The Stack Overflow podcast is back! Listen to an interview with our new CEO.

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
23
votes
1answer
3k views

Group presentation for semidirect products

If $G$ and $H$ are groups with presentations $G=\langle X|R \rangle$ and $H=\langle Y| S \rangle$, then of course $G \times H$ has presentation $\langle X,Y | xy=yx \ \forall x \in X \ \text{and} \ y ...
7
votes
2answers
2k views

What is the center of a semidirect product?

Let $G_1$ and $G_2$ be groups. Let $\varphi:G_2\rightarrow \operatorname{Aut}(G_1) $ be a group homomorphism defining the semidirect product $G_1 \rtimes G_2$. Determine the center $\operatorname{Z}(...
13
votes
3answers
746 views

Lie Group Decomposition as Semidirect Product of Connected and Discrete Groups

I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in this Math Overflow thread, it is mentioned ...
13
votes
1answer
1k views

Describing the Wreath product categorically.

The Details: Let's have a recap of some definitions (taken from "Nine Chapters in the Semigroup Art" (pdf), by A. J. Cain). Definition 1: Let $P$ be a semigroup. The left action of $P$ on a set $X$...
3
votes
3answers
396 views

$\mathbb S_n$ as semidirect product

In this note, I've read that $\mathbb S_n$ is a semidirect product of the alternating group $A_n$ by $\mathbb Z_2$. So I am trying to define a morphism $\rho: \mathbb Z_2 \to Aut(A_n)$ to show that $\...
11
votes
1answer
1k views

Semidirect Products with GAP

I'm wondering how to specify to GAP which homomorphism to use when constructing a semidirect product. I'm trying to have it construct $\left(\mathbb{Z}_p\times\mathbb{Z}_p\right)\rtimes_\varphi S_3$. ...
3
votes
1answer
68 views

Describing the Lie algebra structure of a semi-direct product of Lie groups

Let $G$ and $H$ be Lie groups with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$. Suppose $G$ acts on $H$ by automorphisms, i.e. there exists a lie algebra homomorphism $\phi:G\to Aut(H)$. I want to ...
1
vote
1answer
96 views

When does a group of dilations/scalings exist in a metric space?

Notation: Let $(X,d)$ be a metric space. A similitude will be (by convention) a surjective (hence bijective) map $f: X \to X$ such that for all $x_1, x_2 \in X$, $d(f(x_1),f(x_2)) = r d(x_1, x_2)$ for ...
39
votes
1answer
11k views

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 ...
11
votes
2answers
4k views

What are the subgroups of a semidirect product?

Goursat's Lemma characterizes the subgroups of direct products. Is there a similar characterization for the subgroups of semidirect products? What about if I'm only interested in the normal subgroups?
7
votes
1answer
709 views

GAP semidirect product algorithm

Can anybody guide me towards, or possibly even explain here, the algorithm that GAP uses to compute the semidirectproduct of two permutation groups which outputs another permutation group? EXAMPLE: ...
5
votes
1answer
713 views

Semi-direct product Lie algebra

Given Lie algebras $S$ and $I$ and a Lie homomorphism $\theta:S\to Der I$, we have the semidirect product to be the space $S\oplus I$ with operation $$ (s_{1},x_{1})(s_{2}x_{2}):=([s_{1},s_{2}],[x_{1},...
3
votes
1answer
152 views

Question on notation (topology & fiber bundles)

This is a very elementary question but I can't quite seem to track down a worthwhile source, so I was hoping someone more knowledgeable than I could lend their superiority. In Moore & Schochet's ...
2
votes
3answers
184 views

In what case the semi direct product is abelian?

Only when H$\rtimes$K is direct product and H,K are abelian? How to prove? In other words if the homomorphic from K to Aut(H) is not trivial then the semi direct product is not abelian?
22
votes
5answers
2k views

What is the motivation for semidirect products?

I haven't the slightest idea why (inner or outer) semi-direct group products are an interesting construction. I understand inner direct products, because you're just giving conditions for which a ...
4
votes
0answers
361 views

semidirect product of semigroups

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$. A function $f:S\to\...
5
votes
1answer
822 views

Nonabelian groups of order $p^3$

From a little reading, I know that for $p$ and odd prime, there are two nonabelian groups of order $p^3$, namely the semidirect product of $\mathbb{Z}/(p)\times\mathbb{Z}/(p)$ and $\mathbb{Z}/(p)$, ...
5
votes
1answer
105 views

Why is $H \rtimes_{\varphi} K$ isomorphic to $H \rtimes_{\varphi \circ \phi} K$ where $\phi \in \text{Aut}(K)$?

