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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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 ...
1
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1answer
142 views

If H, N are normal subgroups of G, then do all the commutators lie in the intersection?

Okay, I know that this is elementary, but, ah, well. How do I show that if N and H are normal subgroups of a finite group G with coprime orders, then, $xyx^{-1}y^{-1} \in H\cap N$ for all $x \in H, y ...
3
votes
3answers
439 views

Prove that: the center of any group is characteristic subgroup .

Let $G$ be any group , $Z(G)$ is the center of the group $G$ , prove that : $\forall \tau \in Aut(G) , \tau [(Z(G)] = Z(G)$ My first trial was to prove that the center of any group is the unique ...
2
votes
1answer
110 views

$G$ is $p$-supersoluble iff $|G : M | = p$ for each maximal subgroup $M$ in $G$ with $p | |G : M |$.

Let $G$ be a $p$-soluble group. Then $G$ is $p$-supersoluble if and only if $|G : M | = p$ for each maximal subgroup $M$ in $G$ with $p | |G : M |$.
0
votes
1answer
83 views

$O_{p', p}(G) =\cap C_{G}(H /K )$

Let $G$ b e a group and let $p$ be a prime number. Then $O_{p', p}(G) =\cap C_{G}(H /K )$, where $H /K$ ranges over all principal factors of G with $p | |H /K |$.
8
votes
2answers
110 views

If I know the Conjugacy classes of a group, do I know the group?

I know that a group has Conjugacy classes of size 1, 3, 6, 6, 8 and I know that this matches with the Conjugacy classes of the group $S_4$. But could there be a different group, with the same Congucy ...
1
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2answers
197 views

Let $G$ be a finite group. Show that if $G$ has exactly one nontrivial subgroup, then order of $G$ is $p^2$ for some prime $p$. [duplicate]

Let $G$ be a finite group. Show that if $G$ has exactly one nontrivial subgroup, then order of $G$ is $p^2$ for some prime $p$. I am totally clueless for this problem.Can someone help me to solve ...
5
votes
2answers
175 views

Find a $2$-Sylow subgroup of $\mathrm{GL}_3(F_7)$

We have $|\mathrm{GL}_3(F_7)| = 7^3 \cdot 2^6\cdot 3^4\cdot 19$. I can find the $3,7,19$-Sylow subgroup of it, but failed to find a $2$-Sylow subgroup. Can one help?
1
vote
1answer
104 views

Classification of group extensions

For hours I have been looking for " Claude Archer. Classification of group extensions. PhD thesis,Université Libre de Bruxelles, 2002 " but I found nothing . Is there any replacement for this thesis , ...
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3answers
152 views

simple group theory question

Can someone give an example of a finite (ideally nonabelian) group $G$ and two surjective homomorphisms $\phi_1,\phi_2 : F_2 \rightarrow G$ (where $F_2$ is the free group on the generators $x,y$), ...
2
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1answer
50 views

What could be said about a field extension, if it's Galois Group is fixpointfree?

Let $L/K$ be a galois extension, and let the Galoisgroup be fixpoint-free, what could be said about the field extension.
0
votes
3answers
471 views

Show there is a normal subgroup $K$ of $G$ with $K\subset H$ and such that the order of $K$ divides $n!$ [duplicate]

Let $G$ be a finite group and suppose $H$ is a subgroup of $G$ having index $n$. Show there is a normal subgroup $K$ of $G$ with $K\subset H$ and such that the order of $K$ divides $n!$ . any ...
5
votes
1answer
113 views

Computing eigenvalues from characters

This is a question in Representation theory, a first course, where the authors try to explain why character theory turns out to be so effective for the study of representations of finite groups. In ...
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0answers
95 views

Non-abelian groups of order $p^2q$

Let $G$ be a non-abelian group of order $p^2q$ and $p> q$. i) $p \equiv 1 \mod q$ or $p \equiv -1 \mod q$; ii) The number of elements of order $pq$ in group $G$ is $0$ or $p(p-1)(q-1)$.
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0answers
59 views

Assume simply connectivity without loss of generality

Let $X$ a connected Riemann surface and $G$ a finite group that acts faithfully and holomorphically on $X$. Further, let $x \in X$ a non-trivially stabilized point (we know these points are discrete), ...
4
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1answer
104 views

Subgroups of order 8 in the quasidihedral group of order 16

Why are there only $3$ subgroups of order $8$ in the quasidihedral group $QD_{16}$ of order $16$? (I am not interested in drawing the lattice of subgroups, but rather an argument convincing one that ...
2
votes
0answers
123 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 ...
3
votes
1answer
230 views

Irreducible representations over $\Bbb R$

How to prove that all irreducible representations over $\mathbb{R}$ of finite abelian group have dimension 1 or 2?
0
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1answer
102 views

If the group $Q$ is $\langle x, y \mid x^{9}=y^{3}=1, x^{y}=x^{-1}\rangle$, what is the subgroup of $Q$ generated by its elements of order dividing 3?

