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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175 views

A subclass of metacyclic groups

A group $G$ is metacyclic if it contains a cyclic normal subgroup $N$, such that $G/N$ is also cyclic. $N$ is called the kernel, and the order of $G/N$ is the index of the metacyclic group. There is ...
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143 views

A class of finite groups

Let $G$ be a finite group having the property that for any prime $p$ dividing $|G|$, it has a subgroup H with $[G:H]=p$. What can be said about these groups? I believe I can prove that they must be ...
3
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108 views

Blocks and covering modules

I have a question relating to p109 of Local representation theory by JL Alperin. Let $G$ be a finite group and let $N$ be a normal subgroup. If $B$ is a block of $G$, why must $B$ be a summand of ...
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224 views

[H,K] abelian if K centralizes [H,K]

Here is a paraphrased version of problem 4B.4 in Isaacs's Finite Group Theory: Let $G$ be a group and $X,Y$ subgroups of $G$, such that $Y$ centralizes $[X,Y]$. If $X$ is normal in $G$, show ...
2
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27 views

Characterize all $\mathbb Z[i]$ modules of order $21$ and $65$

I am trying to figure out how many $\mathbb Z[i]$ modules (up to isomorphisms) are of $21$ and $65$ elements. I've done a few similar exercises for finite abelian groups ($\mathbb Z$ modules) but I am ...
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51 views

Every non-regular minimal normal subgroup of a doubly transitive group is primitive and simple

Prove that every non-regular minimal normal subgroup $N$, of a doubly transitive permutation group $G$, is primitive and simple. I proved that $N$ is primitive; but how I can prove that $N$ is ...
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53 views

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ?

If $G $ is a group of order $12$ not isomorphic to $A_4$ then does $G$ have an element of order $6$ ? ( By Cauchy's theorem I can show that there are elements of order $2$ and $3$ but cant proceed ...
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39 views

A Question on the Quotient Group and/or set of cosets

I'm just confused about a somewhat simple fact about quotient groups. If we have: $$H<G/N$$ is a subgroup of the quotient of a finite group $G$ by $N\trianglelefteq G$, and $|H|=n$. Can we ...
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32 views

If $G$ is a finite non-trivial group of odd order, it has an irreducible representation not realisable over the reals.

$\textbf{Lemma }$If $V$ is a representation of a finite group $G$, then $V$ is of real type if and only if $V$ is the complexification of a representation $V_{\mathbb{R}}$ over the field of real ...
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39 views

Possible number of Groups order N

For when is the number of groups of some order n more than n? For example, let say this happens at $n=3$ then that would mean that there are more groups of order 3 than 3.
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22 views

How do I know the cyclic subgroups of G generated by a are equal?

I had a question of understanding that I was hoping you guys could help me with. Suppose there exists a group in the integers. For the sake of my understanding, let's say the group is, G = $Z_{60}$. ...
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62 views

Non abelian group of order $p^n$

Construct a non abelian group $G$ of order $p^n$(infact n>2) such that $G$ is not direct product of any of its two subgroups. I think we have to use semi direct product and the fact that G has at ...
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29 views

Parametrizing a group element

I'm looking at the problem of parametrizing a group element $g \in SL(n,\mathbb{R}).$ I think I understand the concept of parametrization $-$ just a change of coordinates, right? But I don't ...
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45 views

Group's morphisms

I know that the constant map equal to $1$ and the signature are two group's morphisms from the group of permutations $(\mathcal S_n,\circ)$ to the group $(\Bbb C^*,\times)$. My question is: Can we ...
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60 views

Existence of a finite group union of self normalizing subgroups

Does a finite group G union of self normalizing subgroups such that the intersection of any two of these subgroups is equal to the unit of group G exist? I don't think so, but I can't prove it. Thank ...
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78 views

Cokernel of injective endomorphisms of a finitely generated free abelian group

By $\text{GL}^+(n,\mathbb{Z})$ we mean the set of $n×n$ invertible matrices with positive determinant and entries from $\mathbb{Z}$. For given $A \in \text{GL}^+(n,\mathbb{Z})$ let ...
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70 views

Groups such that all elements of even order are in $G'$

We know that $G=A_4$ is a group such that all elements of even order in $G$ are in $G':=[G,G]= V_4$, the klein four group. Are there other examples or classes of groups where all elements of even ...
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20 views

structure of a p-group

Let $G=\langle x_1,\ldots,x_6\rangle$ be a non abelian group of order $p^6$ and exponent $p$. Also we know that $[x_i,x_j]=1$, for $1\leq i, j\leq6$, except $[x_1,x_5]=x_3$, $[x_1,x_6]=x_4$ and ...
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35 views

Fixed point set of orthogonal Transformation

I need some help with this problem. Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that ...
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35 views

Sort-of-multiplicative functions on the group algebra

Let $G$ be a finite group. Which functions $f:G \to \mathbf{C}$ obey the equation $$ \sum_{g \in C_1,h \in C_2} f(gh) = \left(\sum_{g \in C_1} f(g) \right)\left(\sum_{h \in C_2} f(h)\right) $$ for ...
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76 views

Why is the order of the subgroup 3?

