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
39 views

Finding invertible 3x3 matrix A such that Stab(M)A=AStab(N) for given matrices M,N

I'm reading that it is possible to find an invertible $3 \times 3$ matrix $A$ such that for the matrices $M = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 0 ...
0
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1answer
25 views

Suppose that the finite group $G$ acts on the set $\Omega$ of size $n$. let $\alpha \ne$

Suppose that the finite group $G$ acts on the set $\Omega$ of size $n$. let $\alpha \not= \beta$ in $\varOmega$ have stabilisers $A$ and $B$. Prove that $|\alpha^G| |\beta^A| |A\cap B|=|G|$. So I ...
1
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1answer
32 views

Why are the composition factors of Sym(4) and Sym(5) unique?

I have worked out the CS and CF, and I have been given in my notes that they are unique, but no explanation as to why and in a past paper I am asked for reasoning. Is it because Alt(4) and Alt(5) are ...
6
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1answer
46 views

Classify $p$-groups in which all groups of the same order are isomorphic

The answer to “Are two subgroups of a finite $p$-group $G$, of the same order, isomorphic?” is definitely no. Such groups are very rare. How rare? Can you classify all finite $p$-groups $G$ such that ...
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0answers
42 views

Are two subgroups of a finite $p$-group $G$, of the same order, isomorphic?

Given $G$, $|G|=p^n$, $H,K\le G$ s.t. $|H|=|K|=p^k$ for some $k<n$. Can we say that $H\simeq K$? I think it's true and I tried to prove it building by hand the isomorphism $\psi:H\longrightarrow ...
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2answers
20 views

Show that G', the commutator subgroup of G, is normal in G. Prove that any subgroup A with G' $\subseteq$ A $\subseteq$ G is normal in G.

Show that G', the commutator subgroup of G, is normal in G. Prove that any subgroup A with G' $\subseteq$ A $\subseteq$ G is normal in G. So the definition of the commutator subgroup is that; In the ...
2
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1answer
35 views

Non-obvious deduction regarding conjugates in $\text{GL}_2(\mathbb{F}_p)$

Let $\text{GL}_2(\mathbb{F}_p)$ act on $\mathbb{F}_p^2,$ the set of $2$-vectors with entries in $\mathbb{F}_p$, by matrix multiplication. $[\,$Prove that$\,]$ for any ...
0
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1answer
29 views

If $r≤m$, can we deduce that $A\leq B$?

Let $A$ be a finitely generated Abelian group, i.e., $$A≃ℤ^{r}⊕C$$ where $C$ is a finite abelian group. Let $B$ be another finitely generated Abelian group, i.e., $$B≃ℤ^{m}⊕C$$ where $m$ and $r$ are ...
3
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2answers
53 views

The number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$

How to count the number of $n\times n$ matrix over integer modulo $p$ field with determinant equal $1$? I know that the number of invertible matrices is GL$(n,p)$. Have any ideas?
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2answers
108 views

Simple proof of the structure theorems for finite abelian groups

Many proofs of the structure theorems for finite abelian groups first reduce to the problem to $p$-groups, which is fine and is an important technique. However, it seems to me that a simple proof can ...
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0answers
44 views

Question about subgroups of a symmetric group

I have the following question: For $n\ge 5$ show that the symmetric group $S_n$ cannot have a subgroup $H$ with $3\le [S_n:H]< n$. ($[S_n:H]$ is the index of $H$ in $S_n$). This is technically ...
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1answer
37 views

Finite Subgroup of multiplicative group of Field is Cyclic proof

I know this question has been asked before, but I would like to specifically address one part of the proof I'm reading that confuses me. I am following Fraleigh 7th edition, page 213. The theorem is ...
3
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2answers
42 views

Conjugacy of simple system in a root system

I'll set up the problem, then ask the question. Let $V$ be a finite dimension vector space over $\mathbb{R}$ and let $\Phi$ be a root system in $V$, i.e. (1) $\Phi \cap \mathbb{R} \alpha = ...
4
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0answers
20 views

Example of projective rep being used in Clifford theory

I'm trying to understand the use of projective representations in Clifford theory, and I'd like a small example where projective representations really help, and the ingredients are actually ...
3
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1answer
40 views

Showing the existence of an eigenvector using groups

Let $\text{GL}_2(\mathbb{F}_p)$ act on $\mathbb{F}_p^2$ ($2$-vectors with entries in $\mathbb{F}_p$) by matrix multiplication: ...
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0answers
35 views

For $g, g' \in G$, show that $gA=g'A$ if and only if $gA \cap g'A$ is non empty.

