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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-2
votes
1answer
22 views

Order of Automorphis group [on hold]

Let $ G $ is a finite solvable group and $ N $ be a normal minimal subgroup of $ G $ that $ G = MN $ for maximal subgroup $ M $ of $ G $, which $ M \cap N = 1 $. Let $ \vert N \vert = 4 $. Then $ ...
1
vote
1answer
22 views

Endomorphism structure of the Klein four-group

I am reading the Algebra by Grillet, this is ex 17(-18), pag. 22. I understand that $V$ can be viewed as a two dimensional vector space over $\mathbb{F}_2$. Noticed this, it is easy to see that ...
-1
votes
1answer
28 views

Direct product of quotient groups

Let $ G $ is a finite solvable group, Suppose $ H $ and $ N $ are minimal normal subgroups of $ G $. Then $ G/N \times G/H \cong G/N\cap H $ ?
0
votes
1answer
20 views

$ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $, $ G/N_{1}\cap N_{2} \cong G $?

Let $ G $ is a finite group and $ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $ that $ N_{1} \neq N_{2} $. Suppose $ G/N_{1} $ and $ G/N_{2} $ are supersolvable. Then $ G $ is ...
0
votes
2answers
38 views

How can I prove that $g\cdot H \cdot g^{-1}$ is also finite and has the same number of elements that $H$?

Suppose $G$ a group and $H$ is a finite subgroup of $G$ also $\forall g \in G$ the set $g\cdot H \cdot g^{-1}=\{ g\cdot h \cdot g^{-1} : h\in H\}$, is a subgroup of $G$. Prove that $g\cdot H ...
2
votes
1answer
65 views

How to prove that G is a cyclic Group? [duplicate]

Suppose $G$ is a finite Abelian group and, $\forall\ n\in \mathbb{N} $, there exist at most $n$ elements in $G$ which satisfy $x^n=1$. Prove $G$ is cyclic. Thanks for your help.
-1
votes
0answers
21 views

$ G $ is soluble, then every properties of $ G $ is inherited by $ G/N $ ? [on hold]

Let $ G $ be a finite group and $ G $ is soluble. Suppose $ N $ be a normal minimal subgroup of $ G $. Then every properties of $ G $ is inherited by $ G/N $ ?
7
votes
0answers
33 views

$|G:H|=p^n$ means $O_p(H)\leq O_p(G)$?

Let $H\leq G$ (finite group) and $|G:H|=p^n$, ($p$ is a prime number) prove that: $$O_p(H)\leq O_p(G)$$ note: $O_p(G)$ defined as the intersection of all Sylow-$p$ groups in $G$ I try to prove ...
4
votes
2answers
24 views

$G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; $G$ is not simple

Let $G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; then how to show that $G$ is not simple ? I have proceeded by Cayley's theorem , $\ker f$ is ...
2
votes
0answers
16 views

Frattini subgroups and nilpotent groups: bijection? [duplicate]

I have been proved that the Frattini subgroup of a finite group is nilpotent. Now I am wondering: is the converse true? I mean, if $G$ is a finite nilpotent group, is there always a finite group ...
1
vote
0answers
39 views

I need the paper “ J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365” [on hold]

I need this paper " J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365", But i not found. Who can help me to find this paper ?
1
vote
1answer
16 views

Definition of $ p $-supersoluble group.

I was searching for definition of $ p $-supersoluble group but not find definition. Please help me.
1
vote
1answer
15 views

$ K/N $ be a normal subgroup of $ (G/N)^{\prime} $. Now why $ K \cap G^{\prime} $ is a maximal subgroup of $ G^{\prime} $? [on hold]

Let $ G $ be a finite group and $ N $ is a normal subgroup of $ G $. Suppose $ K/N $ be a maximal subgroup of $ (G/N)^{\prime} $. Now why $ K \cap G^{\prime} $ is a maximal subgroup of $ G^{\prime} $? ...
10
votes
3answers
58 views

Smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi direct product of cyclic groups.

