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

3
votes
3answers
112 views

Determining whether these two groups are isomorphic

Consider the following group of matrices with multiplication: $$I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix},\ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \ B = ...
4
votes
3answers
159 views

How to show that $H \cap Z(G) \neq \{e\}$ when $H$ is a normal subgroup of $G$ with $\lvert H\rvert>1$

Let $G$ be a group of order $p^n$, $p$ a prime, $n>1$ and $H$ a normal subgroup of $G$ with $\lvert H\rvert>1$. Show that $H \cap Z(G) \neq \{e\}$.
1
vote
2answers
70 views

Immersion of Quaternions

Does there exist an immersion of the Quaternion Group in the Symmetric Groups $S_6$ and $S_7$? If it does exist, can you give me an explicit description of that immersion?
1
vote
1answer
78 views

Does there exist a group of even order which every element is a square?

Does there exist a group of even order which every element is a square? I know in any group of odd order every element is a square. I am not sure the case of even order. Any suggestion?
3
votes
1answer
64 views

Are quasinilpotent groups a Fitting class?

A finite group is called quasinilpotent if it induces inner automorphisms on all of its chief factors. A solvable group is quasinilpotent iff it centralizes all of its (necessarily abelian) chief ...
2
votes
1answer
33 views

Given $S\unlhd\unlhd G$ (subnormal), where $S\unlhd H\leq G$. How can I construct a subnormal series from $S$ to $G$ that is $H$-invariant?

Given $S\unlhd\unlhd G$ (subnormal), where $S\unlhd H\leq G$. How can I construct a subnormal series from $S$ to $G$ that is $H$-invariant (this is all the elements from the series are normalized by ...
1
vote
1answer
497 views

Groups of order 2k have a normal subgroup of order k and odd permutations

Okay, well, I have to show that, if $G$ is a finite group, such that, $|G|= 2k$, where $k$ is odd, then, translation by its element of order 2 is an odd permutation and $G$ must have a normal subgroup ...
0
votes
1answer
60 views

Definition question of convex orbit of finite group action

Assume that a finite group or discrete group $G$ acts on a manifold $M$. Here what does it mean that orbit $G\cdot x$ is convex ? Thank you in advance.
6
votes
0answers
269 views

Finding the maximum number of subspaces of a vector space over finite field that satisfy these relations

I have a question and I am stuck. I was wondering if anyone has a thought, before I start a brute-force search. For $q$ a prime number and $n =6$, let $\mathbb {F}_{q}^{n}$ be an $n$-dimensional ...
17
votes
1answer
413 views

Probability that two randomly chosen permutations will generate $S_n$.

Every undergraduate learns a fact about the symmetric group that $(1\,2)$ and $(1\,2\,\cdots\,n)$ generate $S_n$. Also, it is interesting to know that for a prime $p$, any 2-cycle and any $p$-cycle ...
1
vote
2answers
99 views

If the order of $x\in G$ is $p$, the smallest prime dividing $|G|$, and $h^{-1}xh=x^{10}$ for some $h\in G$, then $p=3$

Let $G$ be a group and let $p$ be the smallest prime dividing $|G|$. Let $x\in G$ be such that $|x|=p$. If $\exists h\in G$ such that $h^{-1}xh=x^{10}$, then show that $p=3$.
4
votes
1answer
102 views

Possibilities for $[KL:F]$ when $[K:F]=[L:F]$ is prime

Suppose $K/F$ and $L/F$ are extensions of $F$ (contained in some common field) of degree $p$, where $p$ is prime. Standard arguments show that $[KL:F]$ must be in $\{p,2p,\ldots,p^2\}$. But are all ...
0
votes
2answers
93 views

Finding the order of $\,[2]\,$ in $\;\Bbb Z/m\Bbb Z$

Find the order of $[2]$ in $\Bbb Z/m\Bbb Z$ where: $\begin{align} (i) \;m & = 11 \\ \\ (ii)\;\; m & = 17 \\ \\ (iii)\;\; m & = 31 \\ \\ (iv)\;\; m & = 9 \\ \\ (v)\;\; m & = 14 ...
133
votes
1answer
4k views

Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...
1
vote
0answers
52 views

$G$ be a $p$-supersoluble group. and $O_{p'}(G ) = 1$ then $G$ is supersoluble

Let $G$ be a $p$-supersoluble group. If $O_{p'}(G ) = 1$ then $G$ is supersoluble.
1
vote
2answers
154 views

Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$, where $N \unlhd G$ and $ \theta \in Irr(N)$.

Let $N \unlhd G$ and $ \theta \in Irr(N)$. Show that ${\theta}^G \in Irr(G)$ iff $I_G(\theta) = N$. Where $I_G(\theta)$ is the stabilizer of $\theta$ in the action of $G$ on $Irr(N)$ defined by ...
2
votes
2answers
99 views

What is the significance of permutable subgroups? (and $X$-permutable subgroups?)

Let $G$ be a group and $H$, $K$, $X$ be subgroups of $G$. We say $H$, $K$ are permutable if $HK=KH$. or we say $H, K$ are X-permutable if $‎\exists x, x\in X$ such that $H^{x}K=KH^{x}.$ Why are ...
4
votes
3answers
130 views

Group of order $p^2$ is abelian. [duplicate]

Yes, I know that there are tons of solutions of this up here, but I, essentially, wanted to try it a different way and ah, well. Let $|G| = p^2$ for some prime $p$. Consider $x \in G$. So, $|x| = p, ...
1
vote
2answers
75 views

Finite Abelian Groups and Automorphisms of Groups

Let $G$ be a finite group, $T \in \text{Aut}(G)$ such that $T(x)=x$ implies $x = e$. Show that for all $g \in G$ there exists $x \in G$ with $g = x^{-1}T (x)$. Thanks to all.
11
votes
1answer
139 views

Abelian subgroups of $GL_n(\mathbb{F}_p)$

Let $p$ be a prime number, and let $k=\mathbb{F}_p$ be the field of $p$ elements. Let $G=GL_n(k)$. We know that $$|G|=\prod_{i=0}^{n-1}(p^n-p^i)=p^{\binom{n}{2}}\prod_{i=0}^{n-1}(p^{n-i}-1)$$ so ...
1
vote
1answer
80 views

Image of a normal Hall Subgroup under an automorphism

Let $G$ be a group such that $|G|= n = md$, where $\gcd(m,d)=1$. Let $N$ be a normal subgroup of $G$ with order $m$. Further, let us define a subgroup $H$ of order $d$. I managed to prove that, $H ...
1
vote
1answer
1k views

How to find generator in a finite group?what is generator?

Suppose that a group $Z_p=${$1,2,3......(p-1)$} where p is a prime number. How to Determine the generator/generators of this group? what are the possible method of finding it?
1
vote
2answers
774 views

Proof Help: In a group $G$, there exists a $g$ such that $g^2 = e$ [duplicate]

I'm working through an Abstract Algebra book to teach myself, and came across the problem: Prove: If $G$ is a finite group of even order, then there exists a $g\in G$ such that $g^2 = e$ and $g ...
2
votes
0answers
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
vote
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
386 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
107 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 $p | |H /K |$.
7
votes
2answers
108 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
vote
2answers
192 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
172 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
103 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 , ...
1
vote
3answers
151 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
votes
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
463 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
98 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 ...
1
vote
0answers
93 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 $p(p-1)(q-1)$.
1
vote
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
votes
1answer
100 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
120 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
224 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
votes
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
425 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 $G$?
1
vote
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 ...
1
vote
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
votes
0answers
95 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
589 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
votes
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
661 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?