Questions tagged [holomorph]

A holomorph of a group $G$ is a semidirect product $G \rtimes Aut(G)$, where $Aut(G)$ acts naturally over $G$. To be used with tags [group-theory] and [automorphism-group].

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For what $n \in \mathbb{N}$ is $ \operatorname{Hol}(C_2^n)$ complete?

For what $n \in \mathbb{N}$ is $ \operatorname{Hol}(C_2^n)$ complete? Here $ \operatorname{Hol}$ stands for holomorph, and $C_2^n$ stands for direct product of $n$ isomorphic copies of $C^2$. It for $...
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Is the statement that $ \operatorname{Aut}( \operatorname{Hol}(Z_n)) \cong \operatorname{Hol}(Z_n)$ true for every odd $n$?

Is the statement that $ \operatorname{Aut}(\operatorname{Hol}(Z_n)) \cong \operatorname{Hol}(Z_n)$ true for every odd $n$? $Hol$ stands here for group holomorph. This problem appeared, when I ...
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Presentation of the holomorph of $\mathbb Z/5 \mathbb Z$

When I look up the presentation of the holomorph of $\mathbb Z/5 \mathbb Z$ it reads like the following: $\left\langle a,b \mid a^5 = 1, b^4 = 1, bab^{-1} = a^2\right\rangle$ See https://groupprops....
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Automorphism group of Hol($\mathbb{Z_n}$)

I am reading about semidirect product in Dummit,Foots. I wonder the automorphism group of $Hol(\mathbb{Z}_n)$, that is, $\mathbb{Z}_n\rtimes\mathbb{Z}_n^×$ with an operation $(a,b)*(c,d)=(a+bc,bd)$. ...
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Holomorph of a group $G$, then the automorphism of $G$ are inner automorphisms

In my course notes of algebra it says: Let $G$ be a group. Then $\mathrm{Aut}(G)$ acts on $G$ in a natural way through automorphisms. This allows us to consider $A:= G \rtimes \mathrm{Aut}(G)$. In ...
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Question about Proof - Holomorph as Normalizer in $Sym(G)$

I've adapted the following proof from Wikipedia (here). Let $\lambda: G→Sym(G)$ be the homomorphism induced by letting $G$ act on itself by left multiplication. Now let $H$ denote the normalizer of $\...
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The holomorph of $Z_2 \times Z_2$

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 184, Exercise 5): Let $G=\text{Hol}(Z_2 \times Z_2)$ (a) Prove that $G=H \rtimes K$ where $H=Z_2 \...
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Embedding $G$ in its holomorph

Let $H=G\rtimes \operatorname{Aut}G$ be the holomorph of $G$. Obviously $G$ embeds as $g\mapsto (g,1)$, and this embedding is normal. Composing this with any automorphism of $G$ is also a normal ...
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A question about holomorph of groups.

Question- If $G$ is complete, then the holomorph of $G$ is isomorphic to $G$×$G$. I am studying semidirect products for the first time, and in some notes i found this exercise. As far as i know about ...
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Normal closure of the nonnormal factor of Holomorph of a Cyclic group

Let $C_n$ be the cyclic group of order $n$. Then, we can consider the holomorph $G=C_n\rtimes Aut(C_n)$. let $H$ be such that $Aut(C_n)\leq H\trianglelefteq G$. Is it necessarily the case that $H$ is ...