The study of symmetry: groups, subgroups, homomorphisms, and group actions.

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$|A[d]|=d^r \implies A\cong\left(\mathbb{Z}/n\mathbb{Z}\right)^r$

Let $A$ be a finite abelian group of order $n^r.$ Suppose that for every $d|n$ we have $| A[d]|=d^r,$ where $A[d]$ denotes the subgroup consisting of all elements of order $d.$ I want prove that : ...
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18 views

Is it possible that a finitely generated ring has an ideal that is not finitely generated

Sorry if this is duplicated. I couldn't find an exact answer of my question. One definition of Noetherian ring is: A ring $R$ is Noetherian if all its ideals are finitely generated. I know there are ...
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0answers
15 views

group hyper_(Abelien_by_finite)groups has non trivial normal subgroup H of G such that H finite or Abelien [on hold]

Let G be group hyper_(Abelien_by_finite)group, show that G has non_trivial normal subgroup H of G such that H finite or Abelien. hyper_(Abelien_by_finite)groups by definition if it has an ascending ...
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20 views

S is 6 cycle if $ s^i $ is a 6 cycle then find i

Let s be a 6 cycle in $ S_{12}$ then $ s^i $ is also 6 cycle if value of i is 1.2 2.3 3.5 4.12 I think because 6 cycle has order =6 therefore $s^6 $=identity therefor $s^7$=s so for i=7 it is ...
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1answer
21 views

Proving the minimality of an element order

Assume that I have a finite group G of order n with a generator g, and also assume that I want to prove that $\frac{n}{gcd(n,m)} $ is the order of an element $x = g^m \in G$. First , I showed that ...
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1answer
25 views

Intuition of coset of a subgroup

Hey guys I am trying to form the intuition that distinct left coset of subgroups are actually disjoint. I understand the proof constructed but I don't think I get the intuition behind why that the ...
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3answers
58 views

Set $A$ not closed under $\star$ then $A$ not a group under $\star$?

I am currently doing some exercises. I have been through some examples of solutions in other books that questioned me. I know well that $(A,\star)$ is a group if it satisfies the following points, ...
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41 views

Polynomial ring addition in $\mathbb{Z_{6}}$

I know this is a very simplest question ever. But, I need help with understanding it. So here it goes... Let, $f(x) = \bar{1}+\bar{2}x+\bar{3}x^2$ and $g(x) = \bar{4}+\bar{5}x$ $\in \mathbb{Z_{6}}.$ ...
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1answer
27 views

If $G/Z(G)$ is of size $qp$ and $p-1$ is not divisible by $q$ then $G/Z(G)$ is cyclic?

I have $G/Z(G)$ with size $pq$, $p, q$ are prime and $p>q$; $(p-1) $ is not divisible by $q$ How do I deduce from the above that $G/Z(G)$ is cyclic?
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34 views

Group Action and “nice” Approximation!

Studying the action of a group $G$ on a set $X$ is naturally the same as looking at the group homomorphism $\alpha: G \rightarrow Perm(X)$. So, for a given group $G$, classifying all sets $X$, on ...
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1answer
83 views

What is an intuitive way to think of Cauchy's theorem?

I am looking at a problem which involves an understanding of why a finite group $G$ has an element with order $p$ if $p$ is a prime factor of $|G|$. I have looked at several resources and proofs ...
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1answer
20 views

Determining whether ${p^{n-2} \choose k}$ is divisible by $p^{n-k -2}$ for $1 \le k < n$

Let $p$ be an odd prime and $n \ge 3$ a positive integer. I would like to know whether ${p^{n-2} \choose k}$ is divisible by $p^{n-k -2}$ for $1 \le k < n$. It should be noted that one can ...
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1answer
24 views

Isomorphism of finitely generated groups

Let G and H be two groups such that $G=<a,b>$, $H=<c,d>$ and o(a)=o(c), o(b)= o(d). Does that imply that G and H are isomorphic? or some other condition is also required ?
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100 views

Do these groups have a meaning?

Let $G$ be a group. We can say that $Aut(G)\leq S_G$ where $S_G$ denotes the set of all bijection from $G$ to $G$. But $S_G$ is not a good bound for $Aut(G)$ as $S_G$ grows very fast. Let's define ...
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4answers
48 views

$(\mathbb Z_7^{*},\cdot)$ is isomorphic to $(\mathbb Z_6,+)$

Just a short question: Is $(\mathbb Z_7^{*},\cdot)$ isomorphic to $(\mathbb Z_6,+)$? I would say "yes", since both group have order 6, hence the left hand side only can be isomorphic to $S_3$ or ...
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31 views

Prove $G\cong H\oplus \Bbb{Z}^{k}$.

Let $G$ be an abelian group and let $H$ be a subgroup. Let $G/H\cong \Bbb{Z}^{k}$. Prove $G\cong H\oplus \Bbb{Z}^{k}$. What I did so far is: there is an epimorphism from $G$ to $\Bbb{Z}^{k}$ such ...
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1answer
24 views

Does first isomorphism theorem work both sides?

