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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3
votes
2answers
80 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 ...
3
votes
0answers
35 views

Compute the Galois group of $p(x)=x^4+ax^3+bx^2+cx+d$

Compute the Galois group of the following polynomial: $$p(x)=x^4+ax^3+bx^2+cx+d$$ Step 1: Calculate cubic resolvent Step 2: Calculate the discrimimant $\gamma$ of the cubic resolvent Step 3: ...
3
votes
1answer
42 views

Why all irreducible representations appeear in the regular representation?

Let $G$ be a finite group and $R$ the regular representation. That is, as a vector space $R = F(G)$ is the free vector space with basis $G$. If the basis is $\{e_g : g \in G\}$ the action is defined ...
0
votes
1answer
19 views

Local group rings

Let $k$ be a field of characteristic $p$ and $G$ a finite group. How do you prove that if $kG$ is local then $G$ is a $p$-group? (I know how to prove the converse but not this implication).
1
vote
1answer
185 views

Bound on the size of Permutation Set for Isomorphism

$\textbf{Claim :}$ $G, H $ are partitioned into sub-graphs $\{ G_1,G_2 \cdots G_x \}$ and $\{ H_1,H_2 \cdots H_x \} $ . For each $G_i$ we constructed a set permutation, $\beta_i$ such ...
-1
votes
0answers
27 views

Isomorphisms concerning group of units

How is it possible to show that $U_2^k \cong \mathbb{Z}_2 \times \mathbb{Z}_2^ {k−2}$ for $k\geq 3$?
2
votes
1answer
18 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 ...
4
votes
3answers
84 views

Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
2
votes
1answer
22 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 ...
2
votes
1answer
47 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 ...
0
votes
1answer
277 views

Prove that $[G\times H : A\times B]=[G:A][H:B]$ when $A < G$ and $B < H$.

The original question is that: If $A$ is subgroup of group $G$ and $B$ is a subgroup of group $H$, then express $[G\times H : A\times B]$ in terms of $[G:A]$ and $[H:B]$ and prove the result is ...
0
votes
0answers
11 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 ...
3
votes
0answers
35 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 ...
0
votes
1answer
110 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\}, ...
0
votes
0answers
55 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 ...
2
votes
3answers
77 views

Show that $S_5$ does not have a quotient group isomorphic to $S_4$

Show that $S_5$ does not have a quotient group isomorphic to $S_4$. If we to assume that $H$ is such a group, than $H$ must be normal in $S_5$ and $|H|=|S_5|/|S_4|=5$. So $H$ must be isomorphic ...
0
votes
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 ...
0
votes
2answers
44 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 $G$?
1
vote
1answer
26 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 ...
1
vote
1answer
41 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 $3$ $x^4y^2=(yx)^6$.
1
vote
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 ...
0
votes
0answers
60 views

What is the name of the group $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$?

I know, that $\mathbb Z_2\times \mathbb Z_2$ is the Klein four-group. Is there a nice name for $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ too?
0
votes
0answers
34 views

Classification of the irreducible group representations of the dihedral groups

Let $D_n$ be the dihedral group of order $2n$. Show that all irreducible representations have vector space dimension $1$ or $2$, and describe them up to isomorphism. Any hints how to even start? ...
0
votes
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 $N_G(U)$ ...
0
votes
0answers
41 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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
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. ...
3
votes
3answers
177 views

Isomorphism in certain groups.

Let $G = \langle \Bbb Z^3_2,+\rangle $, where as $ \Bbb Z_2= \{0,1\} $, $ \Bbb Z^3_2 = \{(x,y,z)\mid x,y,z \in \{0,1\}\} $, and the operation in $ \Bbb Z^3_2 $ is defined by $ (x_1,y_1,z_1) + ...
13
votes
5answers
4k views

A4 has no subgroup of order 6

Can a kind algebraist offer an improvement to this sketch of a proof? Show that $A_4$ has no subgroup of order 6. Note, $|A_4|= 4!/2 =12$. Suppose $A_4>H, |H|=6$. Then $|A_4/H| = ...
1
vote
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 ...
1
vote
1answer
35 views

Study of specific Quotients of a $p$-group in MAGMA

In Magma, I wanted to do a program, but since I was getting errors in "basic" commands, I am in trouble. Here I will describe a problem to be done in MAGMA, then its interpretation in MAGMA. I would ...
1
vote
1answer
78 views

Number of k-products of disjoint cycles in the symmetric group S(n)

Suppose that $S(n)$ denotes the group of all permutations of the set $\{1,2,...,n\}$ with the usual composition operation. Is there any formula or expression for $n(k)$, where $n(k)$ denotes the ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
0answers
60 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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
3
votes
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 ...
2
votes
1answer
41 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 ...
1
vote
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 ...
1
vote
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 ...
3
votes
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 ...
0
votes
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 ...
1
vote
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 ...