Questions tagged [wreath-product]

In group theory, a method to build a new group out of two existing groups that is based upon the idea of a semi-direct product.

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Associativity of Wreath Product Group Action

Given two groups $H$ and $K$ such that $K$ acts on a non-empty set $\Delta$, we can compute the wreath product as $G = H \wr{_\Delta} K := Fun(\Delta,H)\rtimes K$. Now, suppose that $H$ acts on the ...
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Misunderstanding fusion of characters

In section 3 of this paper the author describes the character $\overline{\psi_i\times\psi_j}$ of $G\wr \mathcal{C}_2$ where $\psi_i\in \hat{G}, i\neq j$, and claims that this agrees with the values of ...
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Exercise 6.2.4 of "Fundamentals of the Theory of Groups".

Let $A$ and $B$ groups and $A\wr B$ is a restricted wreath product. Prove that $$ [A\wr B, A\wr B]=[B, B]\ H $$ where $\mathrm{fun}(B, A)=\{f| f:B\longrightarrow A\quad \mathrm{supp}f<+\infty\}$ ...
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the non-abelian subgroups of the Lamplighter group

The lamplighter group can be defined by the semidirect product: $$ L_2=(\mathbb{Z} _2) \wr \mathbb{Z} \cong \bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{2} \rtimes_\phi\mathbb{Z},$$ where $\phi(1)$ &...
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In the regular wreath product $W = H \wr G$, every subgroup of $G$ is the centralizer of some element

This is Exercise 3A.9 - (b) from M. Isaacs' "Finite Group Theory". It goes as follows: Let $W = H \wr G$ be the regular wreath product and let $C \leq G$ be an arbitrary subgroup. Show that ...
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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 an imprimitive group can be embedded in a wreath product

I came across this theorem here:https://cameroncounts.files.wordpress.com/2014/12/lect2n.pdf. However it is stated without proof and I'm not really sure where to start, any hints would be appreciated! ...
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Wreath product as a permutation group.

I'm working through Dixon and Mortimer's book Permutation groups. I'm currently working on showing that we can turn a wreath product into a permutation group. I've changed the notation used from the ...
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Aschbacher Class $2$ subgroup structure

In $PGL(12,3)$, there should be an Aschbacher Class $2$ subgroup the image of $GL(2,3)^6 \wr{\rm Sym}(6)$. I am trying to locate the image of $GL(2,3)^6$ in Magma using derived subgroup but it doesn't ...
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Minimum base size of product action of wreath product

I've recently started trying to teach myself some basic theory of permutation groups. I wanted to compute the minimum base size of the permutation group $G=S_d\,wr\,S_r$ equipped with the product ...
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Krohn-Rhodes decomposition for transformations over $\{0,1\}$

I'm trying to learn about the Krohn-Rhodes theorem, and I'm struggling to apply it even on incredibly simple semigroups. Notation $C_2 = \langle e,x \mid x^2=e \rangle$ is the cyclic group on two ...
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Generalised Symmetric group and Semidirect Product

$G= \mathbb{Z}_r \wr \mathfrak{S}_n:=(\mathbb{Z}_r)^n\rtimes\mathfrak{S}_n$ is the generalised symmetric group where the element $G$ is denoted by $(f,\pi)$ where $f:\{1,2,\dots n\}\to \mathbb{Z}_r$ (...
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About the intersection of two commutator subgroups

I'm reading an example in Lennox & Stonehewer's book "Subnormal Subgroup of Group". There (p.144/145) they construct the following example. Let $L$ be an infinite elemetary abelian $2$-...
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Is $S_2 \wr S_k$ contained in $(S_a \times S_b) \wr S_{k-1}$?

I'm working on my thesis and I want to prove a theorem but I need the following to be true: $S_2 \wr S_k$ is not isomorphic to a subgroup of $(S_a \times S_b) \wr S_{k-1}$ where $a,b < 2k$. Does ...
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Let $G$ be a non abelian simple group. Show that ${\rm Aut}(G^n)$ is isomorphic to ${\rm Aut}(G) \wr{\rm Sym}(n).$

I got this question but don't know how to answer it. Let $G$ be a non abelian simple group. Show that ${\rm Aut}(G^n)$ is isomorphic to ${\rm Aut}(G) \wr{\rm Sym}(n)$. I already know that ${\rm Aut}(G)...
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Is it true that $F^*\wr S_n$ is a solvable group, when $n\leq 5$?

