Skip to main content

Questions tagged [semidirect-product]

The semidirect product is a construction in group theory generalizing the direct product. It arises as the structure of a group $G$ with a normal subgroup $N$ having a complement $N$.

Filter by
Sorted by
Tagged with
0 votes
1 answer
36 views

Splitting lemma for Non-abelian groups (semi-direct product)

I was reading the wikipedia page to learn about how the splitting lemma partially holds for non-abelian groups. Splitting lemma partially true $$ 0\to A \to B \to C \to 0 $$ If a short exact sequence ...
stoneaa's user avatar
  • 424
5 votes
2 answers
272 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 ...
user760's user avatar
  • 1,670
2 votes
2 answers
47 views

What is the semidirect product we use in Levi Decomposition

So using the Levi Decomposition for any lie algebra $\mathfrak{g}$, there exists a semisimple subalgebra $\mathfrak{s}$ such that: $Rad(\mathfrak{g})$$\ltimes$$\mathfrak{s}$=$\mathfrak{g}$ However in ...
Albi's user avatar
  • 69
1 vote
0 answers
88 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 $M$ ...
ghc1997's user avatar
  • 1,641
3 votes
1 answer
236 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 ...
JAG131's user avatar
  • 917
2 votes
2 answers
141 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 ...
Ryan Zhou's user avatar
  • 110
1 vote
1 answer
82 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 $G \cap K=1$. ...
Juanandete's user avatar
-1 votes
1 answer
62 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 ...
noparadise's user avatar
-1 votes
1 answer
111 views

Character table for a covering group of $\mathbb{Z}_n \times \mathbb{Z}_m$

I’m considering the group $G = \mathbb{Z}_n \times \mathbb{Z}_m$ and its covering group $$G^* = \langle \alpha, \beta, a|\alpha a = a\alpha, \beta a = a\beta, a^p = 1, \alpha^n = 1, \beta^m = 1, \...
slowspider's user avatar
  • 1,065
0 votes
1 answer
45 views

Is the group of 4 by 4 nilpotent triangular matrices a semidirect product?

Let $N$ be the Lie group of 4 by 4 nilpotent triangular matrices with 1 on the diagonal. Let us denote by $E_{ij}$ the square matrix with entry 1 where the $i$th row and $j$th column meet, all the ...
alejandro's user avatar
  • 123
1 vote
1 answer
71 views

Understanding Multiplicative Group Structure On Hom$(\mathbb{Z}/n\mathbb{Z},\text{Aut}(\mathbb{Z}/m\mathbb{Z}))$

Motivation. This post comes in direct response to the fact that $($Hom$(G,G'),+)$ inherits an (additive) group structure whenever $G, G'$ are abelian. In particular, I'm curious what properties one ...
JAG131's user avatar
  • 917
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 ...
mathlover's user avatar
3 votes
2 answers
263 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 ...
Cosine's user avatar
  • 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 ...
NewViewsMath's user avatar
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 ...
Greg's user avatar
  • 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 ...
Ty Perkins's user avatar
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 $b$. ...
Aron's user avatar
  • 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 $...
Siddhartha's user avatar
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 ...
ghc1997's user avatar
  • 1,641
-1 votes
1 answer
97 views

Semidirect product of groups

In recent days I am studying semidirect product of groups and I have come up with the following question which has already been answered here (From semidirect to direct product of groups), but I can't ...
Priya Sarkar's user avatar
2 votes
2 answers
65 views

Using combinational group theoretical perspective on semidirect products, show $\langle r,s\mid r^8, s^2, srs=r^3\rangle$ has two Klein four subgroups

Note: This is an alternative-proof question, since I know how to prove the result but I'm asking for a particular kind of proof. Why? For the fun of it! Motivation: I've been trying to give a reason ...
Shaun's user avatar
  • 45.7k
1 vote
0 answers
54 views

Monolithic Groups

I have recently gone through the semi-direct product of groups. While I was studying monolithic groups (a group is said to be monolithic if it has a unique minimal normal subgroup, and this is ...
Pratina's user avatar
  • 149
2 votes
0 answers
47 views

Showing that a group is isomorphic to a semidirect product

Let's say that we have two groups $ G $ and $ H $ with an epimorphism $ \phi : G \to H $. Denote $ B $ to be the kernel of $ \phi $. If we have another group $ C $ isomorphic to $ B $ and a ...
mathmehmet's user avatar
1 vote
1 answer
71 views

Explicit construction of $(\mathbb{Z}_{3} \times \mathbb{Z}_{3}) \rtimes _{\varphi} \mathbb{Z}_{2}$

I am new to group theory and I wanted to explore further the concept of external semidirect product by constructing $(\mathbb{Z}_{3} \times \mathbb{Z}_{3}) \rtimes _{\varphi} \mathbb{Z}_{2}$ for a ...
serpens's user avatar
  • 344
3 votes
0 answers
115 views

Can we make my analogy between semidirect products and twisted groups of lie type more precise?

