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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3
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0answers
141 views

Can all of them be different?

Edit: Cross-posted to MathOverflow here (and resolved). Let $G=\{g_1,g_2,...,g_n\}$ be a group with $e=g_1$ and $n$ is odd, Set $$a_1=g_1$$ $$a_2=g_1g_2$$ $$a_3=g_1g_2g_3$$ $$a_n=g_1g_2...g_n$$ I ...
2
votes
1answer
125 views

An example concerning some fields

I was trying to understand the following example This is also why I've made some questions today on locally finite fields. I've almost understand everything (thanks also to some of you) but I ...
0
votes
2answers
37 views

suppose $G$ is a finite group, $G$ is nilpotent iff every quotient group of $G$ has no trivial center.

suppose $G$ is a finite group, $G$ is nilpotent iff every quotient group of $G$ has no trivial center. any hint or guidance will be great,I badly stuck in this one,thanks.
0
votes
2answers
28 views

Principal series of a finite super soluble group

Every principal series of a finite super soluble group is a composition series. I have thought a lot about this and no progress, please help me a little and guide me in right path. Thanks a lot.
0
votes
1answer
20 views

show that $D_{2n}=\left \langle a,b \mid a^2=e,b^n=e,aba^{-1}=b^{-1} \right \rangle$ is nilpotent iff $n$ is power of 2.

show that $D_{2n}=\left \langle a,b \mid a^2=e,b^n=e,aba^{-1}=b^{-1} \right \rangle$ is nilpotent iff $n$ is power of $2$. please help me how should start and proceed,any guide or hint will be ...
2
votes
1answer
28 views

Order of cyclic subgroup

This should be easy, but I keep getting stuck on it. Suppose $D$ is a cyclic group of order $m$ (written additively) and then let $D[n] = \{x \in D : nx = 0\}$. I'm trying to show $D[n]$ has order ...
1
vote
1answer
29 views

regular unipotent elements

Let $G$ be a finite simple group of Lie type over a finite field of characteristic $p\neq 2$ and $q$ elements. We know that if $p$ is not bad for $G$ then $G$ contains $q^l$ regular unipotent elements ...
1
vote
1answer
33 views

The Jordan-Hölder Theorem

I want to solve the following exercise from Dummit & Foote's Abstract Algebra (exercise 10 in page 106): Prove part (2) of the Jordan-Hölder Theorem by induction on $\min\{r,s\}$. [Apply the ...
1
vote
1answer
37 views

The subgroup $O_{\pi}(G)$ in a finite group

For a finite group $G$ the set $O_{\pi}(G)$ is the maximal normal $\pi$-subgroup of $G$. Could there anything said about $G / O_{\pi}(G)$? And maybe do you know some more properties around ...
1
vote
1answer
18 views

Radical and Residue in (Finite) Groups

I am currently trying to understand the definition of the $\mathcal K$-residue and $\mathcal K$-radical of a group $G$. The definitions are from the book The Theory of Finite Groups by Kurzweil and ...
0
votes
1answer
49 views

I claim that there is a 'path' connecting $k$ and $1$.

Let $S=\lbrace 0,1,2,...,n\rbrace$. Define $S_0=S-\lbrace 0\rbrace$ and $S_k=S-\lbrace k\rbrace$ for some $k\in S_1$. Let $f:S_0\longrightarrow S_k$ be such that $f(i)=(k+i)$ mod $n+1$. I claim that ...
0
votes
1answer
19 views

On permutable $\pi$-groups and Existence of certain normalized subgroups [Question on proof].

Two subgroups $A,B \le G$ permute if $AB = BA$. All groups considered are finite. I have a question on some points of the following proof of Lemma 2. Lemma 1: Let $H \le G$, and let $\mathcal X$ be a ...
2
votes
0answers
36 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?
0
votes
1answer
24 views

Sylow-Subgroups and arbitrary groups where their order contains the same prime-power.

Let $|G| = p^k m$ with $p$ and $m$ being coprime. Then it is well known that there exists a subgroup $S$ of $G$ with $|S| = p^k$, the so called Sylow-$p$-subgroups. Now let $U \le G$ be some subgroup ...
1
vote
1answer
34 views

Prove that $\psi\in\operatorname{Aut}(Q_8)$

Suppose $Q_8\le G$ where $G$ has an element of order $3$, call it $a$. Let $\psi:Q_8\longrightarrow Q_8$ be defined by the conjugation by $a$, i.e. $\psi:q\mapsto q^a$. Suppose to know that this is ...
3
votes
2answers
96 views

Is there any usage of this sum formula?

Let $\phi$ be the Euler phi function, then $$n=\sum_{d|n}\phi(d)$$ It is one of the most famous formulas in number theory. It can be generalized by using groups. Let $G$ be a group of order $n$ and ...
1
vote
1answer
29 views

Does the Product of conjugates of some subgroups commutes with all elements of another subgroup

Let $U,V \le G$ abelian subgroups such that $V$ is normal and $G$ be finite, does it hold that for $v \in V$ we have $$ u \cdot \left( \prod_{u' \in U} v^{u'} \right) = \left( \prod_{u' \in U} ...
6
votes
0answers
49 views

Can we find a bound so that we can conclude $G$ is a $p$ group?