I want to find the order of the subgroup $\langle ab\rangle$ of $D_3=\langle a,b\mid a^3=1,b^2=1,ba=a^2b\rangle$ According to my notes, the order of this subgroup is 3. But why is it like that? I ...
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38 views

Neccessary Condition involving Sylow-Subgroups for $p$-Solvability

Let $G = AB$ and let $P \unlhd A$, where $P \in Syl_p(G)$. If $P$ permutes with all Sylow $q$-subgroups of $B$ with $q \ne p$, then $G$ is $p$-solvable. Any suggestions on how to proof?
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24 views

Counting homomorphisms by the order of their images

I am trying to count homomorphisms from $\mathbb Z^r$ to $(\mathbb Z/m)^n$ while keeping track of the order of the image of each map. In other words, for each integer $k$ dividing $m^n$, I want to ...
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48 views

Suppose that $|G| = p^aq$, where p and q are primes and a > 0, Then $G$ is not simple?

Proof : We can assume that p$ \neq $ q and $n_p >1 $, so $n_p$ = q. Now choose distinct Sylow p- subgroups $S$ and $T$ of $G$ such that $|S\cap T|$ is as larger as possibe and write $D = S \cap ...
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73 views

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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41 views

An algorithm for generating a finite group with a finite set of generators

Let $A$ be a finite set of permutations on $\Bbb N$ with finite support. Is there a good efficient algorithm to obtain the subgroup $\left<A\right>$ of the symmetric group ...
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28 views

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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56 views

Projectivizing a group: how to go from AGL(n,K) to PGL(n+1,K)

$ \newcommand{\GL}{\operatorname{GL}} \newcommand{\AGL}{\operatorname{AGL}} \newcommand{\PGL}{\operatorname{PGL}} $Given an irreducible matrix group $G_{\infty,0} \leq \GL(n,K)$, I form the group ...
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18 views

Show that the order of the class of $p+1$ in $\left( \mathbb{Z}/p^{\alpha}\mathbb{Z}\right)^{*}$ is $p^{\alpha-1}$

I tried to do that: $$(1+x)^p=1+px+\binom{p}{2}x^2+\dots+\binom{p}{p-1}x^{p-1}+x^p$$ So if $x=0 \quad mod(p^k)$ then $(1+x)^p=1 \quad mod(p^{k+1})$ Now I'm trying to deduce that ...
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107 views

Calculating sylow subgroups of some concrete groups

Question is to : exhibit all sylow $3$ - subgroups of $S_4$ What i have done so far is : Number of elements in symmetric group is $|S_4|=4.3.2=2^3.3$ number of elements of order $3$ in $S_4$ ...
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72 views

Groups of order 8

Ok so I am looking at a proof to all the groups of order 8, I've attached an image which basically is the start of the proof. In particular the part highlighted with yellow is causing me problem, I ...
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56 views

Automorphisms of finite almost simple groups

Let $P$ be a finite nonabelian simple group. Let $G$ satisfy $$ P\leqslant G \leqslant {\rm Aut}(P), $$ where $P$ is identified with $\rm{Inn}(P)$. I am trying to see if $$ {\rm Aut}(G)\cong ...
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37 views

It is true that $\mathrm {Im}(f^{n_{0}})=\mathrm {Im}(f^m)$ for all $m\geq n_0$ implies $\mathrm {Im}(f^{n_{0}})=\{0\}$

Let $G$ be a finite abelian group, and $f: G\longrightarrow G$ an endomorphisme of $G,$ such that $\ker(f)\neq \{0\},$ and $\mathrm {Im}(f)$ a propre subgroup of $G,$ so we have a descending chain ...
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214 views

Maximal subgroups of direct product of solvable groups

Let $G_1$ and $G_2$ be finite solvable groups and $M$ be a maximal subgroup of $G=G_1\times G_2$ show that one of the following holds: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
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35 views

