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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Intuition about the semidirect product of groups

If we have two groups $G,H$ the construction of the direct product is quite natural. If we think about the most natural way to make the Cartesian product $G\times H$ into a group it is certainly by ...
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Question regarding writing a group of order $p^2qr$ using notations

Let $G$ be a solvable, non-nilpotent group of order $p^2qr$, where $p,q,r$ are distinct primes, and let $F$ be a Fitting subgroup of $G$. Then $F$ and $G/F$ are both non-trivial and $G/F$ acts ...
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Find all semidirect products of $(\mathbb{Z}_4,+)$ by $C_2$

Problem: Find all semidirect products of $(\mathbb{Z}_4,+)$ by $C_2$ (the cyclic group of order $2$). My attempt: We know that $(\mathbb{Z}_4,+)$ is a cyclic group of order $4$. To find all ...
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Semidirect product that is not isomorphic to direct product

On Lee's Introduction to smooth manifold, Page 173, there is a problem wants to show that for $n>1$ $O(n)\cong SO(n) \rtimes O(1)$(which has been shown) is not isomorphic to $SO(n)\times O(1)$ as a ...
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Abelianization of $\mathbb{Z}\ltimes_\varphi \mathbb{Z}^n$

i would like to ask how to compute the abelianization of the semidirect product $\mathbb{Z}\ltimes_\varphi\mathbb{Z}^n$ where the action is $\varphi(k)v=A^k v$ where $A$ is a fixed invertible matrix ...
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$N\rtimes_{\phi}H\cong N\rtimes_{\phi\circ\psi}H$ for $N,H$ be groups, $\phi \colon H\rightarrow Aut(N)$ be a homomorphism, $\psi \in Aut(H)$

Let $N,H$ be groups, $\phi \colon H\rightarrow Aut(N)$ be a homomorphism, if $\psi \in Aut(H)$, prove that $$N\rtimes_{\phi}H\cong N\rtimes_{\phi\circ\psi}H.$$ This is mentioned in the original post. ...
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Find all group of order $20$ which is a semidirect product of a cyclic group of order $4$ by a cyclic group of order $5$

Problem: Find all group of order $20$ that are a semidirect product of a cyclic group of order $4$ by a cyclic group of order $5$. My attempt: We knew that a cyclic group of order $n$ is isomorphic ...
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K-theory of semidirect product

Given a semidirect product $G=A\rtimes B$ is there a general way to find the $K$-theory $K_0(G)$ and $K_1(G)$ of the semidirect product from $A$ and $B$?
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What is the dicyclic group of order $12$? (What is $\mathbb{Z}_3\rtimes \mathbb{Z}_4$)

I have come across the dicyclic group of order $12$. I can see that this is generated by three elements subject to some relations. Is there a way to realize this group without talking about generators ...
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Behaviour of restrictions of automorphisms of groups on characteristic subgroup under epimorphisms

Let $G = H \rtimes_\alpha K$, where $H$ is abelian and characteristic in $G$. Let $\phi\in\mathrm{Aut}(G)$, and $\phi'$ is its restriction: $\phi'=\phi\big\rvert_H$. Let $A = B \rtimes_\beta K$, ...
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Problem related to semidirect product

I have a small question regarding the semidirect product. Consider a group $G$ which is the semidirect product $\mathbb{Z}_3 \ltimes (\mathbb{Z}_5 \times \mathbb{Z}_5)$ (internal semidirect product). ...
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How can I show that $D_{2n} \cong C_n \rtimes C_2$

Let $D_8 := \langle a,b \mid a^4 = 1 = b^2, bab = a^{-1}\rangle$ I'm trying to formally show that $$D_{8} \cong C_4 \rtimes C_2 = \langle s\rangle \rtimes \langle t \rangle$$ My book gives as hint ...
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isomorphism between semi-direct products

given two groups H and K, and two morphisms $φ$ and $φ'$ from K to $Aut(H)$. given $σ \in Aut(K)$ such that $φ' = φ ◦ σ$, prove that $H \rtimes_φ K \cong H \rtimes_{φ'} K$. I found this simple ...
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Extension of $\mathbb Z_2$ by $SO(n)$

How to show that the extension of group $\mathbb Z_{2}$ by $\operatorname{SO}(n)$: $$\operatorname{Id} \to \operatorname{SO}(n) \to \operatorname{O}(n) \xrightarrow{\det} \mathbb Z_2 \to 1$$ is a ...
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Semi-direct v.s. Direct products

What is the difference between a direct product and a semi-direct product in group theory? Based on what I can find, difference seems only to be the nature of the groups involved, where a direct ...