Why is $H \rtimes_{\varphi} K$ isomorphic to $H \rtimes_{\varphi \circ \phi} K$ where $\phi \in \text{Aut}(K)$? I can see that $\varphi(K) = \varphi(\phi(K))$, but it is not clear to me how the ...
5
votes
0answers
271 views

Presentations of Semidirect Product of Groups

I have seen here that given two groups $G=\langle X|R \rangle := F(X)/N(R)$ and $H=\langle Y|S \rangle := F(Y)/N(S)$, then their semidirect product can be written as: $$ G\rtimes_\phi H \;=\; \langle ...
5
votes
1answer
696 views

Classify groups of order $pq^2$ using semidirect product

I am struggling with semidirect products and how they can be used to classify groups of a certain order. In particular, I need help with the nonabelian case. This is the problem I am working with.. ...
4
votes
1answer
744 views

Classifying groups of order 60

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 185, Exercise 14): This exercise classifies the groups of order $60$ (there are thirteen isomorphism types)....
4
votes
3answers
753 views

Is semidirect product unique?

This is about semi direct product on Dummit and Foote algebra text book. Why is this statement true? Theorem 12. Suppose $G$ is a group with subgroups $H$ and $K$ such that $H\...
4
votes
3answers
278 views

$G\rtimes_\phi H \cong G\rtimes_\psi H$ when certain automorphisms exist

I'm reviewing for exams and came across this problem from an older exam: Let $G$ and $H$ be groups and let $\phi,\psi : H\to \operatorname{Aut}(G)$ be homomorphisms from $H$ into the group of ...
3
votes
0answers
437 views

Relation between kernels of homomorphisms and the semidirect product of groups

I'm reading Abstract Algebra of Dummit and I got stuck on this point. When we try to build the semi-direct product of 2 group predefined, such as $H$ and $K$, then all the semi-direct product groups ...
2
votes
0answers
62 views

Split extensions of $G/U$ by elementary abelian $p$-group and $H^1(U, \mathbb{Z}/p\mathbb{Z})$

This question is a followup to Serre's Exercise 3.4.1 on Galois Cohomology. Let $G$ be a profinite group, $1 \to P \to E \to W \to 1$ a finite group extension and $f\colon G \to W$ a continuous ...
2
votes
2answers
752 views

Isomorphism types of semidirect products $\mathbb Z/n\mathbb Z\rtimes\mathbb Z/2\mathbb Z$

Let $n = p_1 p_2 \cdots p_k$ be the product of pairwise distinct odd primes. Let $X$ be the set of element in $\operatorname{Aut}(\mathbb Z/n\mathbb Z)$ of order $1$ or $2$. For each $\psi\in X$, the ...
6
votes
1answer
176 views

Exercise 3A.7 of “Finite group theory”, M. Isaacs

Let $G$ finite group and $\sigma \in \text{Aut}(G)$, suppose that at most two prime numbers divide $o(\sigma)$. Show that $\left \langle \sigma \right \rangle$ has a regular orbit on $G$. Suppose $o(\...
5
votes
1answer
348 views

Central extensions versus semidirect products

Consider an extension $E$ of a group $G$ by an abelian group $A$. $$1 \to A \overset{\iota}{\to} E \overset{\pi}{\to} G \to 1$$ Two special kinds of extensions are: Central Extensions: $A$ is ...
5
votes
1answer
231 views

Subgroups of the Semi-Direct Product $\mathbb{Z}/7\mathbb{Z} \rtimes (\mathbb{Z}/7\mathbb{Z})^{\times}$

I want to list all the subgroups of the semi-direct product $\mathbb{Z}/7\mathbb{Z} \rtimes (\mathbb{Z}/7\mathbb{Z})^{\times}$, under the homomorphism $\theta: (\mathbb{Z}/7\mathbb{Z})^{\times} \...
4
votes
2answers
248 views

Existence of a non-abelian group of order $p^n$.

Question: Let $p$ be any prime and $n \geq 3$. Show that there exists a non-abelian group of order $p^n$. Attempt: Take $n = 3$. Writing $\mathbb Z_p \times \mathbb Z_p = \{e, \alpha_1, \ldots,...
4
votes
1answer
187 views

Do homomorphisms $H \to \operatorname{Aut}(K)$ that coincide at the level of $\operatorname{Out}(K)$ induce isomorphic semidirect products?

I recently realized that I don't know of any group that is a nontrivial semidirect product of some symmetric group $S_n$ and another group ($S_n$ being the normal subgroup), except when $n=6$. (For ...
2
votes
0answers
442 views

On monomial matrices (Generalized Permutation Matrices )

A matrix $a\in GL_{n}(F)$ is said to be monomial if each row and column has exactly one non-zero entry. Let $N$ denote the set of all monomial matrices. I have already proved here that the ...
2
votes
1answer
219 views

Where can I find a proof of the Presentation for Semidirect Products?

