# Tagged Questions

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### The tallest possible lattice?

Let O be a complete discrete valuation ring and G a finite group. Recall that a finitely generated O-free OG-module $M$ such that the traces of the invertible endomorphisms of $M$ generate a strictly ...
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### Finite groups for which the element orders form an arithmetic progression

Which are the finite groups $G$ such that the element orders of $G$ form an arithmetic progression? Several remarks: $S_3$, $A_4$ and any $p$-group of exponent $p$ satisfy this property. If $G$ ...
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### More on Moonshine for the Thompson group and weakly holomorphic weight one half modular forms

This question is a follow-up to Monstrous Moonshine for Thompson group $Th$? and is based on various comments to that question, in particular S. Carnahan's mention of the connection to known ...
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Let $G$ be a finite group and $H$ a subgroup. The normal core of $H$ in $G$ is $core_G(H) := \bigcap_{g \in G}g^{-1}Hg$ Definition: $K$ is a normal intermediate subgroup of the inclusion $(H ... 0answers 72 views ### A question about perfect groups 1 G is a finite group, it is not simple, G=G',then we call G is a perfect group. My question is if there is a classification for perfect groupsã€‚ 0answers 203 views ### On the Groups of Order$(p^2+1)/2$A few days ago I asked a question (Groups of order$p(p^2+1)/2$) about a finite group of order$p(p^2+1)/2$and I got a lot of useful information about it. Thanks for the nice and very helpful ... 0answers 75 views ### Normal subgroups of finite solvable groups [migrated] Let$G$be a finite solvable group,$N$a nontrivial abelian normal subgroup of prime exponent$p$. Let$Q$be a$p$-Sylow subgroup of$G$containing$N$. Is it possible that the normal core of ... 5answers 646 views ### Groups of order$p(p^2+1)/2$It seems that when$p>3$is a prime, then each group of order$p(p^2+1)/2$is abelian as I checked by Gap for small$p$. Is it true for each$p$? Thanks for your answers 0answers 604 views ### Monstrous Moonshine for Thompson group$Th$? I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2, $$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ... 0answers 106 views ### Is the direct product of distributive inclusions of groups, modular? Let H a subgroup of G and \mathcal{L}(H \subset G) the lattice of intermediate subgroups (\mathcal{L}( G) if H= \{ e \}). Definitions: A lattice (L, \wedge, \vee) is : - Distributive if ... 2answers 101 views ### Products of maximal inclusions of finite groups with a non-obvious intermediate Let (H_1 \subset G_1) and (H_2 \subset G_2) be core-free maximal inclusions of finite groups. Their product, the inclusion (H_1 \times H_2 \subset G_1 \times G_2), admits four obvious ... 0answers 143 views ### Existence of inclusions of finite groups with a particular lattice property Definition : Let \sim be the equivalence relation on inclusions of finite groups, generated by : (H \subset G) \sim (\phi(H) \subset \phi(G)), with \phi: G \to L a finite group morphism and ... 3answers 405 views ### Classification of automorphism groups of groups of order p^4 For the purpose of classifying another algebraic structure which is parametrised by the choice of a group and of an automorphism I need the classification up to isomorphism of automorphism groups of ... 2answers 163 views ### 3-cocycle representatives for the dihedral group D_{2n}? I am looking for a reference for a complete list of 3-cocycle representatives for H^3(D_{2n},\mathbb{C}^\times), where$$ D_{2n}=\langle a, b\mid a^2=b^2=(ab)^n=e\rangle $$is the dihedral group of ... 0answers 102 views ### Fusion pattern in a cyclic subgroup of order 8 Can a finite simple group G have an element x of order 8 such that x^2 is conjugate to x^{-2} but x is not conjugate to any element of \{x^3,x^5,x^7\}? In other words, does this fusion ... 1answer 91 views ### Index of agemo subgroups in p-groups Having a finite p-group G (p odd). we denote by \Omega_1(G) the subgroup generated by all the elements of G of order dividing p. Is there an example of such a group G, such that ... 4answers 386 views ### Iterated semi-direct products Let G be a finite group. Suppose that we can write G= A \rtimes B and also A = C \rtimes D. Further suppose that C is normal in G (not just in A). Then can we write G = C \rtimes E where ... 2answers 195 views ### Factor subset of finite group Let G be a group of order n and d a positive divisor of n. Is it true that there exists a subset A of G with d elements and a subset B such that G=AB and |AB|=|A||B| ... 1answer 264 views ### Does O'Nan-Scott depend on CFSG? My question is in the title. Some context: there are two versions of the O'Nan-Scott theorem. The first, weaker version, is due to O'Nan and Scott (independently) and gives the structure of the ... 6answers 414 views ### Almost uniquely generated groups This is inspired by this question. Does there exist an infinite finitely generated group having (a) a unique (b) finitely many inclusion-minimal generating set(s) up to ... 1answer 161 views ### Doubly primitive groups with simple socle The classification of doubly transitive groups with simple socle is known. A good account of such classification can be found for example in this paper: Cameron, Peter J. Finite permutation groups ... 