What is the smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi-direct product of cyclic groups? So finite abelian groups are ruled ...
1
vote
1answer
17 views

Let $ G $ satisfying the maximal permutizer condition, then $ G/N $ satisfying the maximal permutizer condition ?

Let $ H $ be a proper subgroup of finite group $ G $. Then permutizer $ H $ in $ G $ is defined by $ P_{G}(H) = \langle y \in G \vert \langle y \rangle H = H \langle y \rangle \rangle $. A group $ G $ ...
1
vote
0answers
22 views

problem in understanding some part of proof of theorem

In proof of theorem: for every finite non abelian p-group $G$ of class 2 there is utter automorphism of calss 2 fixing $\Phi(G)$ by counterexample,If we know 1. $G'=\langle[a,b]\rangle$ 2. $H=\langle ...
1
vote
2answers
37 views

Suppose that $\cdot$ is associative and has an identity element. Show that an element $g \in G$ has at most one inverse

Let $(G,\cdot)$ be a group with $e$ its neutral element. For an element $g\in G$, there exists one inverse element in $G$, denoted by $g^{−1}$, such that $g\cdot g^{−1}=g^{−1}\cdot g=e$. Can this be ...
-1
votes
0answers
38 views

Is there any survey paper or book for “Word Problem”? [on hold]

I found many many papers on this topic and reading them takes long time. I want to know for what kind of groups the word problem can be solved, and any other good general result. Is there any survey ...
0
votes
1answer
20 views

points which are fixed points of a finite group action

consider an open set $\tilde{U}\subset\mathbb{R}^n$ and a finite Lie-group $G$, which acts smoothly on $\tilde{U}$, i.e. we have a smooth map $G\times \tilde{U}\rightarrow\tilde{U}$. Suppose further, ...
3
votes
2answers
32 views

If $G$ is a Finite Group such that $H\le K$ or $K\le H$ for all Subgroups $H,K$ of $G$, then $G$ is Cyclic and of order $p^n$ for some Prime $p$.

Since each subgroup $K$ is contained in some other subgroup $H$, we can list the subgroups of $G$ in ascending order $$\lbrace 1 \rbrace < G_1< G_2 < G_3\cdots< G_{k-1}<G$$ By ...
3
votes
1answer
62 views

How can I identify a group given its multiplication table?

Given the group generated by the matrices $$\begin{pmatrix} -1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{pmatrix},~\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 ...
2
votes
2answers
52 views

Normalizer of a Sylow 2-subgroups of dihedral groups

I can't solve the following exercise which is the last exercise in page 146 of Dummi & Foote's Abstract Algebra: Let $2n=2^ak$ where $k$ is odd. Prove that the number of Sylow 2-subgroups of ...
1
vote
0answers
26 views

Is there a good way to break down the order of the centraliser in a symmetric group?

I recently rediscovered the rather nice formula for the order of the centraliser of a permutation in the symmetric group and its realtionship with conjugacy classes. I wondered whether we could say ...
0
votes
0answers
26 views

finding mistake in $\alpha^2(b)=b$

Let $X=\langle a,b|a^{2^m}=b^2=1,[a,b]=a^2\rangle , m\ge3$ and $\alpha \in Aut(X)$ (automorphis group of X) If \begin{cases} \alpha(a)=a^{2^{m-2}+1}b\\ \alpha(b)=a^{2^{m-1}}b \end{cases} and ...
-1
votes
1answer
26 views

Abelian-by-(finite abelian) [on hold]

hope you all doing fine. I have a question. Is it true that a abelian-by-(finite abelian) group is also (finite abelian)-by-abelian? Thanks.
0
votes
0answers
31 views

Show that the homomorph image of an abelian group is abelian

Since $G$ is abelian, we have that: $$ab = ba \implies \phi(ab) = \phi(ba) \implies \phi(a)\phi(b) = \phi(b)\phi(a)$$ Am I rigth?
0
votes
0answers
21 views

Restriction of some representation of $D_{12}$

Let $D_{12}$ be the dihedral group, $\langle x,y: x^2 = y^6 =1 , xy = y^{-1}x \rangle$ and $K = \langle xy \rangle \times \langle y^3 \rangle $ be a subgroup.Let $\xi^2$ denote a representation of ...
0
votes
2answers
63 views

If $\phi$ is an isomorphism, $\phi(g)^n = 1 \iff g^n = 1$. Doesn't this hold for homomorphisms too?

