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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Spherical functions which are invariant under a finite rotation group

Is there a nice, clean reference which lists a basis in terms of (linear combinations of) spherical harmonics for the $L^2$ space of functions defined on the sphere $\mathbb{S}^2$ which are invariant ...
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0answers
28 views

Showing $PSL(2,9)$ doesn't have a subgroup of order $90$

I got stuck in proving that there is not any subgroup of order $90$ in group $PSL(2,9) = SL(2,9)/Z(SL(2,9))$. Can anyone help?
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2answers
38 views

Generating set of same size must have common elements

Generating set $S_1, S_2$ generates permutation group $G$ where the number of elements in $S_1$ is equal to the number of elements in $S_2$. Prove, $S_1 \cap S_2 \neq \emptyset $ (not considering ...
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32 views

Automorphism and Direct Product of Generating Set

Notation: $H $ are partitioned into sub-graphs $ H_1,H_2 \cdots H_x$ . We see them in the adjacency matrix of $H$ given below- $$H = \begin{bmatrix} H_{(x)} & R_{(x, x-1)} & R_{(x,x-...
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0answers
21 views

How can I prove that $PSL(2,3)\simeq A_4$?

I would like to prove that $PSL(2,3) \simeq A_4$. I know the structure of $SL(2,3)$ and that $A_4=V\rtimes C_3$, where $V$ is the Klein subgroup. How can I proceed? Thanks for the help!
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2answers
61 views

On representations of a nonabelian group of order $pq$

Let $p,q$ primes number s.t. $p>q$ and let $G$ a non abelian group of order $pq$. 1) Determine all degree of irreducible representation 2) Show that $|[G,G]|=p$ (where $[G,G]=\left<ghg^...
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1answer
41 views

Showing a surjective homomorphism maps Sylow $p$-groups to Sylow $p$-groups [duplicate]

If $f:G\to H$ is a surjective homomorphism of finite groups, then $f$ sends Sylow $p$-subgroups to Sylow $p$-subgroups. Here's what I have. Suppose $\vert G \vert=p^km$ with $(p,m)=1$. Let $P\in \...
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4answers
65 views

Number of homomorphisms between two cyclic groups.

Is it true that the number of homomorphisms between any two finite cyclic groups of order $m\,\&\,n$ is $\gcd(m,n)$? I have posted an answer which I believe is true, just wanted to know different ...
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1answer
21 views

Solvability and representation of finite groups

Let $G$ be a finite solvable group, and let $G=G^{(0)}\unrhd G^{(1)}\unrhd...G^{(n)}=1$ be its derived series. Is it true that any irreducible representation of $G$ has dimension at most $n$?
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22 views

Non-Isomorphic Groups generated by a Set of fixed cardinaity

Given a set of permutations $A \subset S_n$. It has $|A|$ (the cardinality of $A$) elements. We can construct a group using $A$. How many non-isomorphic groups(all with the same order) can we ...
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45 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: If $...
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1answer
68 views

Why all irreducible representations appear 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 ...
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1answer
26 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).
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2answers
69 views

Decomposition of a representation into a direct sum of irreducible ones

I'm studying representation theory and in the book (Fulton and Harris) the author makes the following proposition with the following proof: Proposition: For any representation $V$ of a finite ...
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2answers
97 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
24 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
14 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 $ord(G)=...
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1answer
50 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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43 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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1answer
181 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\}, \times\right)...
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59 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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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
53 views

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

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
28 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
70 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
41 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 $N_G(U)$ (the ...
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0answers
42 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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1answer
48 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. Since ...
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1answer
45 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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54 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 http://onlinelibrary.wiley.com/doi/10....
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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
47 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
34 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 G.&...
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1answer
33 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 $[G,G]=\left<ghg^{-...
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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
34 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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63 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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24 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
118 views

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

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: $\gcd(4,12,9)$ is 1. Then I assume that $\mathbb{Z}_4 \times \...
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1answer
42 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 $π \in G$,...
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1answer
39 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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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
82 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 of ...
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1answer
30 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 ...
0
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1answer
21 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 $|\hom(G^{ab}...
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1answer
42 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 \...