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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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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27 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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52 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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17 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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84 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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61 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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53 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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124 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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28 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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55 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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18 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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51 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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94 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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52 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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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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(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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62 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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85 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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67 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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46 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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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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112 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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88 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 ...
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85 views

Schmidt group and maximal subgroups

Let $G$ be a Schmidt group, a minimal non-nilpotent group, so that $G$ is not nilpotent, but every proper subgroup of $G$ is nilpotent. I want to prove $G$ has precisely two classes of maximal ...
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201 views

Characters of double groups

Problem: I want to do some calculations with the character projection operator to investigate the irreducible representations of wave functions. Until now, I did these calculations for simple ...
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54 views

Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)

This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups. More precisely, given discrete groups below (a), (b), (c): ...
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150 views

Cardinality relation between subsets of a group

$G$ is an abelian group, $A$ and $B$ are non empty finite subsets of $G$. Set $A+B := \{a+b\mid a\in A, b\in B\}$ and $H := \mathrm{stab}(A+B)=\{g\in G \mid g+A+B = A+B\}$. Prove that $$ ...
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Separable elements of a finite abelian group

Let $\mathbf G$ be a finite abelian group, let $a, b \in \mathbf G$, and let $\langle a \rangle$ and $\langle b \rangle$ be the cyclic subgroups of $\mathbf G$ generated by $a$ and $b$ respectively. ...
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116 views

Element of a certain order in multiplicative group of residues

Let's say that $G$ is the multiplicative group of residues mod $p$, where $p$ is prime. I know that the order of an element $g \in G$ is the least $k$ for which $g^k \equiv 1 \mod{p}$. How can we go ...
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65 views

the orbit of a root under operations of irreducible crystallographic group?

Suppose we have an irreducible crystallographic coxeter group G acting in a vector space V, how can we show that the orbit of an ...
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174 views

$G$ fin ab group, acts faithfully, transitively on $X$, then $|X|=|G|$

Let $G$ be a finite abelian group. Suppose that $G$ acts faithfully and transitively on a set $X$. Show that $|X|=|G|$. Deduce that the action is equivalent to the action of $G$ on itself by left ...
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79 views

Find the cycle index of cyclic group $C_p$ where $p$ is prime

Find the cycle index of cyclic group $C_p$ where $p$ is prime. No idea where to start...any help pointing me in the right direction would be appreciated.
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50 views

A problem about normalizers in $PSL(V)$

Let $K=\mathbb F_{p^k}$ a finite field, and $V$ a vector space on $K$. Clearly $PSL(V)=SL(V)/SL(V)\cap Z$ acts on $V$ by the following rule ($Z$ is the subgroup of the scalar functions): ...
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characteristic-$p$-type groups, and the Borel-Tits theorem for $PSL(V)$

If $G$ is a finite group and $F^*(G)$ is the generalized Fitting subgroup we say that $G$ has characteristic $p$ ($p$ is a prime that divides $|G|$) if $$F^*(G)=O_p(G)$$ Moreover $G$ is a ...
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73 views

Group extension analysis

Let $\mathbb{Z}_p\lhd H\leq\text{AGL}(1,p),\mathbb{Z}_q\lhd K\leq\text{AGL}(1,q)$ with $p,q$ prime. Let $G=H.K$. Can one show that $G$ contains a normal subgroup of order $pq$? Note: Here $G=H.K$ ...
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108 views

Lie Theory in Finite Groups

Currently, I have interest in Finite groups. I also want to learn Lie Groups, Lie algebras, and their representations. But I do not have any motivation for it. Question What are simple but ...
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70 views

“unitary group” with respect to non-hermitian matrices?

the group $GU(n,q)$ is usually defined as the group of $n$ by $n$ matrices $X$ over $\mathbb{F}_{q^2}$ such that $X^H X=I_n$, where $I_n$ denotes the identity matrix and $^H$ the hermitian transpose ...
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280 views

How to find the number of orbits

In our lecture notes, i am confused on how to determined the number of orbits with the given permutations in S4. If i consider the permutation (12) . I can also rewrite it as cycle decomposi- tion and ...
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39 views

Invariants of parabolic subgroups

Let $G$ be a finite reflection group acting on $R^n$. Then for each point $x\in R^n$ we can look at its stabilizer $G_x$. Since $G$ is a finite reflection group, its ring of invariants is a polynomial ...
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43 views

Product of two squares in finite groups

Let $G$ be a finite group and $g$ an element of order $n$ in $G$. Assume that $g$ is a product of two squares. Moreover, assume that $n$ and $k$ are coprime. Prove that $g^k$ is also a product of two ...
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43 views

Maximal soluble subgroups in a parabolic subgroup of finite classical simple group

I'm new to algebaric groups, so please accept my apology in advance if I post crazy questions. Let $G$ be a classical simple group over a finite field $GF(q)$ and $P$ a parabolic subgroup of $G$ ...
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69 views

normal subgroup + Socle

I am looking for examples of socle and normal-subgroup relations. If $G=S_{4}$, the normal subgroups are $A_{4}$ and $V_{4}$, thus the socle of $G$ is the klein-four group, and for $A_{4} \cap Soc(G) ...
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193 views

Normal Sylow subgroups in a group of square free order

If $|G|=n$ with $n$ square free then there exists at least a normal Sylow subgroup? Any suggestion are welcome. Thanks.
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56 views

A bound for the exponent of the Schur multiplier of group G

Let $G$ be a finite group and $M(G)$ be the Schur multiplier of $G$. Show that the exponent $\exp(G)$ of $G$ divides the product of the exponents $\exp(M(S_p))$ of the Sylow $p$-subgroups $S_p$ of ...
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119 views

Automorphisms of a group and cyclic subgroups

I have the following question: Let $G$ be a group. And $A,B\leq G$ two cyclic subgroups with $|A|=|B|=n$. When does an Automorphism $\alpha\in\mathrm{Aut}(G)$ exists such that $\alpha(A)=B$? If ...
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Connection between: Expand a representation and dual representation

I worked the proof for 3) in this link out, but I have problems with the last step: Let $\sigma$ be a irred. representation of a normal subgroup $H=\langle z\rangle$ of $G$ and $\sigma'$ its dual ...
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118 views

Conjugacy classes in $SL(n,q)$

If $q$ is a prime power, then the conjugacy classes in $GL(n,q)$ are determined completely by minimal polynomials. Now in the subgroup $SL(n,q)$, these conjugacy classes can split, in the sense they ...
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58 views

Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant $\mathbf{Question \;1}$. Let $$ A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right). $$ Then $C= \langle A\rangle$ ...