Note: This is a soft-question. Last week, my PhD supervisor spent a good 45 minutes going over Steinberg endomorphisms and twisted groups of Lie type. There's a chance they'll be important for my ...
Shaun's user avatar
  • 45.7k
0 votes
0 answers
43 views

Open normal subgroups with finite image under extensions

Let $A$ be a (discrete) countable group such that for ever completely metrizable group $M$ and any (not necessarily continuous) homomorphism $f\colon M \to A$ there exists some open normal subgroup $N\...
user12345's user avatar
3 votes
1 answer
67 views

Finite image property under extensions

Let $L$ be a group. We say a group $H$ only admits finite images of $L$, if for every homomorphism $f\colon L\to H$ the image $f(L)$ is finite. Now, assume that $G$ is a group and $H\subseteq G$ be a ...
user12345's user avatar
0 votes
0 answers
31 views

Equivalence definition to semidirect product in the operated context

An operated semigroup (or a semigroup with an operator) is a semigroup $U$ together with an operator $\alpha : U \to U$ that is called the distinguished operator on $U$. Is there any definition ...
Nil's user avatar
  • 1,312
0 votes
0 answers
88 views

Decomposition of Homomorphism

Is it correct that if $\psi \ : G \rtimes_{\phi_1} H \to G \rtimes_{\phi_2} H$, then $\exists f \in Aut(G)\ and \ g \in Aut(H)$, such that $\psi(G \rtimes_{\phi_1} H) \cong f(G) \times g(H)$ What ...
Charlie's user avatar
  • 35
0 votes
0 answers
12 views

Can break the isomorphism between 2 semi-direct products componentwisely?

Consider the isomorphism $$\begin{align*} \psi : G \rtimes_{\phi_{1}} H \to G \rtimes _{\phi_{2}} H \end{align*}$$ where $G \rtimes_{\phi_{i}} H$ is the semi-direct product of $G$ and $H$ with ...
M_k's user avatar
  • 1,921
4 votes
1 answer
135 views

How to prove $\mathbb{R}^{2} \rtimes_{\phi_{n}} SO(2) \not \cong \mathbb{R}^{2} \rtimes_{\phi_{m}} SO(2)$ for different $m,n \in \mathbb{N}$

Let $$\begin{align*} r_{\theta} = \begin{pmatrix} \cos\theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \end{align*}$$ for $\theta \in[0,2\pi]$ . Then all $r_{\theta}$ forms $...
M_k's user avatar
  • 1,921
0 votes
0 answers
44 views

Equivalent definition semidirect products

We were recently taught in lecture the definition of the semidirect product: Definition: A group $G$ is a semidirect product of Subgroups $H,K$ if $H$ is normal and the canonical projection $G \to G/H$...
MathMaestro's user avatar
0 votes
1 answer
64 views

Exact short sequences and semidirect products

In several sources I have read that a group is semidirect product $G = N \rtimes_\phi G$ iff there is a short exact sequence related, of the form $0 \rightarrow N \rightarrow \ G \rightarrow K \...
Emmy N.'s user avatar
  • 1,361
2 votes
0 answers
50 views

Non abelian groups of order $28$ and elements of certain order [duplicate]

I have been asked to say how many non-abelian groups of order $28$ are there so that there is at least an element of order 4. Using Sylow's Theorem, it must be that $n_2 = 1 (\mod \ 2)$ and $n_2 | 7$, ...
Emmy N.'s user avatar
  • 1,361
1 vote
1 answer
63 views

Embedding of $\mathbb{Z}$

In a certain exercise I am asked to prove something that involves semidirect product. I wanted to know if $\mathbb{Z}$ could be embedded in a semidirect product of bigger groups, named for example $A,...
Emmy N.'s user avatar
  • 1,361
0 votes
1 answer
113 views

$N \rtimes H$ vs $H \rtimes N$

When we define the semidirect product for $N \lhd G$ with $H < G$. We assumed $N$ is normal because that makes $f: N \times H \to G$, $f(n,h) = nh$ an isomorphism when we assume $f$ is a bijection. ...
wsz_fantasy's user avatar
  • 1,706
0 votes
1 answer
23 views

Properties of Bicrossed Product multiplcation

I am reading Kassel's Quantum Groups book (the chapter on Drinfeld doubles). In it, there is the following claim: If $H,K\subseteq G$ are groups such that $\forall g\in G$, $\exists!(y,z)\in H\times K$...
Wyatt Kuehster's user avatar
2 votes
0 answers
78 views

Show that the Hyperoctahedral Group is a semi-direct product

I am confused about an exercise from Representations of Finite Groups by C.Musili (Exercise 7.10.1), which asks the reader to show that the hyperoctahedral group $B_n$ is a semi-direct product. First ...
Josh's user avatar
  • 21
0 votes
1 answer
53 views

What is the smallest split-simple non-simple group other than generalized quaternion groups and cyclic groups?

