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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Sylow-p-group of matrices group over finite field.

Let $F$ be a finite field of characteristic $p$ and $U$ a subgroup of $F^*$. Let $G$ be the Group of $n*n$ upper triangle matrices over $F$ with elements of $U$ on the diagonal. Find a Sylow-p-...
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Subgroup of semidirect product

Let $G$ be a semidirect product of a normal subgroup $A$ with a subgroup $B$ and Let $H$ be a subgroup of $G$ such that $H\cap A$ is trivial. Is it true that $H$ is contained in a conjugate of $B$ ? ...
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External Semidirect product and isomorphism

Let G and K be two groups and $\phi_1$ and $\phi_2: G \rightarrow Aut(K)$ be homomorphism. Q1: If $\phi_1$ not trivial homomorphism, can When can semidirect product of G and K using $\phi_1$ ...
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Let $G$ be a group which is the product $G=NH$ of subgroup $N,H\subset G$ where $N$ is normal. Let $N\cap H=\{1\}$. I am trying to show that that there is an iso $G\cong N\rtimes H$, with the ...
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When $G$ is a group, $N$ is a normal subgroup of $G$ and $H$ is another subgroup of $G$ where $N \cap H = \{1\}$, the normality of $N$ suggests that we can write, for $n_1, n_2 \in N$ and $h_1, h_2 \... 1answer 57 views Extension of$\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 ... 1answer 75 views About proving that$\operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n)$[closed] How can I prove that $$\operatorname{Aut}(\mathbb {D}_n) \cong \mathbb {Z}_n \rtimes \operatorname{Aut}(\mathbb {Z}_n),$$ where$\mathbb {D}_n$is the dihedral group. Can someone help me please? ... 0answers 49 views Progressed A problem on semidirect products where one component is cyclic : a specific problem I managed partially but stuck on the rest I was recently presented this in my abstract algebra class and I have managed some of it on my own the rest is still a mystery: Let$ H $be a group and$ K = \langle x\rangle $be a cyclic group (... 2answers 117 views When is a group a semi-direct product with its normal subgroup? Let$G$be an infinite non-abelian group. If we have a normal subgroup$N$of$G$, then can we always construct the subgroup$H$such that$G$is a semidirect product of$N$and$H$? 1answer 581 views Symmetric group isomorphic to semidirect product of Alternating group and Z/2Z I'm having a hard time understanding why$A_n \rtimes \mathbb{Z}_2 \cong S_n$. I understand that$A_n$is normal in$S_n$. But that's about it. What would the$\alpha$:$\mathbb{Z}$$_2$$\...
I have four questions related to Klein four group. and I know the answer two of them. ( the first two) and I want to know answer Does $V_4$ has an automorphism of order 6? What are the orders of ...