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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order of an element in a modulo group under multiplication

Suppose $G$ is the group $ℤ_{37}^\times$ under multiplication. Then is there a way that I can prove the order of the element $2$ in $G$ is $36$ without finding all the powers of $2$ until I get unity?...
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1answer
35 views

Order of $\mathrm{SL}(n,\mathbb{F}_p)$ (Constructive proof)

Most proofs of $$ \vert ~\mathrm{GL}(n,\mathbb{F}_p) ~\vert = \prod_{k=0}^{n-1} (p^n-p^k) $$ I have seen so far, are done by counting the possibilities to build up invertible matrices i.e. counting ...
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27 views

Regarding the general method of the ''Classify groups of order $X$'' question.

Anyone who has had to prepare for an algebra qualifying exam is familiar with the "Classify groups of order $X$" question. To illustrate my general question, which I postpone until the end, consider ...
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2answers
28 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 ...
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36 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
26 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 ...
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33 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 ...
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1answer
38 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 $p^e$ ...
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2answers
52 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
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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
40 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 ...
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1answer
42 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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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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2answers
62 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,...
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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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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(...
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1answer
23 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 ...
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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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17 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 ...
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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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1answer
36 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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3answers
70 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 ...
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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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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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1answer
55 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 $\langle r^{-2}\rangle=\langle r^2\rangle \le D'_{2n}$....
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0answers
18 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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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( {\...
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+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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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 = |...
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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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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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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 ...
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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
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$|G| = ?$ 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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3answers
25 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 ...
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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 ...
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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 ...
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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 $3$ ...
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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 ...
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1answer
57 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 ...
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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 ...
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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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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 ...
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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 ...
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2answers
44 views

Prove /Disprove: Existence of Symmetric Subgroup of Order $n^2-1$ of $S_n$.

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where order of $G$ is $n^2-1$. Problem: Prove (or disporve), Such $G$ exists for $S_n$ for finite $n$. For $n=5,11,71$, ...
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2answers
37 views

Can this be done using Sylow theorems? [duplicate]

Let $p$ and $q$ be distinct primes.Suppose that $H$ is a proper subset of the integers and $H $is a group under addition that contains exactly three elements of the set {$p,p+q,pq,p^q,q^p$}.Determine ...
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1answer
58 views

Need help in understanding the proof of “If $ \vert G \vert$=60 and $ G $ has more than one Sylow 5-subgroup, then $ G $ is simple.”

This proof is from Dummit & Foote text. Suppose by way of contradiction that $\vert G \vert=60$ and $n_5$>1 but that there exists $H$ a normal subgroup of $G$ with $H$ $\neq$ $1$ or $G$. By Sylow'...
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3answers
89 views

Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where $n=5, 11, 71$ and order of $G$ is $n^2-1$. Question: Does $G$ (as defined above for $n=5, 11, 71$) exist? How can I ...
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0answers
49 views

How to find the smallest set of generating elements in a group?

Is there a systematic procedure for finding the smallest set of generating elements of a finite group?
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24 views

Relation between permutation group and algebraic equation. [duplicate]

What kind of relation do algebraic equeation and permutation group have? For example, $Z^n -1=0$ is related to a cyclic group $C_n$. Is there anything else in this kind problem? I have read about ...