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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2answers
26 views

$G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
1
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0answers
31 views

Automorphism group of finite $p$-groups

firstly I apologize for my naive knowledge in group theory. Let $G$ be a finite $p$-group with automorphism group ${\rm Aut}(G)$ and let $N$ be a maximal subgroup of $G$. Let $g$ be ...
6
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1answer
416 views

Proving that a group of order $pqr$ (with conditions on those primes) is abelian.

I'm doing this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Also suppose that $p \not\equiv 1$ (mod $r$), $p \not\equiv ...
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2answers
33 views

Questions about Sylow $p$-groups

Question 1 Is it true that there is only one Sylow $p$-group in an abelian group? Question 2 If there is only one Sylow $p$-group, then it is normal, true? Because if $H\leq G$ is a Sylow $p$-group ...
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2answers
25 views

Trying to find an isomorphism

I'm trying to find an isomorphism from Z31 to itself that maps 13 to 28.I used the extended euclid algorithm to find that 12 is the inverse of 13, but that's as far as I got. What do I do next? How ...
2
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5answers
508 views

A criterion for a group to be abelian [duplicate]

I noted a discussion on groups being abelian under a certain restriction on powers of elements, e.g. http://tiny.cc/chs45. Maybe this result (probably not too well-known) concludes it all. Let $m$ ...
1
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1answer
36 views

Sylow $p$-groups in $GL_3(\Bbb F_p)$

Fix a prime $p$. How many Sylow $p$-groups are there in $GL_3(\Bbb F_p)$? Let $s_p$ be the number of Sylow $p$-groups in $GL_3(\Bbb F_p)$. By the Sylow theorems, $s_p\equiv 1\mod p$ and if $p^e$ ...
0
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1answer
41 views

Sigma and Pi Chemistry/Math Permutation Question

Does anyone know if sigma and pi bonds in chemistry have any mathematical definition? The reason I'm asking this is because I've recently read a lot about cycles and permutations, and they seem to ...
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2answers
49 views

Can the order of $x \in U_{31}$ be $10$?

Does there exist an element $x\in U_{31}$ such that the order of $x$ is $10$? Here $U_{31}$ is the group of units of $\mathbb{Z}/31\mathbb{Z}$.
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3answers
33 views

Unique order $d$ subgroup of $\Bbb Z/n\Bbb Z$ if $d\mid n$ [duplicate]

Let $G=\Bbb Z/n \Bbb Z$ and assume $d\mid n$. There exists a unique subgroup of $G$ of order $d$. The mod $n$ reduction of $n/d\in \Bbb Z$ generates a $d$ element subgroup of $\Bbb Z/n\Bbb Z$, ...
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0answers
39 views

Motivation of indices of all subgroups of symmetric group $S_n$

In 1858 a prize question of the Acad´emie des Sciences was - What are the indices of all subgroups of symmetric group $S_n$ acting on $n$ objects ? Three submissions was submitted in 1860, no ...
2
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1answer
26 views

Cohomology ring of $G$ based on its Sylow.

I have a bunch of notes made from a professor about cohomology that states that If $S$ is a $p$-Sylow subgroup of $G$ ($\vert G \vert <\infty$), then $$H^{\ast}(G,\mathbb{F}_p)\leq H^{\ast}(...
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1answer
53 views

Need help in understanding the example from Dummit & Foote text

This is an example from Dummit & Foote text. Let $D_{2n}=<r,s:r^n=s^2=1,s^{-1}rs=r^{-1}>$.Since [$r,s$]$=$$r^{-2}$,we have $\langle r^{-2}\rangle=\langle r^2\rangle \le D'_{2n}$....
7
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1answer
549 views

groups with same number of elements of each order

When two groups which have the same number of elements of each order are isomorphic? Can we characterize them? I already know the abelian $Z_{p^2}\times Z_p$ and non-abelian $Z_{p^2}\rtimes Z_p$ have ...
0
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1answer
35 views
+250

Upper-bounding a sum over non-identity permutations

EDIT: Question 1 has been settled (below). The bounty is for question 2. Let $n\geq 3$ and consider the following function $f:S_n\backslash\{e\}\rightarrow \mathbb{R}$ $$f(\sigma)=\sum_{i=1}^n\frac{...
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2answers
61 views

Books on Group Theory.

