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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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
66 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
70 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
65 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
43 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
60 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
37 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
74 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 ...
1
vote
2answers
49 views

Can we say more about this structure?

Let $N$ be a subgroup of $G$, One can show that, $N$ is normal in $G$ if and only if for all $xy\in N$, $yx\in N$. Above proposion can be proved by elemantary methods. I wonder the following; Let ...
1
vote
2answers
30 views

Example of a subgroup of index two which contains a non square element

If a finite group contains a subgroup H of index two, then every element of the group which is a square belongs to H. Is there a (simple) counterexample showing that not all the elements of H are ...
4
votes
3answers
85 views

Nonabelian group of order $p^3$ for odd prime $p$ and exponent $p$

I am trying to prove the following: Let $P$ be a nonabelian group of order $p^3$ where $p$ is an odd prime and assume that $P$ has exponent $p$. Then $\text{Out}(P) \cong GL_2(p)$. My teacher have ...
2
votes
1answer
44 views

Question about Sylow Theorem and normalizer

I'm dealing with the following problem. Let $G$ be a finite group, $H$ and $K$ Sylow 3- 5- subgroups respectively of $G$. Suppose that 3 divides $|N(K)|$, show that 5 divides $|N(H)|$. I've ...
0
votes
1answer
46 views

Self-dualiy of the subgroup lattice of finite abelain groups

For each abelain finite group $G$ let $\mathcal L(G)$ be the lattice of all subgroups of $G$. For which abelian finite groups $G$, is $(\mathcal L(G),\subseteq)$ order-isomorphic to $(\mathcal ...
2
votes
1answer
32 views

$C_{S_4}(A_4)=1$

I have to show that $C_{S_4}(A_4)$ is trivial. Now we know that $$ C_{S_4}(A_4)=\{x\in S_4\;:\:yx=yx\;\;\forall y\in A_4\}=\bigcap_{y\in A_4}\{x\in S_4\;:\;yx=xy\}\;. $$ Then every element $\neq1$ ...
0
votes
0answers
17 views

$P\in\operatorname{Syl}_3(A_4)\Rightarrow N_{A_4}(P)=P$

How can I prove that $P\in\operatorname{Syl}_3(A_4)$ implies $N_{A_4}(P)=P$? We know that $|A_4|=3\cdot2^2$ and $n_3(A_4)=4$ and the four $3$-Sylows are $P_1=\langle(1,2,3)\rangle$, ...
7
votes
0answers
48 views

The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
0
votes
1answer
20 views

coxeter graph and root system

I want to show that a coxeter graph $\Gamma$ is connected if and only if its root system $\Phi$ is irreducible. So let $\Delta$ be a simple system of $\Phi$, and $\Delta$ is also our simple system. ...
2
votes
1answer
44 views

Order of $ g= \big(\begin{smallmatrix} \ 1 & 1 \\ 1 & 0 \end{smallmatrix}\big)\in GL_2(\mathbb F_3)\;. $

Let\begin{align*} g= \begin{pmatrix} \ 1 & 1 \\ 1 & 0 \end{pmatrix}\in GL_2(\mathbb F_3)\;. \end{align*} Its minimal polynomial is $P_g(X)=X^2-X-1$ which divides $X^8-1$ in $\mathbb F_3[X]$, ...
2
votes
1answer
89 views

How many normal subgroups does a non-abelian group $G$ of order $ 21$

How many normal subgroups does a non-abelian group $G$ of order $21$ have other than the identity subgroup $\{e\}$ and $G$? 1) 0 2) 1 3) 3 4) 7 I think option 1 is incorrect because every group ...
2
votes
1answer
46 views

Proof in Serre/Fulton's rep. theory of Artin-Wedderburn for $\mathbb C[G]$

I have figured out a proof myself for the following theorem, but in both Serre's "Linear Representation of Finite Groups" and Fulton's "Representation Theory" books, I don't understand their comments ...
2
votes
1answer
29 views

Group order notation?

