Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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Product of subgroups and generating sets

Prove or disprove the following: $(1)$ Let $H$,$K$ be subgroups of a finite group $G$. If $|HK|=|\langle H,K \rangle|$, then $HK=\langle H,K \rangle$, that is, $HK$ is a subgroup of $G$. $(2)$ Let ...
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104 views

Hall subgroup of a finite group?

How did the author get that $L=(L \cap H)(L\cap K)$ in Lemma $5$ below. Remark: All the groups here are finite. $H$ permutes (commutes) with $K$ means $HK=KH$ where $H$ and $K$ are subgroups of some ...
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Degrees of parabolic subgroups

Suppose a finite reflection group $G$ has the degrees $d_1,\ldots,d_n$. Let $G^*$ be a parabolic subgroup of $G$. What are the degrees of $G^*$. Since $|G^*|$ divides $|G|$ it is clear that the ...
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On maximal parabolic subgroups of GL(V)

My professor today has showed the follwing proposition: If $V\cong K^n$ where $K$ is a finite field of characteristic $p$, then the maximal parabolic subgroups of $GL(V)$ are exactly the ...
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What are various approaches of proving that a permutation group is a subgroup of another?

I have two permutation groups $G_1=\langle g_1,g_2\rangle$ and $G_2=\langle h_1,h_2\rangle$ acting on $\Gamma_1$ and $\Gamma_2$ respectively. I want to prove that $G_1 \leq G_2$. What are more ...
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Help to show that if this Group is Metabelian?

Here is my problem: Let $G=\langle a,b|a^l=b^3=1,(ab)^3=(a^{-1}b)^3 \rangle$. Find the order of $\frac{G}{G'}$ and then verify that if $G$ is metabelian. What I have done: I added the relation ...
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reflection groups over finite fields and coxeter groups

Coxeter groups include groups like E6, G2 etc which when defined over finite fields are simple finite groups. Are there coxeter representation for such simple finite groups (like E6(q))?
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Rational cohomology of quotient by group action

Let $X$ be a topological space with a continuous action by a finite group $G$. Hopefully under some assumptions on $X$ one can identify the rational cohomology of $X/G$ with the $G$-invariants of the ...
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61 views

$S_k$ action on $A/I$

Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since $$ R = \left( \dfrac{k[x,y]}{(x+y)} \right)^{S_2} = ...
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70 views

What could the meaning of “invariant of $G$” be?

In a context, the author wrote that for a finite permutation group $G$ acting on a set $\Omega$ of degree $n$, the list of subdegrees is an invariant of the group. What could the meaning of ...
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84 views

Is it true that if $g^{-1} = g$, then $g$ is only conjugate to itself in a finite group $G$?

I have a finite group G, where $Z(G)=1$, and I have an element $g\neq 1$ where $g^{-1} = g$ and $gg=1$. I want to say that $g$ is then only conjugate to itself so that I have a contradiction ($Z(G)$ ...
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103 views

Hall subgroup of a finite group G

If $G$ is a finite group and $x$ is a $p'$ element of G does this imply that there a Hall $p'$ subgroup of $G$ containing $x$?
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57 views

Inner tensor and restriction

Let $\pi$ resp. $\chi$ be a finite dimensional representation resp. 1-diml. rep. of a finite group. We define $\chi \otimes \pi (g) = \chi(g) \pi(g)$ as a rep of $G$. For $N$ subgroup, does hold ...
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Free Product of two finite groups

The question entails that I should choose two finite groups, then construct a 'biregular' tree, and show that the action of the free product of the two finite groups on the biregular tree will have a ...
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69 views

2-Brauer characters of the symmetric group $\mathfrak{S}_3$

In a previous question, I asked how to compute Brauer characters of the alternating group $\mathfrak{A}_3$; the answer to this question provided a solution for all cyclic groups. I would now like to ...
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79 views

Sets of independent vectors

So we're working in Z2k, the group of bit-vectors of length k and componentwise addition modulo 2. Now I'm trying to make a function yj=1..?(vi) assigning elements of Z2k to n vertices, such that ...
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32 views