Let $n_p$ be number of the elements of order $p$ in a group $G$, My motivation is that if $n_2\ge\dfrac 34 |G|$ then $G$ is $2$ group. You can check it from this. Is there such general bound for ...
1
vote
0answers
34 views

Center of a $2$-Sylow

Here we go for the last time with the group $G=GL_2(\mathbb F_3)$. We know that it has three $2$-Sylow subgroups. So let $P\in\operatorname{Syl}_2(G)$. I was searching for its center but I don't ...
4
votes
1answer
56 views

Is it necessarly abelian $2$ group?

Is the following correct ? Let $n_2$ be a number of elements of $G$ of order $2$ if $n_2> \dfrac {|G|}2$ then $G$ is elemantary abelian $2$ group. Edit: we see that the conclusion "if $n_p> ...
0
votes
2answers
94 views

What are the other methods used to prove that a homomorphism is bijective?

The motivation can be found in: Show that $ℤ^{m}$ is a subgroup (and a free abelian group) of $ℤ^{n}$ for all $m≤n$. In a specified problem related to a dynamical system the only possibility is ...
3
votes
0answers
31 views

Simple question on Sylow subgroups.

Given a finite group $G$, $|G|=p^am$, $P,Q\in\operatorname{Syl}_p(G)$, then we know that $Q=P^x$ for some $x\in G$. Thus every element of $Q$ is the image of an element of $P$ under the isomorphism ...
2
votes
1answer
58 views

Two subgroups $N, A$, both abelian, $C_N(a) = 1$ for each $a \in A, a \ne 1$. Show that $A$ is cyclic.

I am stuck with the following proof, and hope you can help me: Let $N$ be a non-trivial abelian normal subgroup of $G$ and let $A \le G$ such that $|N|$ and $|A|$ are coprime. Let $A$ be abelian too, ...
1
vote
1answer
30 views

About $S$-semipermutable Hall $\pi$-subgroups and $\pi$-complements

Two subgroup $A, B \le G$ are said to permute if $AB = BA$. A subgroup $H \le G$ is called $S$-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for primes $q$ not dividing ...
1
vote
1answer
37 views

Groups of order 1814400

I would like to know whether there exists a group with the following structure: $G$ is a non-split extension of a cyclic group of order 3 ($C_3$) by the Janko group $J_2$ such that $G/C_G(C_3)$ has ...
5
votes
2answers
57 views

Show finite group is $p$-group given some structure of group

Let $G$ be a finite group. If there exists an $a\in G$ not equal to the identity such that for all $x\in G$,$\phi(x) = axa^{-1}=x^{p+1} $ is an automorphism of $G$ then $G$ is a $p$-group. This is ...
4
votes
4answers
69 views

$n$th power map is an automorphism implies abelian group?

If $G$ is a finite group and $\phi(x) = x^n$ is an automorphism of $G$ does this imply $G$ is abelian? I've been reading this page. Def: A group $G$ is said to be $n$-abelian if $(ab)^n=a^nb^n$ ...
2
votes
6answers
71 views

A question on cyclic group

I have no trouble proving the following statement. Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove the map $x \mapsto x^k$ is surjective. It is clear by ...
6
votes
0answers
66 views

Product of more than two subgroups [duplicate]

Let $A_1,A_2,...,A_n$ be subgroups of $G$ and $H=A_1A_2...A_n$. Is there any sufficient and necessary condition for $H$ to be subgroup ? When $n=2$, $H$ is a subgroup if and only if $A_1A_2=A_2A_1$. ...
3
votes
2answers
47 views

On the involutions of $GL_2(\mathbb F_3)$

Ok I swear this will be more or less the last topic on the group $GL_2(\mathbb F_3)$! I'm searching for all its involutions. I know that $ \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ is ...
0
votes
0answers
30 views

On the possible subgroups of a Sylow subgroup

Let $G$ be a finite group, say $|G|=p^am$, with $(p,m)=1$, Let $n$ be the number of $p$-Sylow subgroups of $G$. Call them $P_1,\dots,P_n$. Is true that every subgroup of $G$ of order $p^b$ with $b\le ...
1
vote
1answer
44 views

Conjugate relation in a group implies conjugate relation in a subgroup.

Let $G$ be a finite group, let $H$ be its normal subgroup of prime index. Take $x\in H$ such that the centralizer of $x$ in group $H$: $C_H(x)$ be a a proper subgroup of the centralizer of $x$ in $G$: ...
0
votes
0answers
28 views

On $S$-semipermutable Hall subgroups, such that $H^G$ is solvable.

Two subgroup $A, B \le G$ are said to permute if $AB = BA$. A subgroup $H \le G$ is called $S$-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for primes $q$ not dividing ...
0
votes
1answer
40 views

Is there any such example?