2answers 452 views ### Reference for the triple covering of A_6 I would like to ask for a reference (book, paper ...) for the following nice construction, which I have found as an exercise in some notes of a course by R. Borcherds. For n=6 or 7 (and only in ... 2answers 821 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 314 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, ... 1answer 226 views ### Group cochains invariant under the action of the symmetric group Let G be a finite group and A an abelian group. Recall the cochain groups$$ C^k = \{f: G^k \to A\} $$and the coboundary map$$ \delta : C^k \to C^{k+1}  (\delta f)(g_1, \ldots, g_{k+1}) ... 1answer 143 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 491 views ### Examples of finite groups with “good” bijection(s) between conjugacy classes and irreducible representations? For symmetric group conjugacy classes and irreducible representation both are parametrized by Young diagramms, so there is a kind of "good" bijection between the two sets. For general finite groups ... 4answers 334 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 ... 1answer 273 views ### How many finite loops? How many finite loops of order$n$are there? I am interested in the exact values â€‹â€‹of$n$if$n <40$or even reasonable estimates. I am also interested in formulae or bounds for all$n$. Note ... 1answer 162 views ### The compact Lie group contains a finite subgroup$\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$Given a finite Abelian group:$G=\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}$, where${n_1},{n_2},{n_3}$are arbitrary positive integers.${n_1},{n_2},{n_3}$may have or may not ... 1answer 168 views ### Simple group of order 504 [closed] As we know,there are 9 Sylow 2-subgroup in the Simple group of order 504.Can anyone prove it only by Sylow's theorem? (you can't use knowledge about PSL(2,8)) 1answer 355 views ### Are There Always Group Generators Which Give Unimodal Growth? Suppose$G$is a$k$-generated finite group. Is there always a set of$k$elements which generate the group and have a unimodal counting function? Background: The counting function,$f(n)$, is a ... 1answer 83 views ### Flag primitivity of the correlation group of classical projective planes. We know that the full automorphism group of the$\pi_q = PG(2,q)$acts imprimitively on the flags (all flags through a fixed point form a block). But, things change when we consider the action of full ... 2answers 343 views ### Number of isomorphism types of finite groups Are there some good asymptotic estimations for the number$F(n)$of non-isomorphic finite groups of size smaller than$n$? 0answers 377 views ### How many sporadic simple groups are there, really? I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is ... 1answer 144 views ### Simplification problem for finite groups Let$G_1,G_2,H$be finite groups. My question is: if$G_1\times H$is isomorphic to$G_2\times H$, is$G_1$isomorphic to$G_2$? I came to this question while preparing an exercise on finite abelian ... 0answers 120 views ### A connection between nonplanar complete graphs and the alternating group? I didn't get any response on MSE so I though I'd give this a try here (my question on MSE). I went to an undergrad's senior honors thesis presentation a while ago. She was discussing crossing numbers ... 3answers 1k 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 294 views ### A subgroup intersects conjugacy class of every prime power order element Let$G$be a finite group and$H$be a subgroup of$G$. Suppose that for any prime power order element$x$of$G$, there exists some element$g$in$G$such that$x^g$is contained in$H$. Does it ... 2answers 179 views ### On non-split extensions of$\mathrm{SL}_d(q)$Throughout this question, the following notation holds: Let$q$be a power of a prime$p$, and let$d>4$be a positive integer. Let$G$be a finite group with a normal subgroup$E$which is an ... 1answer 136 views ### A finite$p$-group with certain properties Is there a finite$p$-group$G$such that : (a)$G= \langle A,x,y \rangle$, with$G/Z(G)$has exponent$p$,$A$is a maximal abelian normal subgroup of$G$, and$G/A$has order$p^2$(thus it is ... 2answers 326 views ### Pre-images of unipotent elements in$\operatorname{SL}_{n}(A)$The starting point of this question is the (presumably) well-known theorem (the proof I know is from Abelian$\ell$-adic representations and elliptic curves from J-P.Serre in which it is a lemma for ... 1answer 135 views ### Involution centralizers in simple groups I often see lower bounds on the size of centralizers of involutions in finite (nonabelian) simple groups, but is there a general upper bound for the size of an involution centralizer in such a ... 6answers 1k 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 ... 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 ... 1answer 777 views ### Monstrous moonshine for$M_{24}$and K3? An important piece of Monstrous moonshine is the j-function, $$j(\tau) = \frac{1}{q}+744+196884q+21493760q^2+\dots\tag{1}$$ In the paper "Umbral Moonshine" (2013), page 5, authors Cheng, Duncan, and ... 1answer 425 views ### Automorphism group of a finite group I would like to ask if there exists an explicit description of$\mathrm{Aut}(G)$, the group of automorphisms of a finite group$G$, in particular, when$G$is abelian. E.g., if$G = ...
My question is simple: If a group $G$ has the same character table with the semidihedral group $SD_{2n}$, are $G$ and $SD_{2n}$ isomorphic ?