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
2answers
30 views

A normal subgroup so that any homomorphism into a $p$-group is trivial on it.

Problem Let G be a finite group of order $n$ and $p|n$. Show that there is a unique normal subgroup $N$ satisfying the following property: (1)$G/N$ is a $p$-group (I guess it can be trivial group). ...
2
votes
3answers
39 views

A finite group which has a unique subgroup of order $d$ for each $d\mid n$.

Problem Suppose G is a finite group of order $n$ which has a unique subgroup of order $d$ for each $d\mid n$. Prove that $G$ must be a cyclic group. My idea: I try to prove it by induction. Let ...
1
vote
1answer
26 views

Natural action of $S_n$ on $\{ 1,2,\dots,n \}$

From reading online the "natural" action of $S_n$ on $\{ 1,2,\dots,n \}$ is $(g,x) \mapsto gx$. How is this action transitive? As far as I can see if we take $g$ to fix some element we will not get a ...
-1
votes
1answer
25 views

Center of group of a dihedral group

An example from my text ask to verify this: $$Z(D_{n})= \begin{cases} {R_{0},{R_{180}}} & \text{when n is even}\\ {R_{0}} & \text{when n is odd}\end{cases}.$$ How should I begin to verify ...
1
vote
1answer
26 views

How can you tell if a normal subgroup induces a semidirect product?

Suppose I have some (finite) group $G$ and a normal subgroup $N$. I know there's no full characterization of whether $G \cong N \rtimes G/N$, but are there well-known tests I can use to answer the ...
3
votes
2answers
34 views

Fiber product of groups

Let $f: G \longrightarrow K$, $g: H \longrightarrow K$ be group homomorphisms. For the fiber product $G \times_K H$ to exist, is it necessary to require that $f$ and $g$ be surjective? I can't see ...
0
votes
1answer
46 views

A singleton set $\{g\}$ can be regarded as a unary relation in $G$. Why?

Theorem 1.1. A relation $R \subseteq M^n$ is definable if and only if every automorphism of every elementary extension of $M$ preserves $R$. For a proof, the reader can see [4]. Suppose we ...
0
votes
2answers
68 views

Abelian group which is not one of these

Im struggling to find a finite abelian (commutative , associative) group $(G,\circ)$ with some specific conditions: $a\circ b$ isn't naive addition $a+b$ for $a,b\in G$ $G$ is a subset of ...
2
votes
3answers
99 views

Verifying if a multiplication table is from a group

I'm asked to verify which of these multiplication tables form a group. I'm having problems to see which of the axioms for a group are violated in each table. In (a), I couldn't find an element $e$ ...
5
votes
0answers
82 views

Aut(G) is abelian

I've heard of this (open?) problem: Classify groups G such that Aut(G) is abelian. What I discovered: Any characteristic abelian subgroup is cyclic. Center is cyclic. Commutators are cyclic. ...
32
votes
3answers
1k views

If I know the order of every element in a group, do I know the group?

Suppose $G$ is a finite group and I know for every $k \leq |G|$ that exactly $n_k$ elements in $G$ have order $k$. Do I know what the group is? Is there a counterexample where two groups $G$ and $H$ ...
-1
votes
1answer
18 views

Classifying the central product HK of two cyclic groups [on hold]

Let group $H$ be a direct product of cyclic groups $C_1$ and $C_2$ of order $p$ and $p^2$ respectively. Let $D=\{x\in H\mid \text{ord}(x)\leq p \}$. D is generated by $C_1$ and subgroup $E$ of $C_2$ ...
0
votes
0answers
15 views

Finding efficiently finite groups whose set of commutators is not a subgroup

The set of commutators of a group might not be a subgroup of the group. I give here such an example from P.J. Cassidy. It is an infinite group. It is possible to derive from this example one of a ...
0
votes
0answers
38 views

I want to know if the below sentence is true and why?

I want to know if the below sentence is true and why? Let $G$ be an insoluble finite group then there exists $\pi\subset\pi(G)$ such that if $K=O_{\pi}(G)$ and $\bar{G}=G/K$ it follows that ...
2
votes
1answer
23 views

The generating set of Cayley graphs over $Z_n$

Say we have a undirected and connected Cayley graph over $\mathbb{Z}_n$, with generating set $S=\{\pm x_0,\pm x_1,\ldots,\pm x_k\}$. Is it true that we can assume without the loss of generality that ...
-3
votes
0answers
18 views

If $ G $ is finite group and $ K $ is nilpotent subgroup of $ G $. [on hold]

If $ G $ is finite group and $ K $ is nilpotent subgroup of $ G $. is there theorem that said can let $ K = K_{\pi^{\prime}}K_{\pi^{\prime}}$ that $K_{\pi}$ is $\pi$-Hall subgroup of $ K $ and $ ...
1
vote
2answers
32 views