Semidirect products of cyclic groups

Consider $A=\langle a\rangle$, cyclic group of order $9$ and $B=\langle b\rangle$, cyclic group of order $3$. Consider now the following action of $B$ on $A$ via automorphism: ...
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12 views

Coset representatives of upper triangular matrix in $\text{GL}_2(\mathbb{F}_p)$

Let $p$ be prime. Let $G=\text{GL}_2(\mathbb{F}_p)$. Let $U$ be the subgroup of $G$ of upper triangular matrices. By computing cardinalities, I see that $[G:U]=p+1$. Is the a nice description for ...
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Calculate the order of the generalized quaternion group and list its elements

Let $n \in \mathbb N$ and $w \in G_{2^n}$ a primitive $2^n$-th primitive root. Consider the matrices $R=\begin{pmatrix}\omega & 0\\0&\omega^{-1}\end{pmatrix}$ and ...
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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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31 views

Existance of semisimple elements in a torus.

Let $G$ be a finite simple group of Lie type and $T$ be a maximal torus in $G$. Is it true that $T$ contains a regular semisimple element (a semisimple element which $C_G(x)=T$)? If yes, why?
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50 views

Show that the number of elements in the equivalence class is n?

Question is : Let $G$ be a finiote group of order n = $p^{\alpha}$m, where $p$ is a prime number and if $p^r$ | m but $p^{r+1} \nmid$ m Let $\mathcal M$ be the set of all subset of $G$ which have ...
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About some special kinds of group automorphisms

let $G$ be a finite group with $1\neq Z(G) \lneqq G$. Also let $H=\{x_1,...,x_n\}$ be the set of all disjoint representative elements of right cosets of $Z(G)$ in $G$. Is there any non-trivial ...
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prime cycle in permutation groups

I'm trying to use the jordan theorem and for that we need find the prime cycle on permutation group which i don't have idea how to find it . (my English is poor, so sorry for this)
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six transitive permutation groups

If I'm explaining right than please give me some hints about how we prove a permutation group is six transitive. I have proved that it is two transitive because stabilizer of one point acts ...
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44 views

Problem about tensor products and modules over a group algebra

Can anyone help me with hints for these problems? I would appreciate it a lot. 1) Supose $G$ is a group and $k$ is a field. If $U,V$ and $W$ are $kG$-modules then $\operatorname{Hom}_{kG}(U\otimes ...
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30 views

Construction of transitive group of degree $n$

Is there any way to construct all transitive groups of degree 6 with the following block system: {1,2} , {3,4}, {5,6} ?
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40 views

A question about the group of automorphisms of finite simple groups

My question is simple. Can the groups of automorphisms of finite simple groups be caculated?
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dimension of a finie simple group of Lie type.

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p$, namely $F_q$ of size $q$. Suppose $dim(G)=n$. What we can say about $|G|$? Is it true that $|G|=q^ns$, where ...
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A dense subset of a finite group

Let $G$ be a finite group with Zariski topology. Suppose $G=A_1\cup A_2\cup\cdots\cup A_n$, where $A_i$, $1\leq i\leq n$, are pairwise disjoint subsets of $G$ and only $A_1$ is dense in $G$, that is, ...
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Composition Factors of $C_p\times C_p$

I have question that asks me to find the composition series of $C_p\times C_p$, now these are all isomomrphic to the series $\{1\}\lhd C_p \lhd C_p\times C_p$ but the questions wants all the series ...
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31 views

Brauer characters of finite simpl groups of type E8

I would like to know the Brauer characters of finite simple groups $E_8(2)$, $E_8(3)$ or $E_8(5)$. Is there any refrence for this topic? Thanks
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proof for the structure of sylow $2-$subgroup of symmetric group

I know that $Z_2\wr Z_2..\wr Z_2 (r times)$ is asylow $2-$subgroup of symmetric group $S_{2^r}$. But I need a proof for this result, any comment is very appriciated
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Sylow 3-subgroup of Sym(11)

Find a Sylow 3-subgroup of Sym(11). I know we can reduce this question to find Sylow 3-subgroup of Sym(9), but how to find exactly the Sylow subgroups ? And what about Sylow 2-subgroups of ...
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Class of groups having all subnormal subgroups cyclic

Does there exist a class of non abelian groups whose all subnormal subgroups are cyclic? I searched out by myself but I did not found it.
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How to show that $Ind_K^G \mathbb{C}_\chi$ is natrally isomorphic to $\mathbb{C}[G]e_{\chi}$?

