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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1answer
13 views

Proof of unicity of decomposition of a representation

I'm studying representation theory and in the book the author makes the following proposition with the following proof: Proposition: For any representation $V$ of a finite group $G$, there is a ...
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0answers
53 views

$(C_2 )^3$ is not a subgroup of $S_4$

Prove $(C_2)^3$ is not a subgroup of $S_4$. (Using group actions.) I could think of a permutation argument that $(C_2)^3$ is not a subgroup of $S_4$. But I would like to argue it by considering ...
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1answer
19 views

Problems in understanding a passage in the proof of Grün theorem for transfer

This is the statement of the theorem: Let $P$ a Sylow $p$-subgroup of $G$ and $Z$ a subgroup of $Z(P)$ that is weakly closed in $P$. Set $H=N_G(Z)$. Then $P\cap G'=P\cap H'$ and $P/(P\cap G')\simeq ...
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0answers
9 views

Isomorphism classes and invariant factors of abelian group

Let $G$ be an abelian group with $ord(G)=3374=2\cdot 7\cdot 241$. Calculate all isomorphism classes with the invariant factors $k_1\ ...k_n$ sucht that $k_i$ divides $k_j$ $(i<j)$. Since ...
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1answer
45 views

Galois Theory.Subgroups of Galois Group

How to List the subgroups of the Galois group in general.Im not interested in a specific Example but to make it easier. Supposes the galois group $G=Gal[Q(v,i):Q]$ $$v= \sqrt[4]{2}$$ I know how to ...
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0answers
33 views

A group with 3 Sylow 2-subgroup

Let $G$ be a finite group with $3$ Sylow $2$-subgroup(the number of Sylow $2$-subgroups $G$ are $3$), and let for every prime $p$ (not equal to $2$) Sylow $p$-subgroups are normal in $G$. I am looking ...
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0answers
36 views

A finite group theory problem [duplicate]

Let G be a finite group such that $a^{2}b^{2} = b^{2}a^{2}$ and $a^{3}b^{3} = b^{3}a^{3}$ for all $a,b\in G$. Prove that $G$ is abelian. I was wondering if there is any other elegant and general ...
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1answer
80 views

Finding subgroups of the Real Numbers

Find a subgroup of $\left (\mathbb R -\{0\}, \times\right)$ with a finite number of elements, which is not just the trivial subgroup $\{1\}$. Find a subgroup of $\left(\mathbb R − \{0\}, ...
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0answers
52 views

Is there an Unique Generating Set of a Group when Cardinality of the set is fixed?

$A$ is a generating set of permutation-group $G$ (not a p-group), i.e. $\langle A \rangle=G$. $|A| \leq \log_2(|G|)$. $A$ has possible maximum number of elements to generate $G$. It means that the ...
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0answers
29 views

A finitely generated locally finite group

I've understood that there are finitely generated groups which are also locally finite groups (an infinite finitely generated group which has no subgroups of finite index that are no trivial), but I ...
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2answers
43 views

Homomorphism from group of integers modulo $4$ to the Klein four group

Let $G=\mathbb{Z}_4$, the group of integers modulo $4$, and let $H$ be the Klein four group, let $f: G \rightarrow H$ be a homomorphism. Why does the kernel of $f$ contain the element $2$ of $G$?
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1answer
25 views

Necessary condition for a subgroup to be a Hall subgroup

Let $G$ be a finite group and $H \le G$ of order $m$. Assume that we have the following property: P: For all $g \in G$, $o(g) | m$ if and only if $g$ lies in some conjugate of $H$. Under this ...
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2answers
67 views

All groups of order 12; Dic12 and D6

I know this has been asked in various forms before, but so far I have failed to understand those answers properly. I've also read several papers discussing this, but I don't really get it. I have an ...
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1answer
39 views

Show that U is proper subset.

Let $G$ be a finite group and $U$ a subgroup of $G$ such that the order of $U$ is a power of the prime $p$ and $U$ it's not $p$-subgroup Sylow of $G$. Show that $U$ is a proper subset of $N_G(U)$ ...
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0answers
35 views

Construction of Permutation Group (bounded order) from a Set of Permutation

Given a set of permutation $S$. It has $|S|$ (The cardinality of $S$) elements. Consider a subset $A\subset S$. We can construct a group using $A$. One possible algorithm could be- We start ...
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34 views

A question about groups

$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$. $H$ is called $s$-permutably embedded in $G$ if each Sylow subgroup of $H$ is a Sylow subgroup of some ...
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1answer
42 views

The number of Sylow $5$-subgroups in $S_6$

Find the number of sylow $5$-subgroups in $S_6$. First: $ord(S_6)=6!=2^4\cdot 3^2\cdot 5=144\cdot 5$, so $n_5|144$ and $n_5\equiv 1\pmod5$, where $n_5$ is the number of sylow $5$-subgroups. ...
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1answer
39 views

Group of order $210$

Is there a $G$ group of order $210$ satisfy following properties: for some $x,y \in G$, order of $x$ is $5$, order of $y$ is $3$ $x^4y^2=(yx)^6$.
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53 views

Is there a name for the class of infinite groups with no proper infinite quotients?