I need to prove that for an isomorphism $\phi$, the following is true: $$\phi(g)^n = 1 \iff g^n = 1.$$ We know that $$g^n = 1 \implies g\cdot g \cdots g = 1\implies \phi(g\cdot g \cdots g) = ...
9
votes
1answer
55 views

An element of $GL_n(\mathbb F_p)$ cannot have order $p^2$ if $n < p$

I'm preparing for my graduate program's entrance exams, and I came across this problem when studying. Our study group came up with a solution, but I wanted to ask if it was actually correct, since ...
2
votes
2answers
47 views

Determining the minimum dimension required for embedding a finite group

Consider the groups $S_3$ and $S_4$ which are the symmetric groups on 3 and 4 elements respectively. We note that $S_3$ can be realized geometrically as the set of all rotations and reflections of a ...
1
vote
1answer
18 views

$S$ is a non empty set and there are $a$ and $b$ for $c$ and $d$ such that $a\cdot c = d$ and $c\cdot b = d$, prove it is a group

An associative operation $\cdot \ $was defined in $S$ such that $\cdot \ $is associative. Also, for all the pairs $c$ and $d$, there are elements $a$ and $b$ such that: $$a\cdot c = d, \ \ \ \ c\cdot ...
0
votes
1answer
26 views

Verifying if these Cayley tables are from groups

For the first table I noticed that $ab = c \implies abb = cb \implies a = cb$ but in the table, $cb = d$, so this can't be a group For the second table, we have: $ab = c \implies (aa)b = ac ...
-4
votes
1answer
33 views

Two sub-groups of order 3 and 5, prove that the group of order 15 is cylic. [closed]

So all I have is that G is a group of order 15, and there are 2 unique sub-groups, which order is 3 and 5 (I mean there only one sub-group of each kind) and I need to prove that G is cyclic. Dont see ...
1
vote
0answers
39 views

$G$ of finite order $2p$ ($p$ is prime). Prove that $G$ abelian. [duplicate]

I have a group $G$ of order $2p$, where $p>2$ and prime. The additional thing that I also know, that $\exists a\in Z(G)\mid O(a)=2$. I need to prove that G is abelian. But first, before that, ...
0
votes
1answer
38 views

About a proof regarding a property of groups of order $pq$ where $p$ and $q$ are primes

I'm studying right now Automorphisms in Dummit & Foote's Abstract Algebra (Section 4.4). In pages 135-136, the following example is given: and here's Proposition 16 muntionned in the ...
1
vote
1answer
23 views

Index of every maximal subgroup is prime number

Suppose that finite group $G\neq 1$ and $|G : M| ∈ \mathbb P$ for every maximal subgroup $M$ of $G$. Then prove: $G$ contains a normal maximal subgroup. (we all know that a maximal subgroup is normal ...
0
votes
0answers
69 views

$(\mathbb Z /n\mathbb Z)^*$ is cyclic for all $n$ implies every finite abelian group is cyclic. [closed]

Show that, if $(\mathbb Z / n\mathbb Z)^*$ is cyclic for all $n \ge 1$ then every finite abelian group is cyclic. More precisely, prove that the following are equivalent: 1) $(\mathbb Z / n\mathbb ...
1
vote
1answer
20 views

central series of $\frac{G}{Z_2(G)}$

Let $G$ be finite p-group I am trying to make central series for $\frac{G}{Z_2(G)}$ and more inportant what is nilpotency class of $\frac{G}{Z_2(G)}$ since ...
2
votes
1answer
61 views

Set of all inner automorphisms is a normal subgroup

In order to prove this, I first proved that the set of all automorphisms from a group $G$ to $G$ form a group under composition: The identity homorphism is an automorphism because sends $x$ from $G$ ...
-4
votes
0answers
31 views

assume $ M/N $ be a chief factor of $ G $. Why $ M/N $ has prime order or order $ 4 $?

