# Questions tagged [normal-subgroups]

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### Generating set of a group with a unique minimal normal subgroup

I started reading this paper by Andrea Lucchini. Title: Generators for Finite Groups with a Unique Minimal Normal Subgroup. Theorem 1.1(Main Theorem) If $G$ is a non cyclic finite group with a unique ...
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### Classification of the normal subgroups of the discrete Heisenberg group

Let $H$ be the discrete Heisenberg group, i.e., the set of matrices of the form $\begin{bmatrix} 1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{bmatrix}$ where $x,y,z \in \mathbb{Z}$...
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### Algorithm to find a minimal normal subgroup of given group $G$ by matrix group representation

Given a matrix group $G$ by its generators i.e. $G =\langle A_1,A_2,...,A_k \rangle \leq GL_n(q)$, where each $A_i$'s are matrix in $GL_n(q)$ Q. Does there exist a polynomial time (polynomial in ...
• 151
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### Subgroups of infinite solvable groups

I'm looking for results of the form "every infinite solvable group contains <...> as a subgroup". Specifically, I believe: If $G$ is infinite solvable, finitely generated and not ...
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### Every subgroup is isomorphic to a normal subgroup

Let $G$ be a group such that, for every subgroup $H$ of $G$, there exists a normal subgroup $K$ of $G$, such that $H$ is isomorphic to $K$. Under such conditions, can we determine the structure of $G$ ...
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### Defect of subnormality in unit groups of modular group algebras

Let $p$ be a prime number, $G$ a finite p-Group and $K$ a finite field with $char(K)=p$. It is well-known that the group $1+rad(KG)$ is a p-group containing $G$. $G$ is normal in $1+rad(KG)$ if and ...
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### Defect of subnormality and repeated normalizer series

Let $G$ be a a finite group and $S$ a subnormal subgroup of $G$. The lenght of a fastest chain of subgroups $(U_i)_{1\le i\le n}$ such that $U_1=S$, $U_i$ normal in $U_{i+1}$ and $U_n=G$ is called the ...
• 581
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### Normal Fuchsian subgroups

I've been working with Fuchsian groups and from geometrical motivations finding a cocompact normal Fuchsian subgroups of $PSL(2,\mathbb{R})$ would have intresting properties for my research. It is ...
194 views

### Groups whose poset of direct factors are lattices

Let $G$ be a finite group. Denote by $\mathcal{N}(G)$ the modular lattice of normal subgroups of $G$ and denote by $\mathcal{D}(G)$ the subposet of $\mathcal{N}(G)$ whose elements are the direct ...
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### Generalizing a codistributive property of sufficiently disjoint normal subgroups to protomodular categories

In a poset, whenever the meets and joins below exist, their universal properties induce a containment $$(A\vee B)\wedge (A\vee C)\geq A\vee(B\wedge C).$$ This is an instance of codistributivity. In a ...
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### Groups whose normal subgroups form a chain with respect to inclusion

Let G be a finite group. In general, given two normal subgroups N and K of G, we need not have N < K or K < N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So assume ...
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### Groups with all normal subgroups characteristic

Today in my research, I had to use fairly explicitly the rather tautological property of finite cyclic groups that every normal subgroup is characteristic, i.e. fixed by all automorphisms. This got me ...
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### Existence of simultaneously normal finite index subgroups

It is well known that if $K$ is a finite index subgroup of a group $H$, then there is a finite index subgroup $N$ of $K$ which is normal in $H$. Indeed, one can observe that there are only finitely ...
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### Groups with all subgroups normal

Is there any sort of classification of (say finite) groups with the property that every subgroup is normal? Of course, any abelian group has this property, but the quaternions show commutativity isn'...
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