As the title suggests, I would like to know if there is there a name for the class of infinite groups with no proper infinite quotients? I came across this paper ...
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1answer
37 views

$U(st)$ is isomorphic to $U(s)\oplus U(t)$ where $s$ and $t$ are relatively prime.

Suppose $s$ and $t$ are relatively prime.Show that $U(st)$ is isomorphic to $U(s)\oplus U(t)$. I want to show that $\phi$ :$U(st)$ $\rightarrow$ $U(s)\oplus U(t)$ defined by $\phi(x)=(x\bmod s,x\bmod ...
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0answers
45 views

Faithful irreducible character and Sylow subgroup

I am trying to solve the (very nice) exercise 5.25 from Isaacs, character theory. Assume that every Sylow subgroup of $G$ has a faithful irreducible character. Show that $G$ has one also. The ...
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2answers
32 views

Suppose G is a group of order 4 and $x^2=e$ for all $x$ in G. Prove that G is isomorphic to $Z_2\oplus Z_2$. [duplicate]

Suppose $G$ is a group of order $4$ and $x^2=e$ for all $x$ in $G$. Prove that $G$ is isomorphic to $Z_2\oplus Z_2$. My attempt: (1) Show that $G$ is abelian. \begin{align*} \text{Take }x,y\in ...
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1answer
31 views

Question on abelian group, does $G/H$ abelian $\iff [G,G]\leq H$.

Let $G$ a group and $H\lhd G$ a normal subgroup. I have a theorem that say that if $G/H$ is abelian, then $[G,G]\leq H$. I wondered if the converse hold, does it ? I recall that ...
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0answers
22 views

Character and centralizer of a subgroup

I am trying to solve the exercise 2.15 from Isaacs Character Theory. Let $\chi\in Irr(G)$ be faithful and let $H$ be a non-trivial proper subgroup of $G$ such that $\chi_{H} \in Irr(H)$. Show that ...
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1answer
32 views

The Rotation Group of a Cube

Show that the group of rotations of a cube is isomorphic to $S_4.$This proof is from Gallian's Abstact Algebra Theorem $7.3$ Proof:Using the Orbit-Stabilizer Theorem we know that the group of ...
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0answers
59 views

Twisty Puzzle Solving Program

I'm writing a program to help me solve a twisty puzzle. In this case it's the face-turning octahedron. I'm representing the puzzle as a group with face twists as generators. The facelets are in a list ...
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0answers
40 views

Maximum order of element in group of units in a ring

Let $s$ be a natural number and $U(s)$ be a group of units in the ring $\mathbb{Z}/s\mathbb{Z}$. Let $\phi(s) = 2^{k_1}p^{k_2}$, where $p$ is a an odd prime number. I don't understand why the maximum ...
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1answer
34 views

When will the classification of a property of a group be complete

I am studying pronormal Hall $\pi$-subgroups of a finite group $G$ All Hall $\pi$-subgroups of a finite solvable groups are known to be pronormal as this follows from Hall's theorem Recently, it has ...
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0answers
23 views

Suppose n is even positive integer and $H$ is a subgroup of $\mathbb Z_n$ [duplicate]

Suppose $n$ is even positive integer and $H$ is a subgroup of $\mathbb Z_n$. Prove that either every member of $H$ is even or exactly half of members of $H$ are even. Same question have been asked ...
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3answers
94 views

Is cyclic $\mathbb{Z}_4 \times \mathbb{Z}_ {12} \times \mathbb{Z}_9$?

I find on internet this: $\mathbb{Z}_m \times \mathbb{Z}_n$ is cyclic if and only if $\gcd(m,n)=1$. Then I do the next steps: gmc(4,12,9) is 1. Then I assume that $\mathbb{Z}_4 \times ...
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1answer
40 views

Is there a simple solution to these 2 equations without trying all possible values?

Now I have two equations and the computation is in a finite field GF(p), where p is a prime. $x, y$ are unknown, and $a, b$ are known. ($0<y<p-1$, and $0<ax<p-1.$) $\begin{cases} x^y ...
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2answers
25 views

Proving that the index of a subgroup of $S_n$ that keeps a specific set invariant has a certain order

Let $n \in \mathbb{N}, n ≥ 2$, and $k \in \{1, 2, ..., n-1\}$, and let $A \subseteq \{1, 2, ..., n\}$ with $|A| = k$. Furthermore, let $G$ be a subgroup of $S_n$ that fixes $A$, i.e. for all $π \in ...
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1answer
36 views

Question on character theory.