Let $U$ be a linear space over a division ring $D$, $G_1$ a subgroup of $GL(U)$, and $\Gamma$ a subgroup of a symmetric group $S_k$ on $\{1,\ldots,k\}, k>1$. The cartesian product $U^k=V_1$ can ...
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For a group $G$ of order $p^n$, $G\cong H$ for some $H\le\Bbb Z_p\wr\dots\wr\Bbb Z_p$.

This is Exercise 5.3.2 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. The result is mentioned in the following ...
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There exist infinite solvable $p$-groups with trivial centre. (Use a hint.)

This is Exercise 5.2.11 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to this search, it is new to MSE. This is marked as being referred to later on in the ...
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The Frattini subgroup of the standard Wreath product of two quasicyclic groups is the group itself.

This is part of Exercise 5.2.5 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. Here are some previous questions ...
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Does anything special happen when you replace the direct product in the definition for a wreath product with a central product?

In group theory there is a special type of product called the wreath product and is defined as follows: Let $A$ and $H$ be groups, with the group $H$ acting on the set $\Omega$. We define $K$ to be ...
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Irreducible representations of wreath products of cyclic groups.

Let $G= C_m \wr C_n$ be a wreath product of cyclic groups $C_m$ and $C_n$. I am interested to find all irreducible representation of $G$. Update: My thoughts: If I take $H:=\underbrace{C_m \times \...
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Structure of the Brainball group

This is a Brainball: It consists of $13$ numbered pieces arranged in a ring and a core; each piece has one side white and one side yellow. Part of the core, the blue caps in the picture above, can ...
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Number of elements of order $p$ in the sylow $p$-subgroup of $S_{p^2}$

I have the group $G = \mathbb{Z}_p \wr \mathbb{Z}_p$, it's well known that $G$ is isomorphic to a sylow $p$-subgroup of $S_{p^2}$, it has order $p^{p+1}$ and there are 2 possible orders for a non-...
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Showing that the restricted wreath product $\Bbb Z\wr\Bbb Z$ is finitely generated.

This is the first part of Exercise 1.6.15 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this Google search, it is new to MSE. The Details: ...
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Showing $Z(H\wr K)=1$, for abelian $H\neq 1$ and arbitrary, infinite $K$.

This is a part of Exercise 1.6.14 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. The first part of the exercise is here, a ...
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Showing $Z(H\wr K)$, for abelian $H\neq 1$ and arbitrary $K$, is the diagonal subgroup of the base group.

This is Exercise 1.6.14 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0 and this search, it is new to MSE. The Details: The definition of the wreath ...
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A typo on page 42 of Robinson's book on group theory.

There's a typo on page 42 of Robinson's, "A Course in the Theory of Groups (Second Edition)", ISBN 987-1-4612-6442-9. Here's a picture: It reads as follows: If $H$ and $K$ are permutation ...
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Simple example of computing wreath products of groups

I am trying to understand wreath products but cannot find an example of one being computed in depth. I know that $D_{4}$ is isomorphic to $\mathbb{Z}_{2} \wr \mathbb{Z}_{2}$, but I cannot seem to ...
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When is normalizer of wreath product the wreath product of normalizers?

I set out wondering about the $p$-Sylow subgroups of symmetric group $S_n$. First, if $n=p^k$, the $p$-Sylow is the iterated wreath product $C_p^{\wr k}:=C_p\wr C_p\wr\cdots\wr C_p$. This is the ...
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Showing that a Group Extension is Split

I have a group extension $1\rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ that I think is a split extension (so $G \approx N \rtimes Q$), but I'm having trouble showing this. Is there a ...
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"joining" two representations of subgroup to one of group

Consider the (four) one-dimensional representations of $V_4\simeq S_2\times S_2$ the Klein four group, as listed for example here. It is convenient to consider these as four vectors forming a basis ...
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GAP WreathProduct with fixed points

Gap always takes the domain of a permutation group to be the set of points moved by its elements. The documentation of WreathProduct thus includes this comment ...
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Quotient of wreath group by commutator

[Self studying Robinson, ex 1.6.20] Robinson asks us to prove that $G/[B,K] \cong(H/H') \times K$ (where $G=H \wr K$, $K\neq 1$ and $B$ is the base group.) As a hint, we are told to show first that $B'...
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Help identifying this finite group I have characterised (subgroup of $(C_3 \wr C_3)^3$