For $g,g' \in G$, show that $gA=g'A$ if and only if $gA \cap g'A$ is non empty. I have the Lemma that if $g, g'$ are in $G$ then $gH \cap g'H = \emptyset$ or $gH=g'H$ but I am not sure how to apply ...
2
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2answers
65 views

Nilpotent groups are monomial

I'm trying to show that a nilpotent group $G$ is monomial; i.e., that every irreducible representation $\rho$ of $G$ satisfies $\rho = \text{Ind}_H^G(\tau)$ for some $H \leq G$, $\tau$ a one ...
1
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1answer
46 views

Hexagon group theory

I was thinking that $$\alpha\beta=\beta_6$$ $$\beta\alpha=\beta_2$$ $$\beta^2\alpha= \beta_3$$ But I don't know what to do next or is this the right way of solving it,note I have only one example ...
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3answers
123 views

Show that a group of order fifteen has an normal subgroup of order 5

Let $G$ be a group where $|G|=15$, I want to show that $G$ has a normal subgroup of order $5$. I have shown that $G$ must have a subgroup $H$ of order $5$, (and one of order $3$), and I have shown ...
4
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1answer
32 views

Any two $n$-cycles are conjugate in $A_{n+2}$ if $n$ is odd

How would one go about proving the claim in the title? I see that if $\alpha,\beta$ are $n$-cycles and $\alpha,\beta$ permute $A,B\subset \{1,\dots, n+2\}$ respectively then $\overline{A\bigcap B}$ ...
9
votes
1answer
148 views

Is there any group $H\times K$ where $H\times 1$ and $1\times K$ are the only subgroups of order $n$?

$G=H\times K$ where $H$ and $K$ are non-isomorphic groups of order $n$. I am looking for an example such that $G$ has no subgroup of order $n$ except $H\times 1$ and $1\times K$. If anyone can find ...
4
votes
1answer
32 views

$N_Q(R)=N_Q(Q\cap R)$ where $Q, R \le P$, $Q$ normal and $P$ a $p$-group

Let $Q$ and $R$ be subgroups of the finite $p$-group $P$ and suppose that $Q$ is normal (maybe this is not needed). Is it then true that $$N_Q(R)=N_Q(Q\cap R)?$$ Obviously, we have $N_Q(R)\le ...
4
votes
2answers
41 views

What is this notation? Cyclic group $\mathbb{Z}^*_8$

$\mathbb{Z}^*_8$ As I understand it - $\mathbb{Z}_8$ is the group of integers under addition modulo 8. So am I correct in thinking its elements are: $\{0,1,2,3,4,5,6,7\}$? I thought the $*$ meant ...
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2answers
38 views

How to find the elements of $\langle (1,2,3,4) \rangle$ in $S_4$?

How to find the elements of $\langle (1,2,3,4) \rangle$ in $S_4$? The answer is given as $\{\mbox{id}, (1,2,3,4), (1,3).(2,4), (1,4,3,2)\}$. I understand how we got the first $2$ elements. Also ...
5
votes
4answers
110 views

Center of $G/Z(G)$.

Question. Prove or disprove: The center of $G/Z(G)$ is trivial for all finite groups $G$. (Here $Z(G)$ denotes the center of $G$.) Attempt at a Proof. Assume the contrary that $G/Z(G)$ has a ...
3
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1answer
53 views

Group of Order 44.

Question. Let $G$ be a group of order $44$ such that it has a subgroup isomorphic to $\mathbb Z/2\oplus \mathbb Z/2$ and a subgroup isomorphic to $\mathbb Z/4$. Show that $G$ does not exist. ...
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0answers
25 views

Constructing groups from actions

As an example to set the scene, suppose I have a cube and I let its symmetry group act on its faces. Every face admits four transformations which leave the cube invariant and which maps the chosen ...
3
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1answer
26 views

Orbits under action of a subgroup on the set of conjagtes of a second subgroup

i have the following question: Let $A\leq B\leq G$ be finite groups. Then $G$ acts naturally via conjugation on the set of conjugates $A^G$. It's trivial, that there is only one orbit under this ...
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3answers
29 views

Subgroups of cyclic group of order $p^n$

If $H$ and $K$ are subgroups of a cyclic group of order $p^n$ , where $p$ is prime , and $|H|>|K|$ , then is it true that $K \subset H $ ?
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2answers
69 views

Can anyone explain the difference between a ring, a group, and a field in a way so that your average 15 year old can understand? [duplicate]

I can not understand key difference between them, because all on the web uses mathematical notations...Can anyone explain in easy way?...please.Thanks in advance.
0
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0answers
30 views

Generator of cyclic groups $Z_{pq} $

Instead of asking to find the number of generators of $Z_{pq} $ where $p$ and $q$ are prime numbers it was asked to find generators of $Z_{pq}$. This is to find the number s which are relatively prime ...
4
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0answers
51 views

largest permutation group of odd order

For general $n$, what is the largest subgroup of the symmetric group $S_n$ that has odd order? I have a feeling that it might be the Sylow 3-subgroup...ADDED: but it isn't, as Mark Bennet points out ...
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0answers
23 views

On the number of Hall subgroups of a group

A Hall divisor of an integer n is a divisor d of n such that d and n/d are coprime. The number of Hall divisor of n is $2^k$, where k is the number of distinct prime divisors of n. Now a Hall ...
4
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2answers
217 views

On the existence of non-abelian finite groups

Is it true that for any $n\in \mathbb{Z}$ with $n\geq 6$ and $n$ not a prime there exists a non abelian group of order $n$? How can we prove it? If the answer to the above is negative is it maybe ...
18
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5answers
261 views

$\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 $

I came across this result while doing some representation theory of the permutation group $S_n$ $$ \sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
4
votes
2answers
79 views

What do Sylow 2-subgroups of finite simple groups look like?

