# Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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### Some special subgroups of nilpotent groups of nilpotency class 2 [closed]

Let $G$ be a group. Denote by $\mathrm{Z}(G)$ and $G'$ the center of $G$ and the derived subgroup of $G$, respectively. Assume $G'\subseteq\mathrm{Z}(G)$. Then, it is clear that $G$ is a nilpotent ...
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### Classification of natural endomorphisms on finite groups

Any $z \in \widehat{\mathbb{Z}} = \lim_{n} \mathbb{Z}/n\mathbb{Z}$ defines an operation on all finite groups: if $G$ is a finite group and $g \in G$, say $g^n=1$, then map it to $g^{z_n}$. This ...
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### Embedding (Kronecker product) preserves the structure?

In PGL$_{n}(\textbf{C})$, conjugacy classes of toral involutions can be represented by $$s_{i} = \begin{pmatrix} -I_{i} & 0\\ 0 & I_{n-i} \end{pmatrix}.$$ for $1 \leqslant i \leqslant [n/2]$. ...
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### Homogeneity of finite simple groups

According to A complete classification of finite homogeneous groups, the finite simple groups $G$ which are homogeneous (meaning every isomorphism between subgroups extends to an automorphism of $G$) ...
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### Second homology group of a presentation complex

I am trying to learn results related to the presentation complex of a group and I am new to this subject. So I apologize if the questions are silly. Given a finite group $G$, and a presentation $P$ of ...
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### Is this class of groups already in the literature or specified by standard conditions?

In recent work Lifting $N_\infty$ operads from conjugacy data on homotopical combinatorics / $N_\infty$ operads in equivariant homotopy theory, collaborators Scott Balchin, Ethan MacBrough, and I ...
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### Endo reversible words

Let $w$ be a word in free group $F$ on finitely many generators. We will look at $w$ as word map on groups. It is clear that there exists an endomorphism $\phi$ of $F$ such that $\phi(w) = w^{-1}$ if ...
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### "Novelty" maximal subgroups in $S_n$

What are the maximal subgroups $M < S_n$ such that $M \cap A_n$ is not maximal in $A_n$? Maximal subgroups of $S_n$ are described by the O'Nan-Scott theorem and very extensively studied in many ...
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### Finite 2-groups with $(ab)^{2}=(ba)^{2}$

There exist nonabelian finite 2-groups $G$ with the property $(A2)$ : for every $a,b\in G$, $(ab)^{2}=(ba)^{2}$. An example of a such group is given by the quaternion group $Q_{8}$ of order 8. Is ...
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### Groups of orders $7!$ and $\frac{7!}{2}$

In our research, we need to know that whether every group $G$ of order $2520 = 2^3 \cdot 3^2 \cdot 5 \cdot 7=\frac{7!}{2}$ or $5040 = 2^4 \cdot 3^2 \cdot 5 \cdot 7=7!$ has a proper subgroup non-...
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### Splitting of a finite group with no abelian subfactor in composition series

Let $G$ be a finite group with no abelian subfactor in its composition series. Is $G$ obtained from simple groups by iterating semidirect products? (Initially it was asked whether $G$ is a direct ...
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### Finite groups whose polynomials share two common properties with polynomials on commutative groups

This question is motivated by (some available information on) this MO-problem on the largest possible degree of a polynomial on a finite group and this MO-problem on the degree of the constant ...
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### The degree of a constant polynomial on a finite group

A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N=\{1,2,\dots\}$ and elements $a_0,a_1,\dots,a_n\in X$ such that $f(x)=a_0xa_1x\cdots xa_n$ for all $x\in X$. ...
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### Prime divisors of nonabelian simple group and of its outer automorphism group

Let $G$ be a finite nonabelian simple group. Write $\mathrm{Out}(G)$ the outer automorphism group of $G$. For a finite group $H$, let $\pi(H)$ be the prime divisors of the order of $H$. By check the ...
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### Polyextremal groups

A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form $f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
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### Length of representation of $GL_n(\mathbb{F}_q)$ in functions on Grassmannian

Let $G=GL_n(\mathbb{F}_q)$ be the (finite) group of all linear invertible transformations of the vector space $(\mathbb{F}_q)^n$ over the finite field $\mathbb{F}_q$. $G$ acts naturally on the ...
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### $2^2 \cdot U_6(2)$ and $2^2.2^{1+20}U_6(2)$ in $\mathbb{M}$

In the first diagram of this paper, there are conjugacy classes of subgroups of the Monster group which are labeled $2^2 \cdot U_6(2)$ and $2^2.2^{1+20}U_6(2)$, respectively. Can subgroups in the ...
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### Investigating the structure of a group algebra via the derived subgroup

It is well known that each element in the special linear group $\mathrm{SL}_n(\mathbb{H})$ over the real quaternion division ring with $n\geq1$ is a single multiplicative commutator. I am particularly ...
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Let $G$ be a finite group, $A$ some coefficients (e.g. $A = \mathbb{F}_2$ or $\mathbb{Z}$), and write $\mathrm{H}^\bullet_{\mathrm{gp}}(G; A)$ for the (ordinary) group cohomology of $G$ with ...