Let $K \subset G$ be finite groups and $\chi: K \to \mathbb{C}^*$ be a homomorphism. Let $\mathbb{C}_{\chi}$ be the corresponding 1-dimensional representation of $K$. Let $$ e_{\chi} = \frac{1}{|K|} ...
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subgroup of character group

Let $B$ be a subgroup of a finite Abelian group $A$ and $$B^0=\{\alpha \in \hat{A}\mid \alpha(b)=1 \text{ as } b \in B\}$$ Prove a. $B^0$ is subgroup of $\hat{A}.$ And for any subgroup ...
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automorphism group

an endomorphism of a vector space V is a linear operator V → V. and an automorphism is an invertible linear operator on V. When the vector space is finite-dimensional, the automorphism group of V is ...
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isomorphism and subgroups

If we have two finite groups like automorphism groups of vector spaces, in order to check whether two finite groups are isomorph or not, how can we use subgroups of them?
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Sylow subgroups

Let $G$ be a groups of order $385$ 1. Show that $P_7$,$P_{11}\triangleleft G$ 2. Show that $P_7 \subseteq Z(G)$ 3. Show that $Z(G)=P_7$ or that $G$ is cyclic *for $P_i$ - the sylow-$i$ subgroups of ...
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36 views

Definition of K-normal subgroups

"We say that a group K acts semisimply on an abelian group A, if the intersection of all maximal K-normal subgroups of A is trivial" from "Construction of Finite Groups" article by "BESCHE and EICK" ...
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Finding conjugacy classes of $D_{10}$

Looking at the group $D_{10}$, I have found that for some (non-identity) rotation $\rho$ its centraliser has order 5, and for some reflection $\tau$ its centraliser has order 2. By the ...
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If $G/H$ is $p$-group, prove that $G$ is the direct product of $H$ with the centralizer of $H$ in $G$.

Let $H$ be a normal subgroup of the finite group $G$ and assume that $H$ is a nonabelian group of order $pq$ where $p>q$ are primes. If $G/H$ is $p$-group, prove that $G$ is the direct product ...
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Prove that $|K \cap K^g|$ is not divisible by $p$ if $g \notin K$.

Let $G$ be a finite group and $p$ be a prime. Let $K$ be a subgroup of $G$ which contains $N_G(Q)$ for every nonidentity $p$-subgroup $Q$ of $K$. Prove that $|K \cap K^g|$ is not divisible by $p$ ...
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Let $ G$ be $SL_2(\mathbb{F}_5)$. Find a subgroup of $G$ isomorphic to $Q_8$, and an element of order $3$ normalizing it in $G$.

Let $G$ be $SL_2(\mathbb{F}_5)$ i.e. the special linear group of $2\times 2$ matrices $\mathbb{F}_5$. Find a subgroup of $G$ isomorphic to $Q_8$, and an element of order $3$ normalizing it in $G$. I ...
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Irreducible representations of groups of order $pq$: induction from normal subgroups

Consider a group $G$ of order $pq$ ($p$ and $q$ are distinct primes and also $p<q$). It is easy to show that the dimension of each irreducible representation of $G$ is $1$ or $p$. Also, it can be ...
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Extend to a homomorphism.

My question is regarding a step in "p-Automorphisms of Finite p-Groups" by Evgenii I. Khukhro (p. 117 line 7) and would like some response to my argumentation/understanding of it. p-Automorphisms of ...
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Irreducible and regular representation

I was working on representations of $S_3$ and thought about the following problems: The symmetric group $S_4$ acts on $\mathbb{C}^{4}$ by permuting coordinates. Decompose this representation into ...
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Character table of a cyclic group

I am trying to verify the orthogonality relations for characters of a cyclic group $C_3$. First I am trying to prove that all the rows are orthogonal. However if one refers to the character table ...
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67 views

Representation theory

I am trying to study representation theory from the book Algebra by Artin. I came across the following problem which seemed interesting: Prove that the linear operator $T=\sum_{g\in C} \rho_{g}$ is G ...