What do Sylow 2-subgroups of finite simple groups look like? It'd be nice to have explanations of the Sylow 2-subgroups of finite simple groups. There are many aspects to the question, so I envision ...
1
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0answers
28 views

Group Action on $S_n$

Let $S_n$ be the symmetric group on $n$ elements and $\sigma \in S_n$ be a permutation. One can represent $\sigma$ (essentially uniquely) as a set of $m$ disjoint cycles of lengths ...
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votes
1answer
25 views

How many homomorphisms exist from $|G|=70\to |H|=91$?

Given finite groups $G:|G|=70,H:|H|=91$, how many distinct homomorphisms $f:G\to H$ exist? Noting that $|G|=7\cdot5\cdot2,|H|=7\cdot13$, and that $|f(G)| \big\vert|G|$ and $|f(G)|\big\vert|H|$, we ...
2
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1answer
19 views

Question about a non-abelian group of order $p^2q$

Suppose $p<q$, where $p,q$ are primes and we have a non-abelian group $G$ of order $p^2q$. Is it true that it has a subgroup which is not normal? I try to use Sylow's theorems. We take Sylow ...
2
votes
1answer
46 views

A question about the involution in simple groups.

Let $G$ be a finite simple group of Lie type over a finite field ($F_q$) of order $q$ with characteristic $p\neq 2$. Suppose $S$ is $2$-sylow subgroup of $G$ and is not abelian. I have two question ...
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1answer
31 views

If $G$ has more than one nontrivial elements with order 2, how to show that $\prod^n_{x=1}a_x=1$? [duplicate]

Let $G$ be an abelian group of order $n$, and $a_1,a_2,...a_n$ its elements. If $G$ has more than one nontrivial elements with order $2$, how to show that $\prod^n_{x=1}a_x=1$?
9
votes
1answer
118 views

Prove $G$ is abelian if $f(f(x)) = x$?

Let $G$ be a finite group and $f$ an automorphism such that $f(f(x)) = x$, and $f(x) = x$ if and only if $x=e$. Prove that $G$ is abelian and $f(x) = x^{-1}$. My attempt: ...
1
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1answer
36 views

How to find group homomorphisms from one group to another

I am trying to figure out all the homomorphisms from $\mathbb{Z}_2\times\mathbb{Z}_2$ to $\mathbb{Z}_2$. Is there a good process for doing such a think? I am getting lost...
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1answer
40 views

Quotient group of a direct product: $\left(G_1\times G_2\right)/\left(G_1\times \{e_2\}\right)$

Let $G_1$ and $G_2$ be groups, and $G := G_1 \times G_2$. Let $e_2$ be the identity element in $G_2$. Show that $H := G_1 \times {e_2}$ is a normal subgroup of G. Using the homomorphism ...
1
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1answer
32 views

How is this automorphism an inversion? And how is this group abelian?

Let $G$ be a finite group, and let $T$ be an automorphism of $G$ which sends more than three-quarters of the elements of $G$ onto their inverses. Then how to demonstrate that $T(x) = x^{-1}$ for all ...
2
votes
4answers
121 views

General approach for finding how many group homomorphisms are there

So I've asked this type of questions for more than once, and still I don't get the method(s) I've been presented with. What's the general recommended method for finding how many homomorphisms are ...
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2answers
47 views

let G be a non abelian group of order 39. how many subgroups of order 3 does it have? [closed]

let G be a non abelian group of order 39. how many subgroups of order 3 does it have
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1answer
135 views

Abelian group generators and relations

(a) Define what it means for an abelian group to be finitely generated. Explain the terms elementary divisors and rank of $G$ and describe the structure theorem for finitely generated abelian ...
3
votes
2answers
30 views

Average number of distinct values

Let $q$ such that $q < n$. I pick at random $n$ values in $\mathbb{F}_q$. What is the average number of distinct values ? Thank you
1
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1answer
26 views

Modular arithmetic and maximal permutations

I have a research paper about pseudo-random number generators and I need to answer the following: Given $n \in \mathbb{N}$, let's consider the permutation group of $A=\{{0,1,\dots,n-1}\}$. Since $A$ ...