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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1answer
61 views

If $|G/H|=4$ then $G$ is union of three proper subgroups

Let $G$ be a finite group and $N$ a normal subgroup of $G$ of index $4$. Prove that $G$ has a subgroup of index $2$ (hence normal) and if $G/H$ is not cyclic then $G$ is equal to the union of three ...
2
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0answers
62 views

Abelian groups of order 63

I am trying to learn abstract algebra on my own. Unfortunately I am confused and not sure how to proceed with the following question. I want to find all abelian groups of order 63. By theorem of ...
2
votes
1answer
70 views

“Concrete” Realisations of non-abelian finite groups

Many commutative groups could be imagined as something "concrete", for example $\mathbb N$ as an abstraction of operations on sets of objects, and from this $\mathbb Z, \mathbb Q$ and $\mathbb R$ are ...
2
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1answer
55 views

Proof of Lagrange's four square theorem using Cauchy-Davenport Theorem

Cauchy used the Cauchy-Davenport theorem to prove that $ax^2 + by^2 + c \equiv 0 \pmod p$ has solutions provided that $abc \neq 0$. Lagrange used this result to establish his four squares theorem. I ...
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3answers
76 views

Sylow $p$-subgroup of a finite group

I know: Let $P$ be a Sylow $p$-subgroup of a finite group $G$.If $N$ is normal in $G$, then $P \cap N$ is a Sylow $p$-subgrup of $N$. But if $N$ is not normal in $G$ , there is also the issue? ...
0
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1answer
80 views

$G$ be a finite simple group , then every element of $G$ can be written as a product of $n$-th powers of elements of $G$?

Let $G$ be a finite simple group , let $n$ be a positive integer such that not all $n$-th powers of elements of $G$ are identity , then is it true that every element of $G$ can be written as a ...
4
votes
3answers
102 views

Property of odd ordered elements of a Group

I'm (slowly) working my way through "Abstract Algebra" by Dummit and Foote. In the first set of exercises on group theory, the following question is posed: "Let $G$ be a finite group and let $x$ be ...
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1answer
50 views

Finite {2.3}-group with 4 Sylow 3-subgroup

Let $G$ be a finite {$2$,$3$}-group, the number of Sylow $3$-subgroups of $G$ be $4$, and a Sylow $2$-subgroup of $G$ be normal in $G$. Let $N$ be the kernel of the conjugation action of $G$ on its ...
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vote
2answers
53 views

If $\sigma \in S_n$ has order some prime $p$, then is $|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p$? [closed]

Let $\sigma \in S_n$ be such that $o(\sigma)=p$ (some prime). Then is it true that $$|\{1 \le i \le n : \sigma(i)=i\}|\equiv n \pmod p\ ?$$
2
votes
1answer
70 views

Name of the theorem: If $p^k$ divides $|G|$, then $G$ has a subgroup of order $p^k$?

Note: I am not asking for a proof of this theorem or any other theorem or help with a mathematical problem. This question is a reference request. I use the following well-known and somewhat-easy-to-...
3
votes
1answer
71 views

Questions about $\mathrm{SL}_2(\mathbb{F}_7)$

Let $G=\mathrm{SL}_2(\mathbb{F}_7)$, which has order $336=2^4\cdot 3\cdot 7$. And I may assume that $G$ is generated by the two matrices $$\begin{pmatrix}1&1\\0&1\end{pmatrix}, \begin{pmatrix}...
0
votes
1answer
45 views

Image of Sylow $p$-subgroup is Sylow $p$-subgroup

If $f:G\to H$ is an epimorphism between finite groups and $K\subset G$ a Sylow p-group then I want to show that $f(K)\subset H$ is also a Sylow p-group. So we want to show that $|f(K)|=p^m$ where $m$...
5
votes
2answers
462 views

Is it a subgroup?!

