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
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0answers
48 views
+50

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 ...
2
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1answer
35 views

Group of order 175 is Abelian

Question: Prove that any group of order 175 is Abelian. The solution: I am unable to understand why the intersection of the normal subgroups is the trivial intersection. Any help is ...
0
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0answers
21 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 ...
1
vote
2answers
51 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 ...
0
votes
0answers
30 views

$\{g\in G\mid\alpha(g)=g^{-1}\}=\frac34|G|$, find an abelian subgroup of index 2

$G$ is a finite group, $\alpha$ is an automorphism of $G$ and $I=\{g\in G\mid\alpha(g)=g^{-1}\}$. If $|I|=\frac34|G|$, show that $G$ has an abelian subgroup of index 2. Related question I don't ...
4
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0answers
57 views

If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.

I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if $|G|=p_1^{i_1}\cdots p_n^{i_n}$ for $i_j\leq 3$ holds, but I would like to ...
0
votes
1answer
22 views

possible cyclic group from fundamental theorem of finite abelian

Question: Give a representative of each Isomorphism class of Abelian group of order 225. Which ones are cyclic? By the Fundamental theorem of finite abelian group: $\left | G \right |=225=3^{2}...
0
votes
0answers
14 views

Discrete subgroups of $SU(m) \times SU(n) \times U(1)$

Is anything known about which permutation groups are subgroups of $SU(m) \times SU(n) \times U(1)$? Alternatively, how can one possibly go about finding them. I am trying to find discrete groups of ...
1
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3answers
59 views

Is my proof True ? ( about Group theory, direct product )

I have a problem. It states that: Let $G$ is a group and $|G|=mn$, $(m,n)=1$. Assume that $G$ has exactly just one subgroup $M$ with order $m$ and one subgroup $N$ with order $n$. Prove: $G$ is ...
-1
votes
0answers
21 views

How can I prove that the the number of elements of order $k$ in $\mathbb{Z}_n$ is φ(k)? [on hold]

How can I prove that the number of elements of order $k$ in $\mathbb{Z}_n$ is ϕ(k) where $k$ is number that divides n ?
3
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4answers
60 views

If $G$ is a non-abelian group of order 10, prove that $G$ has five elements of order 2.

I'm trying to prove this statement: If $G$ is a non-abelian group of order $10$, prove that $G$ has five elements of order $2$. I know that if $a\in G$ such that $a\neq e$, then as a ...
1
vote
2answers
21 views

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?...
0
votes
2answers
34 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
vote
0answers
36 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
votes
1answer
418 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 ...
0
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2answers
38 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 ...
0
votes
2answers
28 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
votes
5answers
511 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
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 $p^e$ ...
0
votes
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 ...
-1
votes
2answers
53 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 $\mathbb{Z}/31\mathbb{Z}$.
1
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3answers
36 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 $\Bbb Z/n\Bbb Z$, ...
1
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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 ...
2
votes
1answer
28 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}(...
0
votes
1answer
57 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
votes
1answer
550 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
votes
1answer
40 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{...
1
vote
2answers
64 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
votes
3answers
389 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 ...
-2
votes
1answer
49 views

Group with Elements of Order 2 [closed]

How can I prove that if a group, all the elements are from the order of $2$, then is isomorphic to $Z_2+Z_2+Z_2+..+Z_2$.
0
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0answers
41 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(...
3
votes
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 ...
0
votes
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 ...
6
votes
0answers
141 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 $a\in IBr(G)$? [closed]

If $a\in Irr(G/N)$, then $a\in Irr(G)$. How about replacing Irr by IBr?
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0answers
19 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
votes
1answer
37 views

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

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 $x \in N \setminus \{e\}$ ($\operatorname{C}_{G}(x) \subseteq N$, $\...
4
votes
2answers
236 views

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

How do you prove: Given a finite group $G$, with $A,B$ subgroups then $$|AB|=\frac{|A||B|}{|A \cap B|}.$$
4
votes
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 ...
1
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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
votes
1answer
28 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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0answers
64 views

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
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 ...
0
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1answer
28 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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2answers
36 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
votes
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 $S_3$ ...
1
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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
votes
0answers
57 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 $...
1
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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 ...
3
votes
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 ...