How to prove that $a^{|G|}=e$ if a $\in G $

How to prove that; $a^{|G|}=e$ if a $\in G $ if $G$ is a finite group and $e$ is its identity. I think this could be done through pigeonhole principle but I don't want to use the Lagrange ...
-1
votes
0answers
18 views

does dicyclic group of order $2^{\alpha}m$, $(2,m)=1$ have normal subgroup of order $m$?

consider dicyclic group with presentation $$Dic_n=< a,b |a^{2n}=1,a^n=b^2,ba=a^{-1}b>$$ the order of this group is $4n$,suppose that $|Dic_n|=2^{\alpha}m$ where $(2,m)=1$ my question is about ...
1
vote
1answer
53 views

all of subgroups of group

Is the way to gust that in finite group how many subgroup of same order?I ask this question because when draw the lattice diagram of subgroups of group sure that all of them describe. Thanks for hint
1
vote
1answer
36 views

Group of order 10 has an element of order 5, without using Cauchy's or Sylow's theorems

This is almost a duplicate of the following questions (but, read further): Group of order $63$ has an element of order $3$, without using Cauchy's or Sylow's theorems Show any group of order ...
-2
votes
1answer
57 views

Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. [duplicate]

Homework question: Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian. Can you give me some hints ...
-1
votes
1answer
32 views

If $|G| = pqr$ for $p<q<r$ primes and all the Sylow groups are normal; is $G$ abelian? [closed]

Let $G$ be a group with $|G| = pqr$ for distinct primes $p<q<r$. If every Sylow subgroup of $G$ is normal, then is $G$ Abelian? Thank you in advance.
1
vote
1answer
17 views

Proving the following subgroup(verification of logic)

So I was reading the following theorem from dummit that is If $|H| = n <\infty$ then for each positive integer dividing n there is a unique subgroup of $H$ of order $a$. This subgroup is the ...
3
votes
1answer
63 views

Is there a group homomorphism $f:G\longrightarrow G$ for which $G/\operatorname{Im} f \not\cong\operatorname{Ker} f$? $G$ is finite

Can you find a counterexample to the claim that for all group homomorphisms $f:G \longrightarrow G$, $G/\operatorname{Im} f \cong \operatorname{Ker} f$. Let $G = \mathbb{Z}$; $f(n) = 2n$ is a ...
2
votes
0answers
62 views

Need help in understanding the question. Elemntary number theory

I have this question in my home assignment. I contains two parts and I don't quite understand what is the difference between them.The question is: Let $n > 2$ be an integer such that ...
1
vote
1answer
23 views

question about isomorphism involving Dihedral group.

suppose $D_{n}$ is dihedral group with order $2n$, do we have this Isomorphism below? $$D_{2k+1} \times \mathbb{Z}_2 \simeq D_{4k+2}$$ I think it is wrong, I couldn't find any mapping, but I ...
4
votes
1answer
68 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...
2
votes
1answer
27 views

Direct decompositions and quotients of abelian groups

Let $G = \langle a \rangle_{27} \oplus \langle b \rangle_{81}$. Find a direct decomposition $G = \langle 10a + 60b \rangle \oplus ?$. Find the elementary divisors of $G/ \langle 3a + 18b \rangle$. ...
0
votes
1answer
77 views

Herstein Problem No.7 Page 102

let $G$ be a group of order $30$ .How many non-isomorphic groups of order $30 $ are there? Before doing this I have shown that every Sylow 3 and Sylow 5 subgroup is normal in G and G has a normal ...
0
votes
0answers
30 views

Harmonic Analysis of Finite Groups

If I understand correctly, the basic goal of harmonic analysis on finite groups is to find isotypical subspaces of a given set. Why is it important to do so? What are the advantages of decomposing a ...
2
votes
0answers
57 views

Showing the existence of $O^{\pi}(G)$

Assume G is a finite group. I am trying to show the existence of $O^{\pi}(G)$, the unique normal subgroup of G minimal such that $G/O^{\pi}(G)$ is a $\pi$-group, i.e. a group whose order is divisible ...
0
votes
1answer
38 views

Sum of Inverses of the elements in $\mathbb Z_p^*$

If $p $ is an odd prime and if $1+\frac{1}{2}+\cdots +\frac{1}{p-1}=\frac{a}{b}$ where $a,b $ are integers prove that $p|a$. If $p>3\implies p^2|a$ My Try: Can the problem be interpreted as a ...
0
votes
1answer
31 views

information about semi-dihedral groups.

my question is about the elements and the generalized format of caylay table of groups called semi-dihedral groups which have the presentation $$ \langle a,b\mid a^{4m}=b^2=1,ab=ba^{2m-1}\rangle $$ ...
1
vote
0answers
22 views

Class preserving Autpmorphism

This time I am having interest in theory of Class preserving Automorphism and central preserving Automorphism and some topics related to Automorphism of groups. So Could you please tell me some ...
0
votes
0answers
18 views

Looking for an element in a finite abelian group $G$ which is the square of more than two elements of orders $m\geq 3$ in $G$?

