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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1 vote
0 answers
91 views

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 $x,y,z \in \mathbb{Z}$...
3 votes
1 answer
105 views

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 ...
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9 votes
1 answer
250 views

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 ...
  • 2,297
13 votes
1 answer
987 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 $G$ ...
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1 vote
1 answer
268 views

Why $G/F(G)$ is isomorphic to a subgroup of ${\rm Out}(F(G))$?

I know two facts and I’ve managed to figure out how to prove one, but the other one is still a little confusing. Let $G$ be a finite solvable group and $F(G)$ is the Fitting subgroup of $G$. (1) $G/Z(...
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6 votes
1 answer
436 views

Why do we say the Fitting subgroup/generalized Fitting subgroup control the structure of a group?

I’m learning the Fitting subgroup these days. I’m interested in this topic and particularly in the role that it plays in the structure of groups. Many people on MSE mentioned that the Fitting subgroup/...
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1 vote
1 answer
93 views

Restriction of real irreducible 2-Brauer characters to subnormal subgroups

Question: Find a finite group $G$, a subnormal subgroup $H$ of $G$, a real-valued irreducible $2$-Brauer character $\chi$ of $G$ and a real-valued irreducible $2$-Brauer character $\mu$ of $H$ such ...
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2 votes
0 answers
37 views

Does Levi operator always map one-word varieties to one-word varieties?

Suppose $\mathfrak{U}$ is a group variety. Now let’s define $L(\mathfrak{U})$ as the class of all such groups $G$, such that $\forall g \in G$ $\langle \langle g \rangle \rangle \in \mathfrak{U}$ (...
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3 votes
1 answer
59 views

Is $S_\omega/F_\omega$ embeddable to $S_\omega$?

Let $S_\omega$ be the group of bijections $f:\omega\to\omega$, and let $F_\omega = \{\pi\in S_\omega: \exists N\in \omega(\pi(k) = k \text{ for all } k\geq N)\}$. It is easy to see that $F_\omega$ is ...
6 votes
3 answers
351 views

Subgroup generated by a subgroup and a conjugate of it [closed]

Let $H\leq G$ be groups, and $a\in G$ so that $\langle H,a\rangle=G$. Does it follows that $\langle H\cup aHa^{-1}\rangle$ is a normal subgroup of $G$? My hope is that this is true, and my guess is ...
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3 votes
0 answers
185 views

Generalization of normal subgroup

I am wondering whether the following concept appears in the group theory literature under some (perhaps different) name. Let $G$ be a group and let $A,B$ be subgroups of $G$. Definition. Say that $(...
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1 vote
0 answers
29 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 ...
2 votes
1 answer
68 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 ...
5 votes
1 answer
190 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 ...
6 votes
3 answers
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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25 votes
2 answers
2k views

Is the intersection of two subgroups, defined below, always trivial?

Suppose, $G = \mathbb{Z} \ast H$, where $H$ is an arbitrary group. Suppose, $g \in G$ and $g \notin \langle\langle H \rangle \rangle $. Is $\langle\langle g \rangle \rangle \cap H$ always trivial? ($\...
  • 2,608
0 votes
2 answers
880 views

Existence of a cyclic non-normal subgroup in a $p$-group

Let $G$ be a finite non-abelian $p$-group, where $p$ is an odd prime, $N$ be a normal subgroup of $G$ of order $p$, where $\frac{G}{N}$ is non-abelian. Does there exist an element $g\in G$ such that ...
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2 votes
0 answers
89 views

A kind of cancellation ; exchange problem for groups

For which $(m,n,k,l) \in (\mathbb N\cup \{0\})^4$ , with $m\le n ; k\le l$ , does there exist a group $G$ with a finite subnormal series with torsion-free Abelian quotients such that $G \times \mathbb ...
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2 votes
1 answer
111 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 ...
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12 votes
2 answers
427 views

Finite groups with lots of conjugacy classes, but only small abelian normal subgroups?

Denote the commuting probability (the probability that two randomly chosen elements commute) of a finite group $G$ by $\operatorname{cp}(G)$. By a result of Gustafson [2], $\operatorname{cp}(G)=\...
2 votes
0 answers
190 views

Groups with isomorphic quotients [closed]

Assume we have a finitely presented group $G$ and a non-trivial normal subgroup N. How can one decide that $G/N$ is isomorphic to $G$ or not? $G$ is given as a presentation and $N$ as a set of words.
3 votes
1 answer
341 views

A finite distributive lattice which may be represented as the normal subgroup lattice of a supersolvable group

Is there a supersolvable group $G$ with the lattice of all its normal subgroups, order-isommorphic to the 18-element lattice of down-sets of this poset: ? It has been proved that not every finite ...
2 votes
0 answers
213 views

Normal Subgroups of $UT_n(q)$

What is known about normal subgroups of $UT_n(q)$, the group of upper triangular matrices with entries in the finite field $\mathbb{F}_q$ and ones on the diagonal? Is there an interpretation of the ...
2 votes
2 answers
426 views

Does the hyperoctahedral group have only 3 maximal normal subgroups?

An hyperoctahedral group $G$ is the wreath product of $S_2$ and $S_n$, where $S_{n}$ is the symmetric group on $n$ letters, or in other words the semi-direct product $G=S_2^n\rtimes S_n$, w.r.t. the ...
7 votes
4 answers
520 views

Normal Covering of a Finite Group

Suppose $G$ is a finite group and $N_1, N_2, \cdots, N_k$ are proper normal subgroups of $G$. The set $\{ N_1, \cdots, N_k\}$ is called a normal cover for $G$, if $G = \cup_{i=1}^kN_i$. I need to the ...
4 votes
1 answer
401 views

Normal subgroup lattice of the group $U_{6n}$

I need to find the normal subgroup lattice of the group $U_{6n} = \langle a,b | a^{2n} = b^ 3= 1, a^{-1}ba = b^{-1}\rangle$. To the best of my knowledge this group was introduced at first by GORDON ...
12 votes
0 answers
698 views

$2$-group with two isomorphic normal subgroups of index $4$ with non-isomorphic quotients

Hypothesis: Let $P$ be a finite $2$-group with two isomorphic normal subgroups $M$ and $N$ such that $P/M\cong C_4$ (the cyclic group of order $4$) while $P/N\cong C_2^2$. By the lattice theorem, ...
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3 votes
1 answer
2k views

Normal Subgroups of Free Products

Let $G=A\ast \mathbb{Z}$ be the free product of a group $A$ and the cyclic group $\mathbb{Z}$ and suppose $K$ is a subgroup of $G$. By Kurosh Subgroup Theorem we know that $K=F\ast (\ast_{i\in I}(K\...
  • 2,163
7 votes
1 answer
859 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 ...
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16 votes
2 answers
2k views

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 ...
  • 12.6k
29 votes
5 answers
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 ...
  • 93.8k
31 votes
4 answers
9k views

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'...
12 votes
5 answers
4k views

Does every finitely generated group have a maximal normal subgroup?

Given an infinite group which is finitely generated, is there a proper maximal normal subgroup?
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