12}\$?

On some occasion I was gifted a calendar. It displays a math quizz every day of the year. Not really exciting in general, but at least one of them let me raise a group-theoretic question. The quizz: ...