I am looking for a book/note that has a good collection of advance results,theorems on finite group theory. By advance, I mean theorems after Sylow,Lagranges(I am considering theorems of Sylow,...
3
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3answers
384 views

Sylow p-subgroup of a direct product is product of Sylow p-subgroups of factors

Let $G$=$HK$ be a finite group (direct product), $P$ a Sylow $p$-subgroup of $G$. Prove that $P$ = $H'K'$, where $H'$ and $K'$ are Sylow $p$-subgroups of $H$ and $K$ respectively. I am very new to ...
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1answer
47 views

Group with Elements of Order 2 [on hold]

How can I prove that if a group, all the elements are from the order of $2$, then is isomorphic to $Z_2+Z_2+Z_2+..+Z_2$.
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0answers
40 views

Group $G$ with $ord(G)=319$ is a cyclic group

Let $G$ be a group with $ord(G)=319$. Proove that $G$ is a cyclic group. Answer: $ord(G)=319=11*29=n$, the Euler's totient function gives $\phi(n)=\phi(11*29)=\phi(11)*\phi(29)=10*28$. Since $gcd(...
2
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3answers
69 views

If $H$ is a subgroup of $G$ of index $n$, then $g^{n!} \in H \ \forall g \in G$

Let $G$ be a finite group and $H$ a subgroup of $G$ of index $n$, i.e., $[G:H]=n$. Prove that $$\forall g \in G,\; g^{n!} \in H.$$ This is a question I've had in a past exam for Group Theory and I'm ...
0
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1answer
22 views

Number of subgroups of groups with prime power order

Let $G$ be a group of order $p^2$ and put $\mathcal A=\{U\leq G, \#U=p\}$. What is $\#\mathcal A$? If $G$ is cyclic, then $G$ is generated by some element $x$ of order $p^2$. It seems like there ...
6
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0answers
139 views

Is the group $G_{n,\phi} = \langle x_1 , \dots, x_n \mid x_i^2, (x_i x_j)^4, x_i x_{\phi(i)} x_{i+1} x_{\phi(i)} \rangle$ abelian?

I am working on a family of finitely presented groups and I asking me the following question. Let $\phi$ be an application from $\left\{1,\dots,n\right\}$ to $\left\{1,\dots,n\right\}$ (not necessary ...
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0answers
14 views

If $a\in IBr(G/N)$, then $a\in IBr(G)$? [on hold]

If $a\in Irr(G/N)$, then $a\in Irr(G)$. How about replacing Irr by IBr?
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0answers
14 views

How to compute the number of modular/Brauer characters in a p-blocks of a finite groups, for example $A_5$ or $S_3$?

I do not know how to compute the modular character in a p-block of finite group.I want to know some skills for computing the number of modular characters in a p-block of a finite or some material ...
0
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1answer
36 views

A normal subgroup $N$ of $G$ with $\operatorname{gcd}(|N|,|G/N|)=1$ [on hold]

Let $G$ be a finite group and $N$ be a normal subgroup of $G$ such that the centrilizer of $x$ in $G$ is a subset of $N$ for each $x \in N \setminus \{e\}$ ($\operatorname{C}_{G}(x) \subseteq N$, $\...
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votes
2answers
236 views

Given a finite Group G, with A, B subgroups prove the order of AB [on hold]

How do you prove: Given a finite group $G$, with $A,B$ subgroups then $$|AB|=\frac{|A||B|}{|A \cap B|}.$$
4
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1answer
37 views

If $H<G$ is abelian, and $\chi(1)=[G:H]$ for irreducible $\chi$, then $H$ contains a nontrivial normal subgroup?

Suppose $\chi$ is an irreducible character of a finite group $G$, and $H$ is a nontrivial abelian subgroup such that $\chi(1)=[G:H]$. Why does $H$ contain a nontrivial normal subgroup? I understand ...
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2answers
87 views

Group of order $1575$ problem

Problem Let $G$ be a group with $|G|=1575$. If $H \lhd G$ and $|H|=9$, then $H \subset Z(G)$. What I've done so far is $|G|=1575=3^25^27$. I consider $G$ acting on $H$ by conjugation, or, in other ...
0
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1answer
27 views

Cyclic permutation group example ($n>1$)

I have googled around and haven't been able to find any examples of some $S_n$ with $n>1$ that is a cyclic group. This may mean it is a dumb question, any help is appreciated.
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53 views
+100

Growth of the characters of finite permutation groups in the number of symbols

I have the following questions. When can the characters of the irreducible representations of the elements of a finite permutation group increase exponentially in the number of the symbols the ...
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0answers
17 views

Different Representation Matrices from same Generating Set

Motivation: This post. $K \subset S_n, \langle K \rangle =G \leq S_n$. We can create a Representation Matrix $M$ from $K$ that represnts $G$ (Furst. Hopcroft, Luks). Question: Is $M$ unique for a ...
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0answers
35 views

Sifting algorithm for group generated by a set. [closed]

On page 38 of "Lecture Notes in Computer Science" by Christoph M. Hoffmann, there is an algorithm (ALGORITHM 2). I have some confusions. The algorithm needs to go to all column element indexed by ...
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1answer
27 views

Permutation Product

Here is a problem from Algebra by Michael Artin: $p = \left( {\begin{array}{*{20}{c}} 3&4&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&5 \end{array}} \right),q = \left( {\...
1
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2answers
35 views

Group $G$ cyclic as it coincides with the multiplicative group of a finite field

I have a group $G=(\mathbb{Z}_{251}^{*}, \cdot)$ with generator $g=71$ (so, having a generator, I'm given with the fact that is cyclic, right?) further in the example of my study notes I read: "$n = |...
6
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2answers
2k views