The quaternion group $Q_8$ is split-simple, i.e. it cannot be written as an internal semidirect product of proper subgroups. In fact all generalized quaternion groups are split-simple, as are all ...
Keshav Srinivasan's user avatar
2 votes
0 answers
72 views

Semi direct Product question? [closed]

SETUP: Let $G$ be a group such that $G= N \times_\varphi H$ semi-direct product of $H$ and $N$ with respect to the action $\varphi$. Denote: $$p(G):= \max\{ p \in \mathbb{N}^*\: | \: \mathbb{R}^p \...
Yushi MuGiwara's user avatar
0 votes
0 answers
51 views

Classify groups of order 20. [duplicate]

A group of order $20$ must be either a semidirect product of $\Bbb Z/4\Bbb Z$ with $\Bbb Z/5\Bbb Z$ or a semidirect product of $\Bbb Z/2\Bbb Z\times \Bbb Z/2\Bbb Z$ with $\Bbb Z/5\Bbb Z$. There are $4$...
user108580's user avatar
1 vote
1 answer
69 views

Conjugate images induce isomorphic semi-direct products?

See here for the question context. Let $H_5 = \Bbb Z_5\times \Bbb Z_5$. I am trying to prove that for all non-trivial homomorphism $$\varphi:\Bbb Z_3\to \operatorname{Aut}(H_5),$$ the resulting semi-...
user108580's user avatar
3 votes
1 answer
70 views

Isomorphism in quotient spaces of linear spaces

Let $E$ be a linear space and $V$ a linear subspace. No assumption is made on the dimension of $E$. We write $G = \{g \in GL(E) \mid g(V) = V\}$ where $GL(E)$ is the group of invertible linear maps of ...
mathcounterexamples.net's user avatar
0 votes
0 answers
37 views

Semidirect product in Dummit and Foote Abstract algebra

This is regarding Theorem 12 on Page 180 of Dummit and Foote's Abstract Algebra. The theorem asserts that if $G$ has subgroups $H$ and $K$ with $H$ normal and $H \cap K = 1$, then with $\phi: K \to \...
Vanya's user avatar
  • 499
5 votes
0 answers
226 views

Obstruction of semidirect product from being a direct product?

Let $G$ and $H$ be (nonabelian) groups, and $\varphi: G\to \text{Aut}(H)$ be a homomorphism. This defines the semidirect product $H\rtimes_{\varphi} G$. I am wondering, what is the best way to ...
El Rafu's user avatar
  • 608
5 votes
0 answers
143 views

When are two semidirect products of two cyclic groups isomorphic

A semidirect product of two cyclic groups $C_m$ and $C_n$ has the form $$ C_m \rtimes_k C_n = \langle x,y \mid x^m = y^n = 1,\, yxy^{-1} = x^k \rangle, $$ for some $k^n \equiv 1\pmod m$. Now, a ...
Jianing Song's user avatar
  • 1,923
4 votes
0 answers
100 views

If $G=(A×B)\rtimes(C×D);$ $|A|=5,|B|=7,|C|=2,|D|=3,$ $C$ induces inversion on $A×B$, & $D$ acts nontriv on $B$, then ${\rm cs}^*(G)=\{2,6,7,14,35\}$

I'm trying to understand how the author calculates the conjugacy sizes here: Consider $G= (A \times B) \rtimes (C \times D)$, where $A$, $B$, $C$, $D$ are cyclic groups of order 5, 7, 2, 3, ...
Soheil Haghighi's user avatar
1 vote
1 answer
40 views

Are affine symmetries an internal or external semidirect product?

Let $A$ be an affine space with translation space $V$. We may think of it as a nonempty set $A$ endowed with a regular action of $V \times A \to A :(v,p)\mapsto p+v$ (the affine action). The symmetry ...
giobrach's user avatar
  • 7,532
0 votes
1 answer
167 views

Explanation of semidirect products

I have been reading many pdfs and books about abstract algebra, covering almost completely the topic of groups. Fortunately, I have been able to understand almost everything with the exception of ...
Emmy N.'s user avatar
  • 1,361
1 vote
0 answers
21 views

Groups extensions by linear cocycles

Let $0 \to A \to X \to C \to 0$ be an abelian group extension (we require $X$ to be abelian too). Then the group operation on $X$ is described by a $2$-cocycle $c(x, y) = s(x) + s(y) -s(x+y)$ where $s:...
Matteo Casarosa's user avatar

1
2 3 4 5
15