Is this dual-spaced norm based on $L_2$ norm

I am reading the book of Claes Johnson about Numerical Solution of Partial Differential Equations by the Finite Element Method and particularly pages 34 and 98. I wrote these notes to my craft Is ...
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60 views

Finite group with elementary abelian centralizer

Let $G$ be group of order $q(q^{2}-1)/2$ (where $q=p^n$ is an odd prime power) such that $C_{G}(P)$ is elementary abelian for every Sylow $p$-subgroup $P$. Is there any classifications of this type ...
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19 views

isometry group of an integer $n$ as of the corresponding $\Omega(n)$-parallelotope

Let $\prod_{i\in I}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ and let's consider the $\Omega(n)$-parallelotope built with, for all $i\in I$, $a_{i}$ pairwise orthogonal ...
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55 views

Quaternion, Dihedral groups and A-D-E classification

$\bullet$ What is the role of Quaternion group $H$ and dihedral groups $D_n$ in A-D-E classification? $\bullet$ Is Quaternion group $H$ in $A$ (special linear Lie algebra of traceless operators) or ...
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66 views

Suppose that both $N$ and $G/N$ are nilpotent. Let $K$ be a nilpotent subgroup of maximal order, such that $G=NK$. Prove that $N_{G}(K)=K$.

Let $G$ be a finite group, $N$ be a normal subgroup of $G$. Suppose that both $N$ and $G/N$ are nilpotent. Let $K$ be a nilpotent subgroup of maximal order, such that $G=NK$. Prove that $N_{G}(K)=K$. ...
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97 views

Normal subgroups of group with order $p^2 $

I've already established existence of at least one subgroup of order $p$ for G, don't know where to go though. Also, for part b), it seems pretty easy which leads me to believe it may be a typo? ...
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55 views

Characters of subgroups of finite abelian groups

Let $G$ be a finite abelian group. Let $H$ be a subgroup of $G$. Let $\hat{G}$ be the group of characters of $G$. Is there a character $\chi \in \hat{G}$ such that $\chi(g) = 1$ iff $g \in H$?
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76 views

Results on Hall $\pi$-subgroups - proof verification

I would appreciate verification of the following proofs. The problems are all closely related and the proofs are similar, so I think it's only appropriate to post them together (also, they are part ...
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66 views

(some) Dihedral groups have a right transversal isomorphic to some dihedral group.

Let $D_{2n}=\langle a,b;a^n,b^2,(ba)^2\rangle$ denote the Dihedral Group of order $2n$. $H=\langle a^m,b\rangle$ be a subgroup of $D_{2n}$ of even index $m$ such that $m$ divides $n$. Choose a right ...
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64 views

Is a certain subgroup $S\leq G$ in the center of $G$, $S\leq Z(G)$?

All groups considered are finite. Let $A$ be a group such that $A=A'\left<x\right>$, where $A'$ is the commutator group and $\left<x\right>$ is cyclic of order $p\in\mathbb{P}$. How can I ...
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94 views

The matrix form of a representation of $S_3$.

I am going through some notes on group theory, and one problem states: Consider the three-dimensional representation of $S_3$ constructed as follows: Choose a basis $v_1,v_2,v_3$ of ...
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69 views

Partition of Symmetric Group

For symmetric group $S_n$, we need to find a collection of subgroups $G_i$'s such that union of these subgroups is the group $S_n$ and each subgroup found is isomorphic to direct product of cyclic ...
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50 views

On primitive groups with transitive subgroups of smaller degree

Let $G$ be primitive on $\Omega$ and $G_\Delta$ transitive on $\Omega-\Delta=\Gamma$. Let $1 < |\Gamma| \le \frac{1}{2}|\Omega|$. Then $G$ is triply transitive on $\Omega$. In addition, if ...
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68 views

What is the Weyl group of this group?

Let $G$ be the group $GL_{n_1}(q^{l_1})\times GL_{n_2}(q^{l_2})\times GL_{n_3}(q^{l_3})$. Here $GL_{n_1}(q^{l_1})$ is the rational points of $GL(n,\bar {\mathbf{F}}_q)$. My question iis what is the ...
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129 views

Class of $p$-supersolvable group is saturated formation.

I want to show that the class of $p$-supersolvable groups is a saturated formation. I only have the definition of $p$-supersolvable. What do I do? A group is $p$-supersolvable iff every chief factor ...
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106 views

trivial group actions V.s trivial homomorphisms ?!

this question is related to the semidirect product of groups , so let $H,K$ are groups. suppose , $f:K \rightarrow H$ is a homomorphism . so $H\rtimes_f K$ is a semidirect product . the ...