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.

learn more… | top users | synonyms

2
votes
0answers
52 views

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 ...
2
votes
0answers
100 views

Groups not arising from certain centralizers

There's a lot of fuss in certain subfields of algebraic topology about giving fancy interpretations to the rings coming from the cohomology of groups, where "cohomology" is allowed to be taken to be ...
2
votes
0answers
75 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 $G=H.K$ ...
2
votes
0answers
119 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 ...
2
votes
0answers
76 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 ...
2
votes
0answers
405 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 ...
2
votes
0answers
44 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 ...
2
votes
0answers
47 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 ...
2
votes
0answers
45 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 $G$ ...
2
votes
0answers
74 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) ...
2
votes
0answers
225 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.
2
votes
0answers
62 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 ...
2
votes
0answers
137 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 ...
2
votes
0answers
78 views

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 ...
2
votes
0answers
78 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 $C= \langle A\rangle$ ...
2
votes
0answers
52 views

Subgroups of semi-direct products arising from fixed-point-free actions

I am interested in subgroups of semidirect products arising from fixed-points-free actions. Suppose you have a group $A$ acting fixed-point-freely on a group $N$. Can we describe completely the ...
2
votes
0answers
36 views

$k[V]^G = \widetilde{A}$ where $\widetilde{A}$ is the normalization of $A$

Let $V$ be a finite dimensional vector space over $k =\overline{k}$ and let $G$ be a subgroup of $GL(V)$ so that $k[V]^G$ is finitely generated. Let $A$ be a subring of $k[V]^G$ that is finitely ...
2
votes
0answers
130 views

Structure of elementary abelian group by a cyclic group of coprime order

My mind is a little foggy right now, so I would like to ask for some help on this (or an appropriate reference). Suppose $A$ is an elementary abelian $p$-group and $B$ is a cyclic group of order ...
2
votes
0answers
145 views

Are there 16 or 24 automorphisms of $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$?

In this question I said that the automorphism group of $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ has 16 elements because If $\varphi$ is one of this automorphism then ...
2
votes
0answers
77 views

radical layers equal socle layers

I've read that the radical layers of the group algebra $kP$ of a $p$-group $P$ coincides with the its socle layers (char $k = p$). What does this tell me about the structure of the group algebra or ...
2
votes
0answers
111 views

a question about elements of permutation groups

Let $n \in \mathbb{N}$ and let $S_n$ denote the permutation group on $n$ letters. I'm trying to figure out what kind of elements $\sigma, \delta \in S_n$ satisfy the following relations: ...
2
votes
0answers
216 views

Involution centralizer does not determine the group

Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution? The Brauer–Fowler results show that if a finite group has no ...
1
vote
0answers
22 views

Class preserving Autpmorphism

This time I am having interest in theory of Class preserving Automorphism and central preserving Automorphism and some topics related to Automorphism of groups. So Could you please tell me some ...
1
vote
0answers
19 views

Given a space group, how to determine which layer groups are its subgroups?

I am studying various crystals and the two-dimensional materials that could be potentially obtained by cleaving them (isolating a region bounded by two parallel planes). In elucidating the properties ...
1
vote
0answers
20 views

can say every group that satisfy in maximal permutizer condition then satisfy then permutizer condition

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
1
vote
0answers
42 views

An Abstract Characterization of $S_5$ using involutions and their centralizers

This is essentially an exercise from Jacobson's Basic Algebra I. (p.83, ex.10) I've managed to solve all the other part of the proof, except (vi) and (x). I've been thinking about this all day, but ...
1
vote
0answers
18 views

Question about exponents of groups

Okay, I'm trying to understand exponents of groups. I will start with the Set Z_3, where Z_3 is the integers mod3 under addition. Now, I want to set out to find the exponent of this group, but ...
1
vote
0answers
27 views

What does it mean for a subgroup to be self-centralising in terms of group extensions

In texts on group theory I read about subgroups $U \le G$ which fulfill the property $$ C_G(U) \le U $$ (this is called self-centralising, for example the Fitting subgroup in solvable groups fulfills ...
1
vote
0answers
38 views

Subgroups and Direct Products of $\pi$-closed groups are also $\pi$-closed.

Let $K$ be a class of finite groups, then this class is called closed iff i) homomorphic images of groups in $\mathcal K$ lie in $\mathcal K$, ii) subgroups of groups from $\mathcal K$ lie in ...
1
vote
0answers
35 views

Subgroups of the dihedral group D_n modulo Aut(D_n)

This question is related to this math.se question. Consider the dihedral group $D_n = \langle r,s \rangle.$ Two subgroups $G, H \leq D_n$ are said to be ''isomorphic'' if there is an $f \in ...
1
vote
0answers
15 views

Finite subgroups of $O_4(\mathbb{Q})$

I have a problem with the classification of finite subgroups(up to isomorphism) of $O_4(\mathbb{Q})$ (or $GL_4(\mathbb{Z})$). I know about classification of $GL_2(\mathbb{Q})$. Maybe somebody knows ...
1
vote
0answers
38 views

Question on proof that maximal normal abelian subgroup is self-centralising in nilpotent groups

The following is known about finite groups: (*) If $G/Z(G)$ is cyclic, then $G$ is abelian. Proposition: Let $G$ be a nilpotent finite group and $N$ a maximal abelian subgroup of $G$. Then $C_G(N) = ...
1
vote
0answers
36 views

Normalizer of a transitive subgroup in the symmetric group

Let $G$ be a finie group, and $H$ be a core-free subgroup of $G$ (that is to say, there is no nontrivial normal subgroup of $G$ contained in $H$). Denote by $\Omega$ the set of right cosets of $H$ in ...
1
vote
0answers
58 views

How many nonabelian groups up to isomorphism are of the order $p^4q^4$?