I'm working with Dummit and Foote's Abstract Algebra text, and I encountered some notation that confused me. The theorem I saw it in reads as follows. Suppose $\varphi:G \rightarrow H$ is a ...
2
votes
0answers
47 views

Groups of isometries

I recently read the popular scientific book "Symmetry and the Monster" and it emphasizes groups as the sets of symmetries of geometrical objects. So I was wondering, do all groups appear as symmetry ...
0
votes
1answer
36 views

Subgroups of a group of order 60 with a normal subgroup of order 2 (Sylow)

This is the problem 38 of the chapter 24 in the Gallian's Algebra. Suppose that $G$ is a group of order $60$ and $G$ has a normal subgroup $N$ of order $2$. Show that: $G$ has normal ...
1
vote
2answers
59 views

Easy way to get real irreducible characters (reps) from complex irreducible characters?

For plenty of groups, the real irreducible characters/representations aren't the same as the complex irreducible representations. I really enjoy James Montaldi's summary of real representations, for ...
0
votes
1answer
44 views

A finite group with three nilpotent subgroups [closed]

Suppose $G$ is a finite group with subgroups $H_i\leq G$, for $i\in\{1,2,3\}$, such that $H_i$ is nilpotent for all $i\in\{1,2,3\}$, if $i,j\in\{1,2,3\}$ are distinct then ...
3
votes
1answer
61 views

Normal Abelian Subgroup does not imply Abelian Quotient Group

I'm a bit confused and just need some clarification about what I am missing in this: I have $S_4$ with normal subgroup $N=\lbrace(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\rbrace$. I know that $N$ is ...
7
votes
2answers
70 views

Finite groups of which the centralizer of each element is normal.

Recently I noticed that if $G$ is a finite group and $g \in G$ for which the centralizer $C_G(g)$ is a normal subgroup, all of the elements of the conjugacy class $g^G$ commute with each other, and ...
3
votes
3answers
156 views

Isomorphism in certain groups.

Let $G = \langle \Bbb Z^3_2,+\rangle $, where as $ \Bbb Z_2= \{0,1\} $, $ \Bbb Z^3_2 = \{(x,y,z)\mid x,y,z \in \{0,1\}\} $, and the operation in $ \Bbb Z^3_2 $ is defined by $ (x_1,y_1,z_1) + ...
0
votes
0answers
51 views

Is my solution correct? Finite abelian groups are CLT groups.

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Use Cauchy's Theorem and induction to show that a finite abelian group has a subgroup of order $n$ for each ...
2
votes
2answers
50 views

Commutator subgroup having a complement

Suppose we have a finite group $G$ such that $[G,G]$ has a complement in $G$. Then what are some "good" things we can conclude from this situation? I know that "good" is not well-defined, but I am ...
0
votes
1answer
20 views

Order of rotation

For $\alpha \in \mathbb{R}^n$, let $s_\alpha$ be the reflection in $\alpha$, i.e. $s_\alpha$ sends $\alpha$ to $-\alpha$ and fixes pointwise the hyperplane orthogonal to $\alpha$. Then if ...
1
vote
3answers
27 views

$P\in Syl_p(G)\Rightarrow|G:N_G(P)|=|Syl_p(G)|$

Let $G$ be a finite group s.t. $|G|=p^rm$, where $(p,m)=1$. Let then $P$ be a $p$-Sylow subgroup of $G$, i.e. $P\le G$ with $|P|=p^r$. We want to show that $|G:N_G(P)|$ is the number of $p$-Sylow ...
1
vote
1answer
35 views

Subgroups of $\mathrm{PSL}(2,q)$ of order $2q$

Let $q\equiv 1\pmod 4$. Is it true that $\mathrm{PSL}(2,q)$ has a unique class of conjugate subgroups of order $2q$? I looked at the references that appear in this MO question, the only relevant ...