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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13 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 ...
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1answer
40 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
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1answer
29 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 ...
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1answer
53 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 ...
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1answer
21 views

If G = pqr where p,q,r are prime, and all the Sylow groups are normal, then is G is abelian? [on hold]

Let $G$ be a group with $|G| = pqr$ where $p,q$ and $r$ are distinct primes with $p<q<r$. If all the Sylow subgroups are normal, then is $G$ abelian? Thank you in advance,
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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
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1answer
55 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
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0answers
47 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 ...
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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 ...
3
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1answer
59 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
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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
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1answer
76 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 ...
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0answers
27 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
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0answers
55 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
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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
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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 $$ \langle a,b\mid a^{4m}=b^2=1,ab=ba^{2m-1}\rangle $$ ...
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0answers
21 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
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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
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1answer
64 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
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1answer
35 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) ...
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2answers
37 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( ...
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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 ...
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1answer
36 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 ...
4
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1answer
110 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 ...
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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
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1answer
38 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?
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0answers
37 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 $m$ ...
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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 ...
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1answer
49 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
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1answer
80 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 ...
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0answers
43 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
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1answer
40 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 ...
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2answers
44 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 $G$ ...
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1answer
28 views

Existence of non-abelian split metacyclic extensions.

Is there any necessary conditions that must be held in order to guarantee the existence of a non-abelian split metacyclic extension? i.e. for which $m,n\in\mathbb{Z}$ there exist a non abelian split ...
2
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2answers
58 views

Direct product and Sylow subgroups

Let $G$ be a finite group that is equal to inner direct product of its subgroup $P$ and $Q$, where $P$ is a Sylow $p$-subgroup and $Q$ is a Sylow $q$-subgroup of $G$. If $L \le G$, prove that $L$ is ...
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1answer
43 views

Difficulty with a lemma needed to prove $A_n$ is a simple group for $n>4$

The theorem is: For $n \geq 5$, every normal subgroup $N$ of $A_n$ contains a $3$- cycle. The proof starts like this: Let $\sigma$ be an arbitrary element in a normal subgroup $N$. There are ...
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1answer
45 views

On a classification of all the characteristic subgroups of a finite abelian $p$-group.

For any finite abelian group $G$, any $n\mid\exp G$ and any $m\mid\frac{\exp G}{n}$, let $nG[m]:=\{g\in nG\mid mg=0\}$. I wonder if every characteristic subgroup of a finite abelian $p$-group $P$ is ...
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1answer
43 views

Is a finite group which is generated by two characteristic abelian subgroup always abelian?

Let $G$ be a finite group. If there exist two characteristic subgroups $H,K$ of $G$ such that $H$ and $K$ are abelian and generate the whole group $G$. Then can we conclude that $G$ is abelian? All ...
4
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2answers
73 views

Group of order $135$ abelian and not cyclic

I am trying to solve the following: Let $G$ be a group of order $135$. Show that if $G$ has more than one normal subgroup of order $3$, then $G$ is abelian and non-cyclic. What I could do was: ...
0
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1answer
56 views

Finite abelian groups of order 100

(a) What are the finite abelian groups of order 100 up to isomorphism? (b) Say $G$ is a finite abelian group of order 100 which contains an element of order 20 and no element with larger order. Then ...
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2answers
38 views

Group of $2n$ elements, $n$ odd, is not simple

Problem Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. ...
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1answer
47 views

Basic doubt about cosets

Studying some basic group theory I had the following doubt: For $H$ subgroup of a finite group $G$ (doesn't matter invariance of $H$), is it true that $$|G/H|=|\{aHa^{-1}:a \in G\}| \space ...
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2answers
22 views

If $k|n, k \geq 2$, then $D_{n}$ has a subgroup isomorphic to $D_{k}$

Restatement of question: If $k|n, k \geq 2$, then the group $D_{n}$ has a subgroup isomorphic to the group $D_{k}$. My attempt at proving the result stated: Let us say that $D_{n}= \{1, \sigma, ...
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1answer
64 views

Group theory argument

I'm reading Group Theory in Physics by Wu-Ki Tung and on page 69 in the proof of Theorem 5.3 he makes a group theory statement that I don't get. Let me try give some notation and explanation ...
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0answers
28 views

Representation theory and point groups

Hello everyone :) I have a doubt. I have the point group $C_{3v}$, which is the group $$C_{3v}= \lbrace e, C_{3}, C_{3}^{2}, \sigma_{v_{1}}, \sigma_{v_{2}}, \sigma_{v_{3}} \rbrace$$ $C_{3}$ and ...
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0answers
19 views

Given a space group, how to determine which layer groups are its subgroups?

I am studying various crystals and the two-dimensional materials that could be potentially obtained by cleaving them (isolating a region bounded by two parallel planes). In elucidating the properties ...
2
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0answers
26 views

How to find the power of generator defined over finite field , $\mathbb F_{2 ^m}$?

List item Actually ,I am trying to execute the algorithm to find the power of generator of field(group) as shown in table of attached file but when k=7 and onward ,I could not understand what is ...
0
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1answer
61 views

How to find |G| [closed]

Let $G$ be a finite group of odd order $n<60$. Let $H$ be a subgroup of $G$, the order of $H$ is $13$, $|H|=13$. Knowing that there is element $a≠e$ so that $a^6=e$. Find $|G|$ (the order ...
0
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3answers
48 views

Group Theory: How do I determine if an element generates a group?

I was asked if the group $(Z_{17} \setminus \{0\}, \cdot)$ is generated by the element $2$. I understand the concept of generating sub-groups in group theory. If I was given a group $G$ and asked to ...
2
votes
4answers
530 views

Why do Z/7 have no cubic root of 2?

I was reading a textbook and came across the following line: Now we prove there is no cube root of 2 in $Z/7$. By noting that $(Z/7)^\times$ is cyclic of order 6, it will have only two third ...