# Tagged Questions

Questions on group theory which concern finite groups.

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### Rank of a locally free $\mathbb Z[G]$-module

This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free. ...
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### Subgroups of the union of conjugates

This question is an attempt to find a version of this question with a positive answer. In this question, we ask if one can find big subgroups in the union of conjugtes of small subgroups of finite ...
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### For $P$ $\mathbb{Z}G$-projective, $\mathbb{Q}\otimes P$ is $\mathbb{Q}G$-free

I'm looking for a proof of a theorem of Swan [1, Theorem 3]: If $G$ is a finite group and $P$ a finitely generated projective $\mathbb{Z}G$-module, then $\mathbb{Q}\otimes_\mathbb{Z}P$ is a free ...
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### A dual version of a theorem of Øystein Ore in group theory

Let $(H \subset G)$ be an inclusion of finite groups. This post is a dual version for the Generalization of a theorem of Øystein Ore in which it's proved: Theorem: $\mathcal{L}(H\subset G)$ ...
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### A question on $p$-central $p$-groups

Let $p$ be a fixed prime. A group $G$ is termed $p$-central if every element of order $p$ in $G$ lies in the center. Having a finite $p$-group $G$ of rank $k$ (the least integer, such that every ...
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### Character Values for Alternating Groups of degree $\geq 7$

Is it true that in each row and column of the character table of alternating groups with degree $\geq 7$ there are at most two complex values? Any reference will be highly appreciated.
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### Is this group given presentation isomorphic to $\mathbb{Z}_2$, and why? [migrated]

We have the group $G = \langle a, \,b \; | \; a^2b^2ab^{-1}, \, a^3b^4a^{-2}b^{-3}\rangle$. Obviously $G/G' = \mathbb{Z}_2$. Is it true that $G = \mathbb{Z}_2$ or equivalently that $G'=1$? It is ...
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### Radical and Centric not Essential P-group

I'm looking that in the Fusion System categories, the p-subgroups that are essential, are centric (by definition) and radical (by implication of the definition), but I want to know if there is an ...
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### A question about (unicity of certain cycles in a Cayley graph of a) symmetric group

Let $S=\{(1,2),(1,2,3,\ldots,n),(1,2,3,\ldots,n)^{-1}=(1,n\ldots,2)\}$ be a subset of the symmetric group $S_n$. We know that $(1,2,\ldots,n)(1,2)=(2,3,\ldots,n)$, and thus ...
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### Normalizers in symmetric groups

Question: Let $G$ be a finite group. Is it true that there is a subgroup $U$ inside some symmetric group $S_n$, such that $N(U)/U$ is isomorphic to $G$? Here $N(U)$ is the normalizer of $U$ in $S_n$. ...
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### Modularisation on group representations with arbitrary braiding

Applying the modularisation/deequivariantisation procedure to the representation category $\operatorname{Rep}_G$ of a finite group $G$ with trivial braiding gives the fibre functor to vector spaces. ...
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### p-groups as finite union of disjoint normal abelian subgroups

I was interested in knowing if groups with following property have been studied( like what can be said about structure of the group) : "$G$ can be written as disjoint union of a given number of ...
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### Group cohomology of the cyclic group

It is well known how to compute cohomology of a finite cyclic group $C_m=\langle \sigma \rangle$, just using the periodic resolution, $\require{AMScd}$ \begin{CD} \cdots @>N>> \mathbb ...
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### Regular elements in the torus of a group of Lie type

Let $G$ be a simple linear algebraic group, and $F$ a Frobenius map, i.e. some power of $F$ is the standard Frobenius map which raises matrix entries to the $q$-th power. Then $G^F$ is a group of Lie ...
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### Collecting proofs that finite multiplicative subgroups of fields are cyclic

I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
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### Exterior Powers of finite abelian group

Let $A$ be a finite $\mathbb{Z}$-module (i.e., a finite abelian group). My question is: for what $n\in \mathbb{Z}^{n\geq 2}$ the map \begin{align} \alpha_{n}:\bigwedge^nA&\to A^{\otimes n}\\ ...
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### Finite group action on quasi-projective varieties

Let $X$ be a smooth, quasi-projective variety, $G$ be a finite group which acts freely and properly on $X$. Denote by $\alpha:X \to X/G$ the quotient. Is $\alpha$ generically etale? Also, as I am ...
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### Classification of generously transitive groups

A permutation group $G \lt S_n$ is called generously transitive, if for each $i,j$ there exists a permutation that interchanges them. Is there a reasonable classification of such (finite) groups?
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### Must normalizing field outer automorphisms “divide” the dimension?

Imprecise question: To get a normalizing field outer automorphism of order $r$, must we multiply the dimension by $r$? Precise hypothesis: Let $p\geqslant 5$ be a prime, let $q$ be a power of $p$ and ...
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### Is there a characterization of CI-groups of order less than 100?

We know some benefit criterion in articles such as: C‎. ‎H‎. ‎Li‎, ‎On isomorphisms of finite Cayley graphs-a survey‎, ‎Discrete Math.‎, ‎256 (2002) 301-334‎. C‎. ‎H‎. ‎Li‎, ‎Z‎. ‎P‎. ‎Lu‎, ‎P‎. ...
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### Is there a formula for the number of elements in $S_n$ having length $k$ with respect to the generators taken to be the transpositions?