Proof that $S_3$ and $S_4$ are solvable groups

I wish to prove that $S_3$,$S_4$ (permutations on $3,4$ elements respectively) are solvable. I know that $D_6,D_{24}$ ($D_n$=Dihedral group of order $n$) are solvable and if I could prove that $S_3$ ...
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3answers
42 views

Calculate multiplicative inverse of $95$ in group of order $n=101$ which is subgroup of $(\mathbb{F}_{607}^*,\cdot)$

In the notes where I'm studying from there is written: "Let $G=\langle g\rangle$ be a subgroup of $(\mathbb{F}_{607}^*,\cdot)$ with $g=64$ and order $n=101$" but that felt strange to me; since I know ...
4
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0answers
54 views

Does Abelianization Commute with Profinite Completion?

Let $G$ be a group, let $\widehat{G}$ be its profinite completion, and let $G^{\text{ab}}$ be its abelianization. Is is true that abelianization commutes with profinite completion, in the sense that $...
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0answers
31 views

Monolithic quotients in soluble groups.

Let $G$ a finite soluble group. Is it true that if $K$$\vartriangleleft$ $G$ is maximal respect to the condition $G$/$K$ non abelian, then this quotient is monolithic with monolith the derived ...
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3answers
55 views

$|G| = ?$ if its subgroups are $\{e\}$, $G$ itself, and a subgroup of order $7$?

Suppose that a cyclic group $G$ has exactly three subgroups: $G$ itself, $\{ e \}$, and a subgroup of order $7$. What is $|G|$? What can you say if $7$ is replaced with $p$ where $p$ is prime? I ...
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0answers
28 views

Number of elements of Complete Right Transveral in pointwise stabilizer. [closed]

Let $G < S_n$ be a permutation group of degree $n$, and let $G^{(i)},0 < i\leq n$, be the pointwise stabilizer of $\{1,2.., i\}$ in $G$. We set $G^{(0)}= G$. For $0 < i\leq n$, let $U_i$ be ...
1
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1answer
22 views

Distinct coset representative and stabilizing an element.

Let $G < S_n$ be a permutation group of degree $n$, and let $G^{(i)},0 < i\leq n$, be the pointwise stabilizer of $\{1,2.., i\}$ in $G$. We set $G^{(0)}= G$. For $0 < i\leq n$, let $U_i$ be ...
3
votes
3answers
24 views

Trying to show that any group of order four is either cyclic or isomorphic to V

I know the question has already been asked. But I have trouble with the answer. Having a non-cyclic group $\,G=\{1,a,b,c\}\,$, how can I show that $ab=c$? In my attempt, I assume that $ab=1$, and ...
6
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1answer
465 views

Finding the maximum number of subspaces of a vector space over finite field that satisfy these relations

I have a question and I am stuck. I was wondering if anyone has a thought, before I start a brute-force search. For $q$ a prime number and $n =6$, let $\mathbb {F}_{q}^{n}$ be an $n$-dimensional ...
2
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0answers
26 views

Isomorphism of invariant factor decomposition

By the structure theorem, for every finite abelian group $A$, we have an isomorphism $A \cong \mathbb{Z}_{d_1} \oplus \dots \oplus \mathbb{Z}_{d_n}$ for unique $d_i$, s.t. $d_i | d_{i+1}$. My question ...
1
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2answers
47 views

Set acting like a Group

I am a little confused with familiar things, so I am looking for some help. Description: Consider the set, $S_3= \{(123), (132), (213), (231), (321), (312)\}$, a symmetric group acting on $3$ ...
0
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1answer
56 views

Determine abelian groups with 48 elements

I am just doing some revision for my linear algebra exam, and I came across this problem: Determine all abelian groups (up to isomorphism) with exactly 48 elements. I am not sure I have ever ...
3
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1answer
32 views

For prime $p$, let $G$ be a group such that every non-identity element of $G$ has order $p$. Show that if $|G|$ is finite, then $|G| = p^n$.

I've been self teaching myself some topics in preparation for university and thought I'd have a go at some past paper questions from their website. As such I do not have much experience with these ...
2
votes
1answer
29 views

Problem with the proof of abelian finite groups decomposition

I can't understand the proof which says that every finite abelian $p$-group can be written as a direct sum of cyclic $p$-groups. I'm using Lang's book of Algebra. My problem is about the following ...
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votes
1answer
40 views

Surjective group homomorphism $f:A_4\times \mathbb{Z}_2 \to S_3$ [closed]

Is there a surjective group homomorphism $f:A_4\times \mathbb{Z}_2 \to S_3$? I need to find one if the answer is "yes" or explain why the answer is "no". With $\mathbb{Z}_n$ I mean the quotient ...
0
votes
0answers
23 views

Explicitly calculating the group of global isometries of a convex regular polygon (dihedral group)

Instead of an intuitive geometric description of the dihedral groups $D_{2n}$, that one can find in virtually every good book on group theory, I want to calculate the global isometries of a convex ...