Let $Q=\langle x, y \mid x^{9}=y^{3}=1, x^{y}=x^{-1}\rangle$ and suppose we define a subgroup $‎\Omega‎_1(Q)$ to be the subgroup of $Q$ generated by all elements in $Q$ of order dividing 3. Can one ...
3
votes
3answers
429 views

Classifying the groups of order $2013$ (up to isomorphism)

Let $G$ be a group such that $|G|=2013$, how would you classify, up to isomorphism, all groups $G$?
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2answers
42 views

Normal subgroups of $\langle(123),(456),(23)(56)\rangle$

Let $G$ be a subgroup of the symmetric group $S_6$ given by $G=\langle(123),(456),(23)(56)\rangle$. Show that $G$ has four normal subgroups of order 3. I may be missing something, but I can ...
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0answers
50 views

How to build $\operatorname{Hom}(\mathbb{Z}_p^*,\mathbb{Z}_{pq}^*)$ without solving DLP?

For given two distinct primes p and q, I want to construct all homomorphisms from the multiplicative group $\mathbb{Z}_p^*$ to the multiplicative group $\mathbb{Z}_{pq}^*$. Thanks to Jyrki ...
2
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0answers
99 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 ...
8
votes
2answers
635 views

Can the semidirect product of two groups be abelian group?

while I was working through the examples of semidirect products of Dummit and Foote, I thought that it's possible to show that any semdirect product of two groups can't be abelian if the this ...
8
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0answers
132 views

How useful are geometric aspects when studying finite groups?

My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of ...
4
votes
2answers
719 views

Order of products of elements in a finite Abelian group

We want to show that if $a,b\in G$ where $G$ is a finite Abelian group, we have $\operatorname{LCM}(|a|,|b|) = |ab|$ given that $ab \neq e$. How I approached this question was by saying let ...
0
votes
2answers
66 views

How to decide whether a p-subgroup of some sporadic groups is cyclic?

Suppose that H is a subgroup of some sporadic groups (say convey groups Co1, Co2, etc.) and $|H|=p^k$ for some prime p and integer k>1, how to determine whether H is cyclic?
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0answers
89 views

Supersolvable group and nilpotent maximal subgroup

$G$ is a supersolvable group, $|G|=pq^{b}$ ($p$ and $q$ are different prime numbers). $Q$ is a $q$-Sylow subgroup of $G$, $Q=\langle x\rangle$ and $\operatorname{Cor}_{G}(Q)=\langle x^{q}\rangle$. If ...
5
votes
2answers
90 views

Are homomorphisms into PGL related to the Schur multiplier?

I've been trying to understand homomorphisms from a finite group $G$ into $\operatorname{PGL}(n,R)$ for $n$ a positive integer, and $R$ a commutative ring with 1, usually a field. I had been under ...
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2answers
57 views

Application of Cauchy theorem to prove normality of a subgroup

Let $G$ be a group $o(G)=pq$, where $p,q$ are both distinct prime numbers. Let $H<G$ be a subgroup of $G$ and $o(H)=p$. I want to show that $H$ is normal in $G$. My argument goes as follows. First ...
3
votes
1answer
163 views

Generalizations of fitting subgroup

The Fitting subgroup of a group $G$ has two generalizations: the generalized Fitting subgroup $F^*(G)$ of Bender and $\tilde F(G)$ of Schmid. The latter is defined by $\tilde F(G)/\Phi(G) = ...
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votes
2answers
225 views

if $G$ is a group , $H$ is a subgroup of $G$ and $g\in G$ , Is it possible that $gHg^{-1} \subset H$? [duplicate]

if $G$ is a group , $H$ is a subgroup of $G$ and $g\in G$ , Is it possible that $gHg^{-1} \subset H$ ? this means , $gHg^{-1}$ is Proper subgroup of $H$ , we know that , $H \cong gHg^{-1}$ , so if ...
0
votes
1answer
78 views

What is wrong with my thinking, simple groups order $168$

How many elements of order $7$ are there in a simple group of order $168$? I will work on this more but I have seen some solutions out there. My only question is regarding what is wrong what my ...
2
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2answers
105 views

About free group and kernel of homomorphism

I'm now reading textbook in group theory but couldn't understand its briefy explanation below "Let $G=<a,b\mid a^4=e,b^2=e,bab^{-1}=a^{-1}>, S=\{a,b\},F(S)$ be a free group and $N$ be the ...
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0answers
44 views

Maximal subgroups of $G=Z_{3}\ltimes Q_{8}$

Let $G=Z_{3}\ltimes Q_{8}$. How can find Maximal subgroups of $G$ ? $$Q_{8}$$ is Quaternion group of order of 8 and $$Z_{3}$$ is cyclic group of order 3
1
vote
1answer
83 views

Is there other homomorphisms from $\mathbb{Z}_q^*$ to $\mathbb{Z}_{pq}^*$?