I have seen it claimed online that: Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\phi\colon H \to \text{Aut}(G)$, then $$ G\rtimes_\phi H \;...
2
votes
2answers
356 views

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 \...
2
votes
1answer
50 views

What does it mean for a subgroup to fix a vector space?

I was wondering if anyone could explain what is meant by the two statements below? Any help would be greatly appreciated. Let $U=\mathbb{F}_{q}^{m}$ and $W=\mathbb{F}_{q}^{n}$ be two vector spaces of ...
1
vote
1answer
237 views

The normaliser of the left regular image [D&F]

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 186): Let $H$ be a group of order $n$, let $K=\text{Aut}(H)$ and form $G=\text{Hol}(H)=H \rtimes K$ (where $...
1
vote
1answer
141 views

A necessary condition for two semi-direct products to be isomorphic.

Notations and definitions. For all $A\in\textrm{GL}_n(\mathbb{Z})$, $A$ is hyperbolic if and only if none of its complex eigenvalue has module $1$. For all $(A,B)\in\textrm{GL}_n(\mathbb{Z})^2$, one ...
1
vote
0answers
37 views

Upper central series, semidirect products and a lifting property (argument verification)

Let $G$ be a profinite group, and define the lifting property: $(*_p)$ For every extension $1 \to P \to E \to W \to 1$ where $E$ is finite and $P$ is a $p$-group and for every surjective morphism $f\...
0
votes
0answers
55 views

Computing the multiplication of elements to generate the Cayley graph of a semidirect product

I have computed a semidirect product, $s$ of $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$ as below and have drawn a Cayley graph for $s$ with respect to a generating set $S$. But I wanted ...
7
votes
1answer
183 views

What's the smallest non-$p$ group that isn't a semidirect product?

Although every non-simple group can be described as a group extension, it is well-known that not every non-simple group can be expressed as a semidirect product of smaller groups. For example, a ...
6
votes
0answers
86 views

A necessary condition on semidirect products to be isomorphic

Is it true that the semidirect product groups $\mathbb{Z}\ltimes_A \mathbb{Z}^n$ and $\mathbb{Z}\ltimes_B\mathbb{Z}^n$ are isomorphic if and only if $A$ is conjugate in $\operatorname{GL}(n,\mathbb{...
5
votes
1answer
578 views

Isomorphism of semidirect products [D&F]

I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 184 Exercise 6): Assume that $K$ is a cyclic group, $H$ is an arbitrary group and $\varphi_1$ and $\...
4
votes
2answers
248 views

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 ...
4
votes
2answers
572 views

Is $G$ always a semidirect product of $[G,G]$ and $G/[G,G]$?

If $G$ is a finite group, it is not true in general that $G$ is the semidirect product of a normal subgroup $N$ and the quotient group $G/N$. It is also not true in general that there is a subgroup ...
3
votes
1answer
157 views

Automorphisms of the affine semilinear group $A\Gamma L(1,2^{n})$

In this question, it is mentionned that the group of automorphisms of the semilinear group $A\Gamma L(1,2^{n})$ is the group itself. Do you have a short proof of this fact?
2
votes
1answer
134 views

When is $A\rtimes_{\phi_1} B \cong A\rtimes_{\phi_2} B$?

Suppose $1\to K \stackrel{m}{\rightarrow} G \stackrel{f}{\rightarrow} H \to 1$ is short exact sequence of groups. The followings are equivalent: $(1)\ G\cong K \times H;$ $(2)$ The sequence right ...
2
votes
1answer
36 views

Semidirect product of subgroups

Let $G$ be a group with $N \unlhd G$ such that there is $K \leq G$ with $KN = G$ and $K \cap N = 1$. It is well known that in this case $$G \cong N \rtimes_\phi K$$ where $\phi: K \to Aut(N): k \...
2
votes
1answer
73 views

General formula for exponent of a semidirect product

This page states that Exponent of semidirect product may be strictly greater than lcm of exponents But it doesn't give any proof for that. Could anyone provide a general formula for the ...
2
votes
1answer
76 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? ...
2
votes
1answer
120 views

Is the semidirect product of normal complementary subgroups a direct product.

If $G$ is a group with $K$ and $N$ as normal complementary subgroups of $G$, then we can form $G \cong K \rtimes_\varphi N$ where $\varphi:N \to Aut(K)$ is the usual conjugation. But someone told me ...