Let $\phi :G \to \text{Aut}(V_0)$ be the regular representation of $G$. Define a representation $\Theta$ of $G \times G$ on $V_0$ by $\Theta(g_1,g_2)e_g=e_{g_1gg^{-1}_2}$, i.e it maps the element ...
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1answer
38 views

If a finite group $G$ has a normal subgroup $N$ of order $p$ where $p$ is the smallest prime divisor of the group order, then $Z(G)$ is nontrivial

Let $G$ be a finite group and $p$ the smallest prime number with $p \mid |G|$. I want to show: if $G$ has a normal subgroup $N \trianglelefteq G$ of order $p$, then $G$ has a nontrivial center. My ...
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0answers
33 views

If $G$ is a subgroup of $S_n$ and $ G \not\subseteq A_n$, is it true that $ G A_n=S_n$ [closed]

If $G$ is a subgroup of $S_n$ and $ G \not\subseteq A_n$, then is it always true that $ G A_n=S_n$.
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1answer
26 views

Determining the size of a subgroup of H

Given that $|GL_{2}(Z_{5})|=480$ find the index of $H$ in $GL_2(\mathbb{Z}_5)$. $$ H=\begin{pmatrix} a & b \\ 0 & c \\ \end{pmatrix} \\a,c \neq 0 , \\a,b,c\in \mathbb{Z}_5 $$ I proved in ...
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1answer
79 views

Counting homomorphisms of groups

How many homomorphisms are there from $A_{5} × S_{3} × A_{4}$ to $\mathbb Z/2\mathbb Z$? I know that $Z/2Z$ is generated by ome element and of order two. I know the respective order of the rest ...
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1answer
28 views

Isomorphism between finite groups with specific property is unique

I am looking for a clever justification for a statement I believe to be true. I would like to show that given an isomorphism, $\Phi$, between two finite groups, if it is known that $\Phi(a) = b$, then ...
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1answer
20 views

$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]|$ is even

Let $G$ be a finite group. My problem is: $$|\hom(G, \Bbb Z/2)|>1 \iff |G:[G,G]| \mbox{ is even}.$$ I know that $\hom(G, \Bbb Z/2)=\hom(G^{ab}, \Bbb Z/2)$, where $G^{ab}:=G/[G,G]$. If ...
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1answer
39 views

Character theory - exercise 5.16 from Isaacs

Hi I am trying to solve the following exercise. Let $H$ be maximal subgroup of a finite group $G$ and let $\chi=(1_H)^G$. Let $\psi$ be a non-principal irreducible constituent of $\chi$. Then $Ker ...
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0answers
24 views

Character theory - exercise 3.4 from Isaacs

Let $G$ be a simple group and suppose $\chi\in Irr(G)$ with $\chi(1)=p$ a prime, Show that a Sylow $p-$subgroup of $G$ has order $p$. The indication provided is: if the Sylow $p-$subgroup $P$ is ...
2
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1answer
38 views

Prove that $\alpha$ is an automorphism of $Z_n$.

Let $ r \in U(n)$[We define $U(n)$ to be the set of all positive integers less than $n$ and relatively prime to n]. Prove that the mapping $ \alpha : Z_n \rightarrow Z_n $ defined by $ \alpha(s)=sr$ ...
3
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1answer
20 views

Show that a set is a right transversal for $S_3$ in $S_4$

I am given the set A ={e,(14),(24),(34)} I'm supposed to show that this set is a right transversal for S3 in S4, meaning that every right coset of S3 contains exactly one element of A. I'm getting ...
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0answers
14 views

Number of conjugacy classes or bounds for the index of stabilizers on finite Coxeter groups.

For a finite group $G$ denote by $\operatorname{conj}\left(G\right)$ the number of conjugacy classes of $G$ and let $q\left(G\right)$ be the quotient: \begin{align} ...
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41 views

What is the smallest number $n$ with $gnu(n)=2016$?

$gnu(n)$ denotes the number of groups of order $n$ upto isomorphy. What is the smallest number $n$ with $gnu(n)=2016$ ? The numbers upto $n=2047$ do not satisfy the given property. I do not know ...
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32 views

A number theory problem?

Let $p$ be a prime number. $k>1,n$ be integers and $0<λ<p^k$ be integer. Is there any $λ,n$ s.t the following relation satisfies: $$(2^n−1)(p^k−λ^2)=p^k(p^k−1).$$ I guess there isn't but I ...
2
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1answer
41 views

$p$-groups have normal subgroups of each order [duplicate]

Suppose $|G|=p^n$. Then $G$ has a normal subgroup of order $p^m$ for every $0\le m\le n$. By induction. It is clearly true for $n=0$. Now suppose $k<n$ and $H_i$ is a normal subgroup of $G$ of ...
3
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0answers
23 views

Explicit computation of $H^2(\mathbb{F}_p^n, \mathbb{R}/\mathbb{Z})$.

I'm interested in the computation of the second cohomology group of the elementary abelian group $\mathbb{F}_p^n$ with coefficients in $\mathbb{R}/\mathbb{Z}$: $$H^2(\mathbb{F}_p^n, ...
2
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1answer
48 views

Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$

I want to determine how many subgroups does the additive group $G:=\Bbb Z_{14} \oplus \Bbb Z_{6}$ have? There are many related posts in our site, for instances: here and there. However, it ...
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0answers
60 views

Chief factors and local formation

Every thing below is concerned with finite groups. My question is about this paper A class of groups is a collection $\mathcal{X}$ of groups with the property that if $G \in \mathcal{X}$ and if $H ...