Please do we have an element in a finite abelian group $G$ which is the square of more than two elements of orders $m\geq 3$ in $G$? For example in $Q_8$ which is non-abelian, all the elements in ...
2
votes
1answer
65 views

Is there a cyclic subgroup C of S8 such that the interval lattice [C,S8] is distributive?

I've checked by hand that for any $n \le 7$, there is a cyclic subgroup $C$ of $S_n$ such that the intermediate subgroups lattice $\mathcal{L}(C \subset S_n)$ is distributive. Question: Is it the ...
0
votes
1answer
38 views

Consequences of $G$ having exactly $3$ irreducible characters

Suppose a finite group $G$ has exactly irreducible characters $\chi_1 = \mathbb{I}, \chi_2,\chi_3$. i) Show that G is soluble and deduce it has a non-trivial one-dimensional character $\chi_2$. ii) ...
1
vote
2answers
38 views

A subgroup of $\textrm{GL}(3,q)$ of order $q^2(q-1)$

Let $q$ be a prime power. Consider the multiplicative group $\textrm{GL}(3,q)$ of the $3 \times 3$ matrixes with coefficients in $\mathbb{F}_q$ which are invertible. The matrixes $$ M_{a,b,c} = \left( ...
1
vote
1answer
45 views

$S_4$ is not supersolvable? Why am I wrong?

I read that $S_4$ is an example of a solvable group who is not a supersolvable group. In order to prove it is solvable, we see that: $\{e\}<\{(1),(12)(34)\}<K<A_4<S_4$ where $K$ is the ...
1
vote
1answer
37 views

Product of a character with an irreducible character a non-negative integer

Why is the inner product of a character with an irreducible character a non-negative integer? I can see that by properties of the inner product it will be non-negative but I cannot see why it would ...
8
votes
2answers
182 views

Maximum number of Sylow subgroups

I've been studying Sylow-$p$ subgroups, and I've come across this problem. Let $G$ be a finite group. Show that the number of Sylow subgroups of $G$ is at most $\frac{2}{3}|G|$ . ($|G|$ is the ...
1
vote
1answer
19 views

Group Determinant Independent of Labeling of Elements

Let $G$ be a finite group with elements $g_1, g_2, \ldots, g_n$. We define the group matrix by $$X_G = [x_{g_ig_j^{-1}}].$$ We then can define the group determinant as $$\det X_G = \Theta_G.$$ ...
4
votes
1answer
40 views

For a group ring, finding if a subset is an ideal. [closed]

For the ring $R=SG$, the group ring of a finite group G over an integral domain S, and a subset $I=(g-1|g \in G)$, is this subset an ideal? Is it prime? How about maximal?
0
votes
0answers
38 views

Finite groups generated by m involutions (where $\;m\ge3\;$)

We know from basic group theory that if a finite group $G$ is generated by two involutions, then $G$ is dihedral. Please is there any version of this result for finite groups generated by $m$ ...
0
votes
3answers
40 views

'Inverse' property of a group and the special case that makes a group an Abelian group

One of the property for which a set must have in order to be a group is to possesses the 'inverse' property. What this says is that for each element $a$ in $G$, there is an element $b$ in $G$ with ...
0
votes
1answer
50 views

Can a group be non-empty by definition of 'group'?

By the definition of a group, "A group is a set combined with a binary operation". By this definition, would a non-empty set constitute as being a group? By virtue of the definition of what a group ...
3
votes
1answer
81 views

Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
0
votes
0answers
44 views

A finite group which is isomorphic to $PSL(2,p)$

Let $p$, $q$ be a odd prime numbers such that $p=2q+1$. Let G be a finite group of order $\frac{p(p^2-1)}{2}$. If all Sylow 2-, p-, and q-subgroups are not normal, G is isomorphic to $PSL(2,p)$. The ...
2
votes
1answer
42 views

Is O(1) a Lie Group?

In reading Georgi (Lie algebra in particle physics) I reaf at page 43 the following definition of Lie Gruoup: "a lie gruoup is a group whose elements depend smoothly on a set of continuous ...
1
vote
2answers
45 views

If $K$ is subgroup of $H$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $G$

Question 1. Let $K\subseteq H\subseteq G$ and if $K$ is subgroup of $H$ and $H$ is a subgroup of $G$, then $K$ is a subgroup of $G$. 2. Let $K\subseteq H\subseteq G$ and if $K$ is a subgroup of $G$ ...