All Questions
Tagged with group-theory semidirect-product
608
questions
5
votes
2
answers
268
views
Is every group the semidirect product of its center and inner automorphism group?
For every group $G$, we have $$G/Z(G)\simeq \operatorname{Inn}(G).$$
I wonder whether the quotient projection has a right inverse. I suspect it doesn’t have one in general. But I can’t find a counter ...
- 1,662
1
vote
0
answers
84
views
How fast does the number of "fixed" points grow compared the the size of the ball in the following group?
Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb Z ^5 $. Let $e_1, e_2, e_3, e_4, e_5$ be the standard basis of $\mathbb Z^5$. Let $e_j^{(i)}$ be the element in $M$ ...
- 1,581
3
votes
1
answer
234
views
Non-Simple, Centerless Group With Exactly One Non-Trivial Normal Subgroup
Question. (Is this statement true?) Given non-simple, centerless group $G$ such that there exists exactly one non-trivial normal subgroup $N\triangleleft G$, then $G/N$ must be isomorphic to some ...
- 905
2
votes
2
answers
135
views
Trouble with understanding classifying groups with semi direct products
I'm trying to understand the following strategy on classifying groups of a particular order from Dummit & Foote's Abstract Algebra (p.181):
Let $G$ be a group of order $n$.
You find proper ...
- 110
1
vote
1
answer
78
views
Relation between centerless groups and semidirect products
Let $H$ be a group, $G \triangleleft H$ a normal subgroup and $K \leqslant H$ a subgroup. We say $H$ to be a semidirect product of $G$ and $K$, denoted by $H= G \rtimes K$, if $H=GK$ and $G \cap K=1$. ...
- 21
2
votes
0
answers
162
views
Examples of abelian-by-cyclic groups whose fixed points of the defining automorphisms grows exponentially
Suppose we have a finitely generated group $G = K\rtimes_\phi \mathbb{Z}$ with $K$ being abelian. Let $T$ be a finite subset of $K$ such that $S = \{ (0,1), (k,0)\mid k \in T \}$ is a generating set ...
- 1,581
-1
votes
1
answer
60
views
Alternative definitions outer semidirect product [closed]
I am working with outer semidirect products and have come across two definitions that are slightly different.
Let $N,H$ be two groups and $\theta: H \to \text{Aut}(N)$ a group morphism. Then in both ...
- 415
1
vote
0
answers
29
views
Finding irreducible representations of $D_{2n}$ using Mackey little group method
Let $D_{2n}$ be the dihedral group on 2n elements, consisting of n rotations and n reflections. I know the group of n rotations form a normal subgroup of $D_{2n}$ and $D_{2n}$ is a semidirect product ...
- 11
3
votes
2
answers
261
views
If G is a product of two subgroups, must one of them be normal? [closed]
I have the following problem with semidirect products which seems like a basic question in group theory, but I could not find the answer.
Let $G$ be a group, and let $H_1$ and $H_2$ be two subgroups ...
- 412
1
vote
1
answer
62
views
What is the action on $\mathrm{Sp}_2(q^2)$ which makes $\mathrm{Sp}_2(q^2)\colon 2$ a maximal subgroup of $\mathrm{Sp}_4(q)$ for an even power of $q$?
I know that there's a homomorphic embedding of the finite field $\mathbb{F}_{q^2}$ into $2\times 2$ matrices over the field $\mathbb{F}_q$. But I cannot determine the action of the semi-direct product ...
- 169
2
votes
1
answer
89
views
Sylow subgroups of semidirect products [closed]
Suppose that $G = A \times B$ is a direct product of finite groups $A$ and $B$. Let $P$ be a Sylow $p$-subgroup of $G$. We have an epimorphism from $G$ to $A$ so that the image of $P$ in $A$ is a ...
- 422
1
vote
2
answers
64
views
Deducing there exists exactly $5$ isomorphism classes of groups of order $12$.
There's a substantial amount that's been written about the semi-direct products of a group of order $12$ on this website. However, there's something that seems to be taken for granted each time the ...
- 730
0
votes
1
answer
40
views
Number of conjugacy classes in each coset of a semidirect product is the same.
Let us consider a semidirect product $X=G \rtimes \langle\sigma\rangle$, where $\langle\sigma\rangle$ acts on $G$ via some automorphism. Assume all groups are finite and that $\sigma$ has order $b$.
...
- 263
1
vote
0
answers
81
views
About semi-direct product of two cyclic groups
The following question is related to seeing semi-direct products as subgroups:
Let $G$ be a non-nilpotent group and $U = \langle x \rangle$ be a cyclic characteristic subgroup of the Fitting subgroup $...
- 808
1
vote
0
answers
25
views
metabelian groups can be represented by matrices, do we know the exact representation?
We know that every finitely generated metabelian group can be represented by matrices (e.g. see https://link.springer.com/article/10.1007/BF02219822 ). I am particularly interested in how finitely ...
- 1,581