Questions tagged [normal-subgroups]
The normal-subgroups tag has no summary.
50 questions
5
votes
1
answer
334
views
Existence of maximal subgroups of even order which are not normal
Let $G$ be a finite non-solvable group. Does $G$ always have a maximal subgroup of even order which is not normal in G?
Attempt: As $G$ is non-solvable, $|G|$ is even and has an element of order $2$, ...
- 529
0
votes
1
answer
258
views
Normal subgroups of $S_\omega/(\text{fin})$ [closed]
$\newcommand{Sfin}{S_\omega/(\text{fin})}$By $S_\omega$ we denote the group of bijections $f:\omega\to\omega$, and we let $(\text{fin})$ be the collection of bijections with finite support, that is $$(...
- 54.4k
2
votes
1
answer
187
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Get the Sylow 2-subgroup of $H$ from that of $G$
It is known that $G:=\operatorname{GL}(4,17)=16.\operatorname{PSL}(4,17).2.2$ and there exists $H:=2.\operatorname{PSL}(4,17).2$ which can be constructed in Magma. I hope to get the Sylow 2-subgroup ...
- 281
0
votes
0
answers
134
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Double, triple, and multi-cosets of subgroups
Due to the definition of the double cosets of two subgroups (see https://en.wikipedia.org/wiki/Double_coset), one may naturally ask for three or more subgroups.
Let $H_1,\cdots,H_n$ be subgroups of $G$...
- 1,011
5
votes
1
answer
551
views
Groups with no proper non-trivial fully invariant subgroup
Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be characteristic if $\phi(H)\subseteq H$, $\forall \phi \in \operatorname{Aut}(G)$ and fully invariant if $\phi(H)\subseteq H$, $\forall \...
- 79
3
votes
1
answer
193
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Normal subgroup of the geometrical fundamental group is the normal subgroup of the arithmetic fundamental group
I asked some questions on a descending lemma in Lawrence-Venkatesh 4 days ago, but it has not received any answer. I understood (2) now but I'm still confused on (1).
I want to ask a new question here....
- 151
10
votes
1
answer
386
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About the normal subgroups of Burnside groups
I was reading "On periodic groups of odd period $n\ge 1003$" of V. S. Atabekyan. He found that the Burnside group $B_n$ with $n\ge 1003$ has uncountably many normal subgroups. However, I was ...
- 111
1
vote
0
answers
145
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Center of factors of a finite $p$-group, obtained from a minimal normal subgroup
throughout a research problem about finite $p$-groups,
I have a challenge as follows,
Let $G$ be a finite non-abelian $p$-group, where $p$ is odd and $Z(G)$ is non-cyclic.
($Z(G)$ denotes the center ...
- 133
6
votes
0
answers
210
views
Normality and small doubling
Suppose that $A$ is a finite, generating subset of a group $G$, and that $H$ is a subgroup such that $A^2$ is a union of left $H$-cosets; moreover, $H$ is maximal subject to this property. Is it true ...
- 23.5k
12
votes
2
answers
968
views
Generators of a group and normal subgroups
Can we say anything about a minimal generating set of a finite group based on its normal subgroups? For example, can we bound their order, or say whether they come from the same conjugacy class?
An ...
- 185
2
votes
0
answers
206
views
Doubly transitive groups in which a point stabilizer has an abelian normal subgroup
Let $G$ be a finite doubly transitive group in its action on the set $X$, such that a point stabilizer $G_x$ ($x \in X$) has an abelian normal subgroup $N_x$.
I have read that if $\vert N_x \vert$ is ...
- 4,837
0
votes
1
answer
515
views
Commutator group and conjugacy classes
Let $G$ be a finite solvable group which is not nilpotent, and let $H=[G,G]$ be the commutator subgroup of $G$. Does the following hold for $G$ and $H$?
"There exists $g \in G \setminus H$ and $h ...
- 227
0
votes
0
answers
344
views
Order of elements in amalgamated free products
Reading the book "A Course in the Theory of Groups" by D. J. S. Robinson, I was looking at the proof of 6.4.3 (iii), which states (suppose we are in the case of two groups): if $G_1$ and $...
user508460
1
vote
1
answer
248
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Finite subgroups of $\mathrm{O}_n(\mathbb R)$ from finite subgroups of $\mathrm{SO}_n(\mathbb R)$
Let $G$ be a finite subgroup of $\mathrm{SO}_n(\mathbb R)$. We also assume $G$ to be "maximal" in the sense that for every $g\in\mathrm{SO}_n(\mathbb R)\setminus G$, we have that $\overline{\...
- 409
1
vote
1
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Subgroups of $\operatorname{PGL}_n$
As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, $\operatorname{GO}_{2n}$ is a closed normal subgroup of the conformal orthogonal group $\operatorname{CO}_{2n}$. ...
- 491
1
vote
0
answers
553
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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}$...
- 1,029
3
votes
1
answer
318
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 ...
- 227
9
votes
1
answer
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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 ...
- 2,569
14
votes
1
answer
1k
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$ ...
- 181
1
vote
1
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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(...
user121195
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1
answer
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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/...
user121195
2
votes
1
answer
146
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 ...
- 1,100
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votes
0
answers
55
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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}$ (...
- 2,658
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votes
1
answer
183
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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 ...
- 54.4k
6
votes
3
answers
518
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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 ...
- 61
3
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0
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230
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 $(...
- 1,349
1
vote
0
answers
48
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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 ...
- 591
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votes
1
answer
144
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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 ...
- 591
5
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1
answer
241
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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 ...
- 53
6
votes
3
answers
270
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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 ...
- 933
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2
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2k
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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?
($\...
- 2,658
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votes
2
answers
1k
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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 ...
- 517
2
votes
0
answers
128
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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 ...
user111524
2
votes
1
answer
163
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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 ...
- 10.7k
12
votes
2
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577
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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)=\...
- 492
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votes
0
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214
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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.
- 93
3
votes
1
answer
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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 ...
- 1,165
2
votes
0
answers
292
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
609
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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 ...
- 3,412
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votes
4
answers
801
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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
502
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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
841
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$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, ...
- 3,395
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votes
1
answer
3k
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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\...
- 2,253
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votes
4
answers
2k
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Being a subgroup: proof by character theory
Let me first cite a theorem due to Frobenius:
Let $G$ be a finite group, with $H$ a proper subgroup ($H\ne (1)$ and $G$). Suppose that for every $g\not\in H$, we have $H\cap gHg^{-1}=(1)$. Then
$$...
- 53.1k
8
votes
3
answers
2k
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Normal subgroups of braid groups
There is a lot of normal subgroups in braid groups (for example there is an action of $B_n$ on unitriangular bilinear forms on $R^n$ over arbitrary commutative ring $R$: $b_i\colon e_j\mapsto e_j$, $j\...
- 1,452
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votes
1
answer
1k
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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 ...
- 307
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votes
2
answers
2k
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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 ...
- 13.8k
29
votes
5
answers
4k
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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 ...
- 120k
34
votes
4
answers
11k
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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'...
- 4,840
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votes
5
answers
5k
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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?
- 525