# Tagged Questions

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### left regular representation of a group thru group action

Let G be a group and let $g\cdot:G\to G$(i.e.$g\cdot g'=gg'$) . This induces a permutation representation of the group. I was trying to walk thru the problems in dummite and foote. One of the problem ...
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### The comultiplication on $\mathbb{C} Q$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=Q$ the ...
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### The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
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### Elements whose orders are multiple of $p$ [closed]

Let $G$ be a non-solvable group, $N$ an abelian minimal normal $p$-subgroup of order $p^r$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. By these assumption we can conclude that $G$ has ...
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### Computing values of centralizers in a non-solvable group with a given property

A finite group G satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
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### The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
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### Constructing all inequivalent faithful irreducible projective representations of finite abelian groups

Let $G$ be a finite abelian group, and $\alpha \in H^2 (G,\mathbb{C}^*)$ a 2-cohomology class. It is known (in Karpilovsky's multi-volume tome or elsewhere) that a finite abelian group admits a ...
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### Derivation and application of Newton's identity

How is the following identity derived? $$\sum_{\ell =0}^{n-1}(-1)^\ell e_\ell s_{n-\ell}+(-1)^nne_n=0$$ Is there an example demonstrating the context in which this might be applied?
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### Conjugacy class name of the product in ATLAS

Well, I'm trying to read "ATLAS of Finite Groups". To be more precise, I'm interested in character tables of some Weyl groups. Is it possible to determine the conjugacy class name of the product of ...
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### The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
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### Proof in Serre/Fulton's rep. theory of Artin-Wedderburn for $\mathbb C[G]$

I have figured out a proof myself for the following theorem, but in both Serre's "Linear Representation of Finite Groups" and Fulton's "Representation Theory" books, I don't understand their comments ...
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### Easy way to get real irreducible characters (reps) from complex irreducible characters?

For plenty of groups, the real irreducible characters/representations aren't the same as the complex irreducible representations. I really enjoy James Montaldi's summary of real representations, for ...
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### Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
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### Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
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### Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
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### What are the finite groups with 8 or 16 conjugacy classes?

What are the list of finite groups with 8 or 16 conjugacy classes? I learned that dihedral groups $D_{10}$ and $D_{13}$ have 8 conjugacy classes. (Here the order of these groups are $|D_{10}|=20$, ...
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### Summing the traces of matrix powers

Let $G=\langle h\rangle_n\subset{\rm GL}(m,\mathbb{C})$ be a cyclic group of order $n$. I wonder if there is a good formula for calculating the sum $\sum_{g\in G}{\rm Tr}(g)$ via ${\rm Tr}(h)$, for ...
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### Number of degree-$d$ representations of a perfect group?

It seems to be a standard result that the number of degree-1 representations of a group $G$ is equal to $[G : G']$ where $G'$ is the commutator subgroup (e.g. Lemma 6.2.7 in the 2012 textbook ...
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### Rational representation of a 136 order group

Let G be the group of order 136 = 8 * 17 with presentation $$<x,y : y^{8}=x^{17}=1 \quad yxy^{-1}=x^{4} >$$ Find the simple summands of the group ring Q[G].
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### The use of conjugacy class and centralizer?

This is more or less for a conceptual and better-understanding question in group theory and in representation theory: (1) Why are conjugacy class and centralizer important concepts in the group / ...
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### Matrix coefficients of representations of finite groups

In finite-dimensional complex representations of finite groups, I would like to understand what I can learn by looking at a single matrix coefficient. In particular, I would like to look at "diagonal" ...
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### When would a finite group be cosidered as Fp G -Module?

What conditions are necessary to think of a finite group as Fp G -Module ?
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### Irreducible representations (over $\mathbb{C}$) of dihedral groups

Find number of complex irreps of the group $D_n$. Find dimension of the irreps. I know that The number of complex irreps of a finite group is equal to the number of conjugacy classes of the ...
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### On the graph of induction-restriction for group-subgroup representations

Let $G$ be a finite group, and $H$ a subgroup. Let $(V_i)_{i \in I}$ and $(W_j)_{j \in J}$ be the irreducible complex representations of $G$ and $H$ (up to isom.). Consider the graph ...
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### character of a representation of the group $S_n$

Let $\Phi$ be a representation of the group $S_n$ in the space with basis $(e_1, \ldots, e_n)$ such that $\Phi(\sigma)e_i=e_{\sigma(i)}.$ Find character of $\Phi$ This is looks like a regular ...
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### Let G be a finite group and V be the regular CG-Module. Find a submodule of V which is isomorphic to the trivial CG-Module

As in the question. I have read through different books and articles and they seem to set W=<$\sum_{g \in G}$ g> as a submodule. I can understand that this is unique, but I fail to see how this is ...
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### Existence of intermediate subgroups and representations theory.

Let $G$ be a finite group, $V$ an irreducible representation, $H$ a subgroup. Let $V^H$ be the subspace of vectors of $V$ invariant under the action of $H$. Suppose that $dim(V^H)>1$. Then ...
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### A question about Character Degrees of G/N

Let $G$ be a finite group such that $p_1p_2\mid |G|$, where $p_1$ and $p_2$ are two primes. We know that there exists an irreducible character $\chi\in Irr(G)$ such that $p_1p_2\mid \chi(1)$. We know ...
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### Permutation representations

I need a source for Permutation representations of general linear groups over finite fields. Can anyone introduce some sources?
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### Representation of $S_4$

Is there a general method to work out all irreducible complex representation of a group? Describe all the the irreducible complex representation of the group $S_4$. $S_4$ is the symmetric group ...
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### Irreducibility of complex 2-dimensional character of the group $S_3$
Let $\chi$ be 2-dimensional complex character of the group $S_3$. Prove that $\chi$ is irreducible character iff $\chi((123))=-1$. There is hint in my book: " Use the Maschke's theorem and ...
Let $A$ be the additive group of n-dimensional vector space $V$ over the field $\mathbb{F}_p$. Let $\chi$ - a nontrivial irreducible complex character of A. Prove that a subset $\{a \in A \mid ... 1answer 73 views ### Reading a Character Table in Magma These are the outputs of two character tables from Magma. The first is for$A_5$. The second is for$GL_3(2)$. What is the significance of the '$+$' and '$0$' symbols? I can produce more tables if ... 1answer 80 views ### representations of the dihedral group Let$\rho_\epsilon(a)=\begin{bmatrix}\epsilon & 0\\0 & \epsilon^{-1}\end{bmatrix}$and$\rho_\epsilon(b)=\begin{bmatrix}0 & 1\\1 & 0\end{bmatrix}$I can prove that$\rho_\epsilon$is ... 1answer 24 views ### irreps of$p^3$-group is faithful representation Let$A$be an irreps of$p^3$-group. Prove that$A$is faithful representation. I know that$p^2$-group and$p$-group are abelian. I have to show, that$Ker A=e$I have no idea how to start it 1answer 41 views ### decompose a real representation of a group$C_2\times C_2
Let $\Phi$ be a real representation of the group $C_2\times C_2=\{e,a\} \times \{e,b\}$ such as $\Phi(a)=\begin{bmatrix}5 & -4 & 0\\6 & -5 & 0 \\0 & 0 & 1\end{bmatrix}$ and \$ ...