Let $ G $ is a soluble group and $ \Phi(G) $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $. Let $ M $ be the smallest normal ...
1
vote
0answers
21 views

minimal normal subgroup of a chief factor of soluble group $ G $ is a minimal normal normal subgroup of $ G $?

Let $ G $ is a soluble group and $ \Phi(G) $. Let $ K/L $ be a chief factor of $ G $. Let $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent and $ M \neq 1 $. assume $ M/N $ ...
0
votes
0answers
20 views

Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal subgroup of $ K $ such that $ K/M $ is nilpotent

Let $ G $ is soluble group with $ \Phi(G) = 1 $ and assume that each minimal normal subgroup has prime order or order $ 4 $. Let $ K/L $ be a chief factor of $ G $ and $ M $ be the smallest normal ...
-2
votes
0answers
17 views

IS $ F(G) $ a direct product of some minimal normal subgroups of G?

Let $ G $ is a finite group and $ F(G) $ is the fitting subgroup of $ G $. IS $ F(G) $ a direct product of some minimal normal subgroups of G? Why ? $ F(G) $ is the largest nilpotent normal subgroup ...
1
vote
1answer
31 views

Let $ H/N $ is nilpotent quotient subgroup. Then is $ N $ characteristic in $ H $? If no, then what condition need?

Let $ G $ is finite solvable group and $ H $ is normal subgroup. Let $ H/N $ is nilpotent quotient subgroup. Then is $ N $ characteristic in $ H $? If no, then what condition need?
3
votes
1answer
52 views

Action of ${\rm Aut}(G)$ on $G$

Let $G$ be a finite group and consider the natural action of ${\rm Aut}(G)$ on $G$ and let there are two orbits under this action. How could we show that $G$ is an (elementary) abelian group? Is ...
2
votes
1answer
25 views

Problems with P. Hall theorem proof (The problem involves the use of Frattini's argument)

Theorem Let G be a solvable group of order $ab$, where $(a,b)=1$. Then $G$ contains at least one subgroup of order $a$, and any two such are conjugate. Details The proof the book presents involves ...
2
votes
1answer
21 views

How many subgroups are there in an elementary-$p$ group

$C_p\times C_p$ has $1+1+(p-1)=p+1$ subgroups of order $p$ (all of which $\cong C_p$), so it has $1+(1+p)+1=3+p$ subgroups totally. But how many subgroups in $C_p\times C_p \times C_p\times C_p$? ...
4
votes
0answers
57 views

Is the monster group a characteristic quotient of $F_2$?

Let $F_2$ be the free group on two generators, and $M$ the monster group. It's known that every finite simple group is 2-generated, so let $F_2\rightarrow M$ be a surjection with kernel $N$. Let ...
0
votes
1answer
30 views

A group of order 2p (p prime) and other conditions - prove abelian.

I have G where $|G|=2p$ ; p is prime. $\exists a\in Z\left(G\right);\:a^2=e$. I need to prove that G is abelian. Now, let's translate it into math. To prove that G is abelian, is in other words ti ...
1
vote
1answer
40 views

p-element centralizing a Sylow p-subgroup

Let $G$ be a finite group, $P$ a Sylow $p$-subgroup for a prime $p$ and $g$ a $p$-element with $gxg^{-1} = x$ for all $x \in P$. Then $g \in Z(P)$. Is this true? How can i prove that $g \in P$ ...