# 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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### 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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### 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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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 $... • 196 0 votes 0 answers 47 views ### Structure of non abelian finite p-groups I am familiar with the concepts of direct products, semi-direct products, wreath products and central products of groups. After seeing the classification of finite$p$-groups upto order$p^4$,(Theory ... • 196 4 votes 1 answer 133 views ### How does$H$act on$G^t$in the wreath product$G^t \wr H$? I'm reading this expository paper about group theory in the Rubik's cube. I'm a little confused by the definition of the wreath product in this paper. Example 3.12 on page 12 states that the elements ... 1 vote 0 answers 60 views ### Relating Galois group of evenized-reciprocalized version to original Galois group Let$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)$. ... • 61.5k 2 votes 1 answer 212 views ### Wreath product, semi-direct product, and partitions Any help with the question which follows will be greatly appreciated. I'm working through Dixon and Mortimer's Permutation Groups and have a question regarding a particular semi-direct product ... • 678 7 votes 1 answer 2k views ### Centralizer of symmetric group Let, an element of symmetric group$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$. Its known that the centralizer of this element is ... • 89 2 votes 0 answers 173 views ### What is the center of$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_2r$times, here$Z_2$denotes a cyclic group of order 2? When$r=2$, I know that$Z_2\wr Z_2\cong D_4$... • 541 2 votes 0 answers 146 views ### Why is the free group on two generators not a subgroup of$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$for example,... 4 votes 1 answer 73 views ### Are positional notation systems for natural numbers wreath products of semigroups? Suppose we are given the finite cyclic group$\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 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 $C_2$, ...