Define the length of a permutation as the minimum number of adjacent transpositions needed to describe it. We know that there is only one element of length $0$ in $S_n$ and $n-1$ elements of length ...
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### A finite supersolvable group with generators of prescribed order

Let $G=\langle a,b\rangle$ be a finite supersolvable group. Is there any special information about the structure of $G$ when $o(a)=2$ and $o(b)=2^k > 2$?
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### Transitive permutation groups which all of their proper subgroups are intransitive

Let $G$ be a transitive permutation group on a finite set $\Omega$. It is clear that if $G$ is regular, then every proper subgroup of $G$ is intransitive. Is there any other class of groups with this ...
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### Gramian of a permutation group orbit

let $W\in R^{d\times k},d>k$. Suppose that the associated gramian has the following structure: $$W^TW=(P_{1}t,\cdots,P_{k}t)$$ with the set $\{P_{i},i=1,\cdots,k\}$ forming a group of ...
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### 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 ...
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### 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 ...
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### Lifting automorphisms of quotient groups

I am concerning here a natural question: Problem: Let $G$ be a finite group, and let $N$ be a characteristic subgroup of $G$. When can an automorphism $\varphi\in\mathrm{Aut}(G/N)$ be lifted to an ...
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### Finite solvable groups with abelian Fitting subgroup

Let $G$ be any finite solvable group with Fitting subgroup $F(G)$. Which conditions on $F(G)$ makes $G$ to be supersolvable? (It is well-known that any finite solvable group with cyclic Fitting ...
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### Cayley graphs of $A_n.$

Consider the Cayley graphs of $A_n,$ with respect to the generating set of all $3$-cycles. Their properties must be quite well-known, but sadly not to me. For example: what is its diameter? Is it an ...
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### Positivity of the alternating sum of indices for boolean interval of finite groups

Let $G$ be a finite group and $H$ a subgroup such that the interval $[H,G]$ is a boolean lattice. Let $L_1, \dots , L_n$ be the maximal subgroups of $G$ containing $H$. Let the alternative sum ...
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### 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 ...
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### Generating finite simple groups with $2$ elements

Here is a very natural question: Q: Is it always possible to generate a finite simple group with only $2$ elements? In all the examples that I can think of the answer is yes. If the answer is ...
Consider the abelian free group $G = \mathbf{Z}^n$ of rank $n$ and a finite subset $A \subset G \setminus \{0\}$. Since $G$ is residually finite, there is a subgroup $H \subset G$ such that $A \cap H ... 3answers 226 views ### Signs in Chevalley's commutator formula I am trying to understand presentations of twisted groups of Lie type (specifically$^2D_5$) over finite fields using Steinberg's presentations (for instance from Gorenstein, Lyons and Solomon, Number ... 1answer 102 views ### Intransitive finite irreducible linear groups whose orbits are all large I am interested in intransitive irreducible linear subgroups$G\subseteq\mathrm{GL}_n(\mathbb{F}_p)$acting on$V-\{0\}=\mathbb{F}_p^n-\{0\}$in the natural way, such that all of the orbits are very ... 0answers 92 views ### A question about$p$-solvable groups$H$is called$s$-permutable in$G$if it permutes with every Sylow subgroup of$G$.$H$is called$s$-permutably embedded in$G$if each Sylow subgroup of$H$is a Sylow subgroup of some ... 1answer 366 views ### Perfect group of order 190080 I need to know some properties of the perfect group of order$190080$which is the Schur cover of the Mathieu group${\rm M}_{12}$, but when using ... 0answers 155 views ### Detecting a module for the free group algebra on a finite quotient Let$F_2$be the free group on two generators$x,y$and let$R$be the group algebra$\mathbf{Q}[F_2]$. Let$a,b,c$be integers. Then define a right$R$-module$M = R / (ax + by + c) R$. I am ... 1answer 196 views ### Are there familiar expressions for (finite) joins of finite groups? Milnor construction of the classifying space of a topological group$G$is given in terms of infinite joins of$G$. Schwarz then proved that the$k+1$iterated self join of a group$G$classifies ... 0answers 178 views ### Cohomology of central extensions of groups Let G be a central extension of a finite group H by$Z/2$. I need an explicit description of the differentials$d_2$and$d_3$in the Lyndon-Hochschild-Serre spectral sequence which converges to the ... 0answers 181 views ### Weyl Groups as Galois groups I am looking for explicit examples (for all positive integers$n \ge 5$) of degree$2n$even polynomials$f(x)=h(x^2)$over the field$\mathbb{Q}$of rational numbers such that the Galois groups of ... 1answer 166 views ### A question on the commutativity degree of the monoid of subsets of a finite group The commutativity degree$d(G)$of a finite group$G$is defined as the ratio $$\frac{|\{(x,y)\in G^2 | xy=yx\}|}{|G|^2}.$$It is well known that$d(G)\leq5/8$for any finite non-abelian group$G$. If ... 1answer 116 views ### A decomposition of$w_0$which is similar to the reduced decomposition Some basic definitions about reduced decomposition: In the symmetric group$S_n$, let$s_i$denote the adjacent transposition$(i,i+1),i\in \{1,2,\cdots,n-1\}.$Since$S_n$is generated by adjacent ... 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 ... 3answers 435 views ### Which groups are Galois over some p-adic field? Suppose I have some finite$p$-group$G$, or a little extension of it. How do I know if there exists a prime$l$and a finite extension$K$of$\mathbb{Q}_l$such that$G$is the Galois group of ... 1answer 242 views ### Does the Fano plane “embed” in the complex projective plane?$PSL(2,7)$acts on the projective plane over$\mathbb{F}_2$(the Fano plane) through its identification with$GL(3,2)$. It also acts on the projective plane over$\mathbb{C}$through either of its ... 0answers 113 views ### Cocycle condition for 2-groups I know that if$\omega_d(g_1, \ldots, g_d)$is an d-cocycle characterized by$H^d(G,U(1))$, it satisfies the co-cycle condition$(d\omega_d)(g_1, \ldots, g_{d+1}) = g_1.\omega_d(g_2,\ldots,g_{d+1}) + ...
A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n-1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...