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 $N$.
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Presentation of non Abelian group (C7⋊C3)⋊C2 . [closed]
I want the Presentation of non Abelian group (C7⋊C3)⋊C2 of order 42.
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 $G \to G/H$...
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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 \...
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Non abelian groups of order $28$ and elements of certain order
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 $n_2 | 7$, ...
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answer
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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 $A,...
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$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. ...
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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 $\forall g\in G$, $\exists!(y,z)\in H\times K$...
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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 ...
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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 ...
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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: $$p(G):= \max\{ p \in \mathbb{N}^*\: | \: \mathbb{R}^p \...
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Classify groups of order 20. [duplicate]
A group of order $20$ must be either a semidirect product of $\Bbb Z/4\Bbb Z$ with $\Bbb Z/5\Bbb Z$ or a semidirect product of $\Bbb Z/2\Bbb Z\times \Bbb Z/2\Bbb Z$ with $\Bbb Z/5\Bbb Z$.
There are $4$...
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Conjugate images induce isomorphic semi-direct products?
See here for the question context.
Let $H_5 = \Bbb Z_5\times \Bbb Z_5$. I am trying to prove that for all non-trivial homomorphism $$\varphi:\Bbb Z_3\to \operatorname{Aut}(H_5),$$ the resulting semi-...
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Isomorphism in quotient spaces of linear spaces
Let $E$ be a linear space and $V$ a linear subspace. No assumption is made on the dimension of $E$.
We write $G = \{g \in GL(E) \mid g(V) = V\}$ where $GL(E)$ is the group of invertible linear maps of ...
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Semidirect product in Dummit and Foote Abstract algebra
This is regarding Theorem 12 on Page 180 of Dummit and Foote's Abstract Algebra. The theorem asserts that if $G$ has subgroups $H$ and $K$ with $H$ normal and $H \cap K = 1$, then with $\phi: K \to \...
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Obstruction of semidirect product from being a direct product?
Let $G$ and $H$ be (nonabelian) groups, and $\varphi: G\to \text{Aut}(H)$ be a homomorphism. This defines the semidirect product $H\rtimes_{\varphi} G$. I am wondering, what is the best way to ...
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When are two semidirect products of two cyclic groups isomorphic
A semidirect product of two cyclic groups $C_m$ and $C_n$ has the form
$$
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 ...
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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,
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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 ...
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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 ...
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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:...
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Abelian subgroups of the group of automorphisms of a finite group
This is a follow-up question from my post here, which has been moved according to a comment. For context, here is the setup.
Let $G$ be a nontrivial finite group. In his book "Finite Group ...
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Size of $p$-subgroups of $\operatorname{Aut}(G)$, where $p$ divides the order of $G$
Let $G$ be a nontrivial finite group. In his book "Finite Group Theory", M. Isaacs proves the following two results:
Corollary 3.3: Let $\sigma \in \operatorname{Aut}(G)$. Then, $o(\sigma) &...
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Semidirect Products of $A_n$ and $\mathbb{Z}/2$ up to isomorphism [closed]
Several posts on this site mention that the symmetric group $S_n$ is the semidirect product of $A_n$ and $\mathbb{Z}/2$. Is it true that $S_n$ is the only nontrivial semidirect product $A_n \rtimes \...
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what will be the exponential map from the Lie algebra $sl(2, C) \ltimes_{ad} sl(2,C)$ to its Lie group $Sl(2, C) \ltimes_{Ad} Sl(2,C)$.
Let $G:=Sl(2, C) \ltimes_{Ad} Sl(2,C)$, where $Ad_X(Y)=XYX^{-1}$ for all $X, Y \in SL(2,C)$. It is known that $G$ is a Lie group and its Lie algebra is given by $g:=sl(2, C) \ltimes_{ad} sl(2,C)$, ...
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Associativity of a semidirect product
I have the following problem. Let
$$
0\to A\to G\to Q\to 1
$$
be a central group extension with $A$ abelian. Assume that this extension splits, i.e., $G\cong A\rtimes Q$.
Now consider an action of ...
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The semidirect product of $\Bbb R^*$ and $\Bbb R$ is isomorphic to the group of upper-triangular real 2-by-2 matrices with determinant 1.
I want to present the group of matrices $$G=\{
\begin{pmatrix}
a & b \\
0 & \frac1a \\
\end{pmatrix}; a,b\in\Bbb R, a\ne0
\}.$$ as a semidirect product of $\Bbb R^∗$ and $\Bbb R$ with ...
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Quadratic extension with Galois group semidirect product
Let $K$ be a quadratic extension and let $F/K$ be an abelian Galois extension with Galois group $H=Gal(F/K)$. Assume that $F/\mathbb Q$ is Galois with Galois group given by the semidirect product $Gal(...
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Another identity implied by the Jacobi identity for semidirect product of Lie algebra
I'm working through Introduction to Lie Algebras by Erdmann and Wilson. In Chapter 3, Exercise 3.9 asks the reader to develop the semidirect product of Lie algebras. I got very stuck on the ...
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Non-Abelian group of exponent $3$ and Nilpotent class $2$ .
Let $G$ be a non-Abelian group such that $G^3 = 1$ and $G$ is a nilpotent class $2$ group, with order $3^{32}$. Our task is to determine the structure of the group $G$ or identify any information ...
