# Questions tagged [normal-subgroups]

The tag has no usage guidance.

29 questions
Filter by
Sorted by
Tagged with
531 views

### 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 $G$ ...
219 views

28 views

### 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 ...
50 views

### 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 ...
175 views

### 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 ...
170 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 ...
1k views

103 views

### 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 ...
388 views

715 views

### 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 ...
3k views

### 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 ...