All Questions
Tagged with finite-groups normal-subgroups
19 questions
35
votes
4
answers
2k
views
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
$...
- 52.3k
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 ...
- 13k
12
votes
2
answers
698
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 ...
- 175
12
votes
2
answers
488
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)=\...
- 492
12
votes
0
answers
775
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, ...
- 3,291
8
votes
4
answers
658
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 ...
7
votes
1
answer
958
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 ...
- 307
6
votes
3
answers
428
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 ...
- 61
6
votes
3
answers
213
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 ...
- 933
6
votes
1
answer
651
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/...
user121195
5
votes
1
answer
364
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 \...
- 361
2
votes
2
answers
527
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 ...
- 3,362
2
votes
1
answer
112
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 ...
- 1,090
2
votes
0
answers
174
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 ...
- 4,547
1
vote
1
answer
317
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(...
user121195
1
vote
0
answers
109
views
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 ...
- 111
0
votes
1
answer
339
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 ...
- 217
0
votes
1
answer
183
views
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{\...
- 245
0
votes
2
answers
956
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 ...
- 487