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What am I missing here? Trying to learn the semidirect product
I'm trying to understand the concept of the semidirect product of two groups.
An application is this answer to a question, where for $p>q$ primes, $q \mid p-1$, there is a non-abelian group of ...
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Consider the group $\mathbb{Z}_2 \wr \mathbb{Z}$, what is $\mathbb{Z}_2 \wr 2 \mathbb{Z}$?
Consider the (Lamplighter) group $(\bigoplus_{n=-\infty}^{n=\infty}\mathbb{Z}_2) \rtimes_\phi\mathbb{Z}$, where $\phi(1)$ "shifts" every element in $\bigoplus_{-\infty}^{\infty}\mathbb{Z}_2$ ...
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Complements in wreath product
Let $A$ be a finite abelian group, and consider the wreath product $A\wr \mathbb{Z}/2= (A\oplus A)\rtimes\mathbb{Z}/2$.
Is it possible to describe all the complements $K$ of the subgroup $A\oplus 0\...
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Proving that there exists a non-trivial semidirect product $\mathbb{Z}/41\mathbb{Z} \rtimes \mathbb{Z}/20\mathbb{Z} $
I want to prove that there exists a non-trivial semidirect product $\mathbb{Z}/41\mathbb{Z} \rtimes \mathbb{Z}/20\mathbb{Z} $. I know this must be given by a non-trivial group homomorphism $\phi: \...
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Showing that $GL(2,p) = SL(2,p) \rtimes H$
I want to show that $GL(2, p)$ is the semidirect product of $SL(2, p)$ and the subgroup $H$ of $2 \times 2$ diagonal matrices with elements on the diagonal being $1$ and $\alpha$, where $\alpha \in \...
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If $|G|=2^2\cdot 5 \cdot 19\Rightarrow n_5=n_{19}=1$
By using the Sylow Theorems we need $n_5\in \{1,2^2\cdot 19\}$ and $n_{19}\in\{1,20\}$.
First Case: If $n_5=2^2\cdot 19$ and $n_{19}=20$, the group cannot stand the pressure: we have too many elements....
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Let $G=P\rtimes Q,P=\Bbb Z_7=\langle b\rangle,Q=\Bbb Z_3=\langle a\rangle.$ Let $\theta:Q\to{\rm Aut}P$ be $\theta(a)(b)=b^4$. Classify $G/G'$.
Let $G=P\rtimes Q$ with $P=\mathbb{Z}_7=\langle b\rangle$ and $Q=\mathbb{Z}_3=\langle a\rangle.$ Take $\theta: Q\rightarrow {\rm Aut}P$ to be $\theta(a)(b)=b^4$. Classify up to isomorphism $ G/G'$.
...
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Is a finitely generated non-abelian group embedded in a semidirect product of a torsion group and a torsion-free group?
Let $G$ be a finitely generated group and $Tor(G)$ the subgroup generated by all torsion elements of $G$. Then $G/Tor(G)$ is torsion-free. In the abelian case we have
Every finitely generated abelian ...
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$N \cong (\mathbb{Z}/p\mathbb{Z}) \rtimes_{\varphi} (\mathbb{Z}/p\mathbb{Z})^{\times}$ where $N$ is the normalizer of Sylow $p$-subgroup of $S_p$
Question: Suppose $p$ is prime, $P \subset S_p$ is a Sylow $p$-subgroup of $S_p$ (the symmetric group on $p$ elements) and $N = N(P)$ is the normalizer of $P.$ Show that
$$ N \cong (\mathbb{Z}/p\...
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Group Ring $F(G\rtimes H)$ .
Is there any relation between group rings $F(G\rtimes H)$ and $F(G)\rtimes F(H)$, where F is a finite field and $\rtimes$ is semi direct product of finite groups $G$ and $H$?
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A problem about semi-direct product which is a direct product
I have been stuck on this problem for a while now. I tried to look up similar problems with possible hints but couldn't find anything. The problems is as follows:
Let $N\rtimes_\theta H$ be the semi-...
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Structure Description of p-groups of nilpotency class 2, where p is an odd prime.
Suppose $G=\langle a,b \mid [a,b]^{p^\gamma}=[a,b,a]=[a,b,b]=1, a^{p^{\alpha}}=[a,b]^{p^\rho}, b^{p^{\beta}}=[a,b]^{p^\sigma}\rangle$, where $\alpha>\beta\geq \gamma\geq1$ and $0\leq\sigma<\rho&...
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When are elements of a (perfect) semidirect product simple commutators?
I have a semidirect product and I have shown that it is perfect. However, I would like to know whether every element is a simple commutator (rather than a product of commutators). I am not aware of ...
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1
answer
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Expressing matrix group as a semidirect product: matrix multiplication is reversed
I am trying to determine the group that the set of matrices of the form
$$
\begin{bmatrix}
1 & 0 & a \\
0 & 1 & b \\
0 & 0 & c
\end{bmatrix}
$$
with $a, b, c \in \mathbb{F},...
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1
answer
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Do all unitary simple groups $U_{2n+1}(2)$ have maximal subgroups of the form $3^{2n}:S_{2n+1}$?
In the ATLAS, the unitary simple groups $U_5(2)$ and $U_7(2)$ have maximal subgroups of structures $3^4:S_5$ and $3^6:S_7$, respectively. It seems that they are subgroups of the generalized symmetric ...
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