I'm working on a group for my master's thesis and I've found a fairly precise characterisation of it, but I can't see how to describe it as a combination of standard groups. I have characterised it as ...
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On wreath product of finite $p$-groups

Actually, I was studying about the group $ C_p \wr C_p$, where $C_p$ is cyclic group of order $p$ and $\wr$ denotes the wreath product. I understood that it is isomorphic to the Sylow $p$-Subgroup of $...
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Structure of non abelian finite p-groups

I am familiar with the concepts of direct products, semi-direct products, wreath products and central products of groups. After seeing the classification of finite $p$-groups upto order $p^4$,(Theory ...
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How does $H$ act on $G^t$ in the wreath product $G^t \wr H$?

I'm reading this expository paper about group theory in the Rubik's cube. I'm a little confused by the definition of the wreath product in this paper. Example 3.12 on page 12 states that the elements ...
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Relating Galois group of evenized-reciprocalized version to original Galois group

Let $Q\in{\mathbb Z}[X], Q=\sum_{k=0}^m q_kX^k$, with degree $m$ and Galois group $S_m$ (over $\mathbb Q$). Consider the evenized-reciprocalized polynomial $P(X)=X^{2m}Q\bigg(X^2+\frac{1}{X^2}\bigg)$. ...
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Wreath product, semi-direct product, and partitions

Any help with the question which follows will be greatly appreciated. I'm working through Dixon and Mortimer's Permutation Groups and have a question regarding a particular semi-direct product ...
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Centralizer of symmetric group

Let, an element of symmetric group $S_N$ is given by $g=(1)^{N_1}(2)^{N_2}....(s)^{N_s}.$ Here $N_n$ denotes the number of cycles of length $n$. Its known that the centralizer of this element is ...
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What is the center of $Z_2\wr Z_2\wr\cdot\cdot\cdot\wr Z_2$, here $Z_2$ denotes a cyclic group of order 2?

What is the center of $Z_2\wr Z_2\wr\cdot\cdot\cdot\wr Z_2$ which is the wreath product of $Z_2$ $r$ times, here $Z_2$ denotes a cyclic group of order 2? When $r=2$, I know that $Z_2\wr Z_2\cong D_4$ ...
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Why is the free group on two generators not a subgroup of $G$?

Let $G$ be generated by the elements $g$ and $\{e_i\}$ for $i\in\mathbb{Z}$, having the relation $ge_ig^{-1}=e_{i+1}$. It seems to me like there is a subgroup $\langle e_i,e_{i+1}\rangle$ for example,...
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Are positional notation systems for natural numbers wreath products of semigroups?

Suppose we are given the finite cyclic group $\mathbb{Z}/b\mathbb{Z}$ and the monoid of natural numbers $\mathbb{N}$, both of which are semigroups. Does the restricted wreath product $(\mathbb{Z}/b\...
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What are elementary examples of elements and operations on the lamplighter group?

What are some elementary examples of elements and operations on the Lamplighter group $L$? I have the definition above which is the infinite sum of infinitely many copies of the cyclic group $C_2$, ...
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Finite p-group involving a wreath product

I have trouble with the following exercise: $\textbf{Exercise}$ Let $H$ be a finite $p$-group, let $A \trianglelefteq H$ be a normal subgroup which is an elementary abelian $p$-group and let $x ...
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Wreath product is associative but regular wreath product is not associative.

I am currently reading An Introduction of Theory of Groups by Rotman. In page 174, Theorem 7.26 states that wreath product is associative. But in the next page, regular wreath product $W=D \wr_r Q$ ...
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Using wreath products to find stabilisers of a partition of a set

I have the following example, that uses the wreath product to find the stabilisers of a partition. I don't understand how the wreath product does this though. I can recite the definition of a wreath ...
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Normaliser of Wreath Product of Matrix Group

I was reading and doing some problems on wreath products. The groups are matrix group in my case. Let $G_s=GL(2^s,R)$ denotes the group of invertible matrices over $R$ and suppose $M\le G_s$ for ...
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Finitely generated subgroup of wreath product with $\mathbb Z$

Consider a countable group $H=\{a_0,a_1,a_2, ... \}$. We define the map $f: \mathbb{Z} \to H $ , $f(2^n):=a_n $ and $f(k)=1$ for $k \notin \{2^0 , 2^1, 2^2, ...\}$. We consider $f$ as an element of ...
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