I am looking for an example that $H,K,L$ be subgroups of $G$ such that $HKL$ is a group but prodoct of any two is not. Thanks.
1
vote
0answers
25 views

Question to Corollary on $S$-semipermutable subgroups

Two subgroup $A, B \le G$ are said to permute if $AB = BA$. A subgroup $H \le G$ is called $S$-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for primes $q$ not dividing ...
1
vote
2answers
61 views

Given a Cayley table, is there an algorithm to determine if it is a dihedral group?

Showing that it is a group is simple enough, but is it possible to determine if it is a dihedral group or not just by looking at the Cayley table?
1
vote
3answers
38 views

Subgroups which contain all $p$-Sylowsubgroups for some fixed prime $p$

Is it true that if some subgroup $H \le G$, $H\ne G$ contains all $p$-Sylowsubgroups for some fixed prime $p$, then $H$ contains some non-trivial normal subgroup of $G$?
2
votes
2answers
68 views

Permutable subgroups are subnormal

Let $G$ be a finite group. A subgroup $H \le G$ is called permutable, if for each other subgroup $U \le G$ we have $UH = HU$. Show that if $H \le G$ is permutable, then $H$ must be subnormal. Does ...
1
vote
1answer
19 views

For permutable $\pi$-group's, the normal closure is a $\pi$-group.

A subgroup $H \le G$ is called permutable, if $UH = HU$ for every subgroup $U \le G$. Show that if $H$ is a permutable $\pi$-group, then its normal closure $H^G$ is a $\pi$-group. Deduce that if $H$ ...
0
votes
0answers
40 views

Request for three foundational papers by Oystein Ore.

Does anybody know how to get or how can I find the following foundational papers on (permutable) groups: O. Ore: Contributions to the theory of groups of finite order (ProjectEuclid Link) O. Ore: ...
2
votes
1answer
35 views

Conjugacy class name of the product in ATLAS

Well, I'm trying to read "ATLAS of Finite Groups". To be more precise, I'm interested in character tables of some Weyl groups. Is it possible to determine the conjugacy class name of the product of ...
1
vote
1answer
41 views

Solvable groups of order 25920

I would like to prove the following statement: Let $G$ be a finite solvable group of order $2^6.3^4.5$. If $O_{5^\prime}(G)\neq1$, then $G$ has an element of order $18$. Also, I would like to know ...
6
votes
1answer
85 views

$|HK|=\frac{|H||K|}{|H∩K| }$ where $H,K$ are finite subgroup of $G$

I know the theorem (and proved) $$ |HK| = \frac{|H| |K|}{|H \cap K|} \text{where $H,K$ are finite subgroups of $G$} $$ NOW I'm wondering about a generalization of this statement. In my 1st attempt: ...
1
vote
3answers
60 views

Maximal subgroups of $S_4$

I have to prove that $$ \operatorname{Frat}(S_4):=\bigcap_{M\stackrel{\max}{\le} S_4}M=1 $$ but I don't know how to compute it since I don't know what are the maximal subgroups in $S_4$. EDIT: ...
2
votes
1answer
40 views

Subgroups of $(\mathbb Z_n,+)$

The problem is to define all subgroups of $(\mathbb Z_n,+), n \in \mathbb N$. My guess is if n is prime number, then there is only trivial subgroups. If n is not prime, then I can factorize it, and ...
4
votes
0answers
75 views

If $H$ is a normal subgroup of group $G$ of odd order and $|H|=5$ . show that $H \subseteq Z(G)$ [duplicate]

If $H$ is a normal subgroup of group $G$ of odd order and $|H|=5$ . show that $H \subseteq Z(G)$ Attempt: If $|H|=5$ and if $H$ is normal, then $H$ must be a normal cyclic subgroup. $\implies H = ...
4
votes
1answer
76 views

Finite Group with $n$-automorphism map

If $G$ is a finite group and $\phi(x) = x^{p+1}$ is an automorphism of $G$ with $order(\phi) |p$ then $G$ is a $p$-group...? If the order of $\phi$ is $1$ then $\phi(x) = x = x^{p+1} = x^px ...
0
votes
0answers
30 views

$C_G(Q_8)=Z(G)$ where $G=GL_2(\mathbb F_3)$

I have to prove that $\operatorname{Aut}(Q_8)\simeq S_4$. In order to do that, I considered $$ \gamma:G\longrightarrow\operatorname{Aut}(Q_8)\\ g\mapsto\gamma_g:q\mapsto q^g $$ which is well defined ...
2
votes
1answer
49 views

Cyclic subgroups of $GL_2(F_q)$

Let $F$ be a finite field with $q = p^f$ elements for $p$ a prime. I know that $G = GL_2(F_q)$ contains a cyclic group of order $q-1$. It is the set of matrices of the form $\begin{pmatrix} x & ...
3
votes
1answer
81 views

Normal subgroups in $GL_2(\mathbb F_3)$

I'm searching for all the normal subgroups of $G:=GL_2(\mathbb F_3)$. Till now I found $N:=SL_2(\mathbb F_3)$ subgroup of index $2$, $Q_8$ subgroup of index $6$ and the center $Z:=Z(G)=\{\pm\mathbb ...