Let $G$ be a finite group, $A$ an its subset and put $A^{-1}=\{ a^{-1}:a\in A\}$. Is it true that if $A$ is symmetric (i.e., $A=A^{-1}$), $G=AB$ and $|AB|=|A||B|$, for some $B\subseteq G$, then $A$ ...
1
vote
1answer
67 views

Showing a polynomial of degree 7 is not solvable by radicals. [closed]

Show that the polynomial of $x^7-10x^5+15x+5$ is not solvable by radicals. For a polynomial of degree 5, we simply consider the derivative and determine the number of real and complex roots from ...
1
vote
0answers
31 views

Is there a simple way to find the conjugacy classes of $A_n$? [duplicate]

For the symmetric groups $S_n$ there's a quite simple way to find all the conjugacy classes. We find partitions of $n$, that is, we write $$n = n_1+\cdots+n_k,$$ and then for each such partition ...
2
votes
3answers
34 views

Let $G=\langle \mathbb{Z},+\rangle $ and $H=\{6n|n \in \mathbb{Z}\}$. Find all the distinct left and right cosets of $H$ in $G$.

I have an exercise where I am supposed to find the left and right cosets. But how do I generate the cosets? As I have understood it you are supposed to pick a number that is not in the set $H$ and ...
2
votes
2answers
95 views

Counting Circular Sequence (Burnside Lemma?)

How many distinct circular binary sequences of length $n$ are there? How many distinct circular binary sequences of length $n$ containing a given pattern, e.g., $110$ are there? The same questions as ...
0
votes
1answer
52 views

Minimal Generating Set of Group $G$ with size $k$

$G$ is a permutation group. So, $G < S_n$ where $S_n$ is a symmetric group acts on $n$ object. $G$ is not isomorphic to any symmetric or alternating group, i.e. $G \neq S_t , A_t$ for $1 <t \...
0
votes
1answer
28 views

multiple of a crossed homomorphism from finite group to a divisible one is principal

Let $\pi$ be a finite group, $\left|\pi\right|=n$ , acting on an abelian, torsion-free, $n$ -divisible group $D$ (i.e., every element of $D$ is divisible by $n$ ). Consider a crossed ...
0
votes
1answer
25 views

Example of a group $G$ that has a $\pi$-subgroup $U$ which isn't contained in any $\pi$ Hall subgroup

I would like to know if it is possible to find a group $G$ such that: $G$ has a $\pi$-subgroup $U$ but $G$ has no $H$ $\pi$-Hall subgroup such that $U$ is contained in $H$, where $\pi$ is a set of ...
5
votes
1answer
58 views

Order of a Subgroup

Let, $A \subset S_n$, $S_n$ is a symmetric group. $|A| \leq \log (n!)$. $A$ generates a subgroup $G$ of $S_n$. i.e. $\langle A \rangle=G < S_n$. What is the order of $G$? Can it be bounded by $|...
0
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1answer
34 views

Recognizing Permutation of Group with different Label

Problem description: Assume, a group, $G \leq S_{26}$ , $S_{26}$ is a symmetric group. Each permutation of $G$ is labeled using $1,2,....26$ as usual. Suppose, $f$ is a function that changes label ...
2
votes
1answer
54 views

Do arbitrary $\log_2(|G|)$ elements generate a group?

A set $S$ has $\log_2(|G|)$ distinct elements (arbitrary) of permutation group $G \leq S_n$. $S_n$ is symmetric group. i.e. $S \subset G$ and $|S| = \log_2(|G|) $. Is it a generating set of $G$? i....
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vote
0answers
22 views

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?
3
votes
2answers
40 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 ...
2
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0answers
34 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!
4
votes
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^...
2
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1answer
45 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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vote
4answers
70 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 ...
0
votes
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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0answers
23 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 ...
3
votes
0answers
46 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 $...
3
votes
1answer
71 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 ...
0
votes
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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votes
2answers
71 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 ...
3
votes
2answers
98 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 ...
2
votes
1answer
25 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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votes
0answers
16 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)=...
2
votes
1answer
55 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 ...
3
votes
0answers
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 ...
0
votes
1answer
190 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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0answers
62 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
31 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
56 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 $G$?
1
vote
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 ...
1
vote
2answers
71 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
1answer
43 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 ...
0
votes
0answers
45 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 ...