For given two distinct primes p and q, is there other homomorphisms from the multiplicative group $\mathbb{Z}_q^*$ to the multiplicative group $\mathbb{Z}_{pq}^*$, except the following two maps: ...
1
vote
1answer
104 views

Is there non-trivial homomorphism from $\mathbb{Z}_q^*$ to $\mathbb{Z}_p^*$?

For two distinct primes $q$ and $p$, is there non-trivial homomorphism from $\mathbb{Z}_q^*$ to $\mathbb{Z}_p^*$? Here, $\mathbb{Z}_q^*$ and $\mathbb{Z}_p^*$ mean the multiplication groups with ...
4
votes
2answers
293 views

Presentation of a non-abelian group of order $pq$.

What is the presentation of the non-abelian group of order $pq$ where $p$ and $q$ are primes and $q\mid(p-1)$? Thanks in advance.
16
votes
1answer
395 views

Books to understand the construction of all groups of a specific order

The algorithms introduced by Besche–Eick (1999) were used to construct (or count) the groups of order up to 2000 in Besche–Eick–O'Brien (2002), yet I find the algorithms somewhat inaccessible. How ...
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1answer
57 views

How to choose a proper binary operation in a semigroup?

I am interested in generating a finite commutative semigroup which is not a group. And by generating I mean choosing a number $n$ (number of elements in a semigroup) and then defining a $n \times n$ ...
8
votes
2answers
238 views

Known bounds for the number of groups of a given order.

The number of nonisomorphic groups of order $n$ is usually called $\nu(n)$. I found a very good survey about the values. $\nu(n)$ is completely known absolutely up to $n=2047$, and for many other ...
0
votes
1answer
49 views

Schmidt group and Permute 2-maximal subgroup with 3-maximal subgroup

$G$ is Schmidt group With abelian Sylow subgroup Then every $2$-maximal subgroup of $G$ permuts with all $3$-maximal subgroup of $G$.
2
votes
3answers
140 views

A solvable group with order divisible by exactly two primes contains a normal subgroup of prime index.

$G$ is solvable group then $G$ has a normal subgroup $N$ of $G$ such that $|G: N|$ is a prime.
1
vote
1answer
40 views

$G$ be a non-nilpotent group and every $2$-maximal subgroup Per with all $3$-maximal subgroup

Let $G$ be a non-nilpotent group. If $|G|=p^{\alpha}q^{\beta}r^{\gamma}$ where $p$,$q$,$r$ are primes (two of them maybe are same) such that $\alpha + \beta +\gamma \leq 3$ then every $2$-maximal ...
0
votes
1answer
36 views

Solutions of $ 0 = x^2 -ay^2 -1$ in $\mathbb F_q$ where $a$ is not a square.

Assume $F = \mathbb F_q$ where $q = p^r$ for $p$ prime and $r > 0$. I have to count $$ \{(x,y) \in F^2 \mid x^2 -ay^2 -1 = 0\} $$ where $a$ is not a square in $F^*$. The equation is equivalent to ...
6
votes
4answers
369 views

If $(|G|, |H|) > 1$, does it follow that $\operatorname{Aut}(G \times H) \neq \operatorname{Aut}(G) \times \operatorname{Aut}(H)$?

Let $G$ and $H$ be finite groups. If $|G|$ and $|H|$ are coprime, then $$\operatorname{Aut}(G \times H) \cong \operatorname{Aut}(G) \times \operatorname{Aut}(H)$$ holds. What about when $(|G|, |H|) ...
0
votes
1answer
108 views

Problem 5.15, I. Martin Isaacs' Character Theory

Isaac's Character theory of finite groups book, Problem 5.15: Let $H \subseteq G$ and suppose $\phi$ is a character of $H$ with $det(\phi)=1_{H}$. Let $\chi={\phi}^{G}$ and show ...
3
votes
3answers
215 views

Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$

What can we say about the intersection of the $p$-sylow and $q$-sylow subgroups of group $G\;$? It's not necessary that $p=q$. Is there general statements about the intersections of sylows subgroups ...
4
votes
3answers
239 views

If $H$, $K$ are characteristic subgroups of $G$, when is $\operatorname{Aut}(H\times K) = \operatorname{Aut}(H) \times \operatorname{Aut}(K)$ true?

If $H$, $K$ are characteristic subgroups of $G$, when is $\operatorname{Aut}(H\times K) = \operatorname{Aut}(H) \times \operatorname{Aut}(K)$ true? What if $H$, $K$ are not characteristic subgroups? ...
1
vote
1answer
127 views

Questions about cosets, conjugate classes etc

Some questions about subgroups, normal subgroups, conjugate classes etc, just to make sure I understand it :-) The index of a subgroup $H$ in $G$, written as $[G:H]$ is defined as the number of left ...