For distinct primes $p$ and $q$, how many nonabelian groups up to isomorphism are of the order $p^4*q^4$? We can say that there are nontrivial subgroups with cardinality $p,p^2,p^3,p^4,p*q,..,q^4$. ...
1
vote
0answers
22 views

Finding Bijection between subgroups of group $G = NU$.

Let $N \unlhd G$ and $U \le G$ with $G = NU$. Then there exists an inclusion-preserving bijection from the set of all subgroups $X$ with $U \le X \le G$ on the set of all $U$-invariant subgroups ...
1
vote
0answers
34 views

Enumerating double coset representatives in the symmetric group on a vector space

Let $V$ be a finite vector space over the finite field $\mathbb F_q$. Is there an algorithm to enumerate representatives for the double cosets $\mathrm{Aff}(V)\backslash S_V/\mathrm{Aff}(V)$, where ...
1
vote
0answers
45 views

Considering a permutation representation of a transitive $G$-set

Suppose $X$ is a transitive $G$-set, where the size is greater than $1$, and $\pi=\pi_X$ the associated permutation representation. What is its character $\chi$? I thought that the permutation ...
1
vote
0answers
56 views

Axiomatisability of the class of finite groups

I'm trying to prove that the class of all finite groups is not finitely axiomatizable. I was trying to do this in the same way one proves that the class of finite sets is not axiomatizable, i.e. Let ...
1
vote
0answers
71 views

My try to prove this lemma: $H\trianglelefteq G \iff \exists\ \phi:G \to G'$ with $\phi$ a group homomorphism such that $\ker(\phi)=H.$

I wanted to share with you the proof I made for this lemma (it has been given to us as an exercise to do on home). Here it goes: Lemma: Let $G$ be a group. Let $H \leq G$. Then: ...
1
vote
0answers
27 views

History of Dihedral Groups?

I'm trying to write a brief historical outline for a project concerning Dihedral Groups, but It has been to hard for me to find information about the first time these groups appeared on literature. If ...
1
vote
0answers
32 views

Number of the subgroups of a $p$-group with order $p^k$ is congurent to $1$ modulo $p$

Let $G$ be a $p$ group of order $p^n$ and $k\leq n$. Theorem:Number of the subgroups of $G$ with order $p^k$ is congurent to $1$ modulo $p$. I have found a proof of this theorem in Rotman's book ...
1
vote
0answers
25 views

General Classification of finite simple ternary groups?

Define a ternary group as an algebraic set endowed with a 3-ary operation f: that maps 3 elements onto another in the set. Furthermore for any three elements a,b,c there exists a unique 4th element ...
1
vote
0answers
30 views

Independent components of a group cocycle

Fix a finite group $G$. An $n$-cochain of $G$ with coefficients in a $G$-module $M$ is a function$$b:G^n\rightarrow M$$To determine $b$, one must specify the values $b(g_1,\ldots,g_n)\in M$ for all ...
1
vote
0answers
42 views

Show that the group of permutations of {1,2,3,4} is equal to the product of it's subgroups…

Show that the group of permutations of {1,2,3,4} $$\sigma_4$$ is equal to the product of it's subgroups $$C_2\times C_2 $$ and$$D_6=(x^3=y^2=1, yx=x^2y)$$ I'm not sure whether to just multiply the ...
1
vote
0answers
37 views

I need example to satisfy in this theorem (Hall Subgroup)

I need example to satisfy in this theorem: let $H$ be a subgroup of $G$ such that $\mid G : H \mid$ is a $\Pi$-number.If there is a nilpotent subgroup $K$ of $G$ such that $G=HK$ then $G=HK_{\Pi}$, ...
1
vote
0answers
64 views

Intuition behind the link between coding theory and group theory

I am trying to find an easy link between group theory and coding theory. The usual path that most of the texts follow is that they present introductory material on groups, fields, rings, etc., and ...
1
vote
0answers
52 views

Direct products of simple non-abelian groups 2

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ a simple non-abelian subgroup of $G$ such that $Z(G)\leq N$ and $\frac{G}{N}\cong A$ be a non-abelian simple group. Is it true that $G=N\times A$ ...
1
vote
0answers
42 views

Abelian minimal normal subgroup in a finite non-solvable group

Let $G=G^{'}Z(G)$ be a finite non-solvable group, $N$ an abelian minimal normal subgroup of $G$ ( $|N|=p^d$ for some integer $d$ and prime $p\neq 2,3,5$) such that $N=C_G(N)$, $Z(G)\leq N$ and ...
1
vote
0answers
99 views

What does $aba^{-1}b^{-1} \notin H$ imply?

​I am working on a problem on commutator subgroup of finite group. Long story short, I was given $H < G$ and $H' \neq H$ and am aiming to prove $H \lhd G$. As you know that $H'$ is commutator ...
1
vote
0answers
97 views

Why do Sylow $p$-groups in finite simple group have trivial intersection?

I'm looking for an argument for the fact that two Sylow $p$-subgroups in a finite simple group have trivial intersection. In the case that the order of the Sylow subgroups is the prime $p$ it is easy, ...