The theorem says that if I have a group homomorphism, then the kernel is normal and the image is isomorphic to the domain group modulo the kernel. Now, suppose I have $G/K \cong{H}$ where $G$ and ...
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2answers
37 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
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2answers
47 views

Centre of a group and normalizers

Let $G$ be a group and let $A \subset G$ be a non empty subset of $G$.Define the following subsets of $G$ $$Z(G) = \{z \in G \space | \space zx =xz \space \space \forall x \in G \}$$ $$N_G(A) = \{h ...
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1answer
26 views

Is this extension of $Sp(4,2)$ a semidirect product?

Somebody I trust has been insisting to me that a certain extension of $Sp(4,2)$ is actually a semidirect product, and I'm inclined to believe him, but I haven't been able to convince myself he's ...
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1answer
26 views

$r$-cycle to a power $k$ is also an $r$-cycle if and only if $\gcd(k, r) = 1$

Let $\sigma$ be an $r$-cycle in $S_n$ and let $k\in\Bbb Z$. Show that $\sigma^k$ is also an $r$-cycle if and only if $\gcd(k,r)=1$.
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2answers
79 views

$x=x^2$ in a sub group?

I have a set E defined in ℝXℝ (E=ℝXℝ) and the operation * defined like this ...
2
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1answer
21 views

The group action of $S_n$ given a partition of $n$

We know that irreducible representations of $S_n$ are given by partitions of $n$. I would like to know if there is a way to explicitly write down the action of some $g \in S_n$ on the representation ...
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1answer
69 views

Is it possible to divide a cube into $5772$ cubes of varying sizes?

Is it possible to divide a cube into $5772$ cubes of varying sizes? I'm pretty sure this riddle has to do with algebra and group theory, but so far everything I've tried has led me nowhere. Any help ...
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23 views

Why use class multiplication in Homotopy groups?

This is a related to a physics question Why use class multiplication to describe topological entangling and merging?. In physics, the homotopy theory is used to describing topological defects in order ...
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1answer
50 views

Find a group $G$ which contains the elements $a,b,c$ such that $a\ne b$ and $ac=cb$

The title says it all. I'm trying to find a group $G$ which contains the elements $a,b,c$ such that $a\ne b$ and $ac=cb$. I didn't have an idea how to construct the group $G$ in a smart way so I was ...
2
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1answer
16 views

Generators of $Sp(2n)$

Let $\sigma =\begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix}$. Define $J_{2n} = \underbrace{\sigma \oplus \cdots \oplus \sigma}_{\text{$n$ copy}}$. We define a $2n \times 2n$ real matrix matrix ...
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51 views

How the centers of a group and its subgroup are related?

Let $G$ be a group and $H$ be a subgroup of $G$. What can I say about the centers of $G$ and $H$. How are they related? For example, if I know the center of $GL(n,\mathbb{R})$ is ...
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22 views

How many permutations of a linear equation

How many strictly positive integer solutions does the equation $x_1+x_2+···+x_n = k$ have? (Hint: Consider the equation $y_1+y_2+· · ·+y_n = k−n$ with variables $y_i \ge 0$.) I believe the ...
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1answer
29 views

Are Sylow p-subgroups in conjugate?

Let $G$ be an infinite group and $p$ be a prime number. Let $\mathscr{C}$ be a chain of p-subgroups of $G$ ordered by inclusion. Then, for every element of the union of $\mathscr{C}$ has an order ...
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2answers
40 views

Prove that $|A\cap B| \le \frac {1}{2} |A|$ where $A,B$ are two subgroups of $G$

Suppose $G$ is a finite group, $A,B$ are subgroups of $G$ and $A$ isn't a subgroup of $B$. Prove (by using Lagrange's theorem) that $|A\cap B| \le \frac {1}{2} |A|$. $ $ This is what I have so far: ...
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3answers
47 views

Infinite group with finite order elements [on hold]

Can you give me an example of an infinite group in which every element has order $3$ (except identity) ?
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72 views

Exercise 1: Galois Theory (J. Rotman)

Definition: Let $F$ a figure in the plane, its symmetry group is defined by $\Sigma(F):=\{\sigma \in O(2,\Bbb R)\mid \sigma(F)=F\}$. Here $O(2,\Bbb R)$ denotes the real orthogonal group. Exercise 1: ...
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67 views

order of non abelian group can't be what?

Let $G$ be a non abelian group; then its order can be: $25$ $55$ $35$ $125$ I think the order cannot be $25$ and $35$. But from option $55$ and $125$ which one is not possible? Why not $25$ ...
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1answer
31 views

Questions in Abstract Algebra

I have two question which I couldn't solve: Let $G$ be a group of size $40$. a. Show the $5$-Sylow subgroup in $G$ is Normal - this part was easy, I just showed that $n5=1$ and then $P5$ is ...
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1answer
30 views

$\langle s \rangle \sim \langle t \rangle$ and $\langle s \rangle \cap \langle t \rangle$, where $s$ and $t$ are permutations in $S_6$

Let $s=(12)(345)$ and $t=(123456)$ be permutations in $S_6$ how to know if $\langle s \rangle$ and $\langle t \rangle$ are isomorphic or not? Also what about $\langle s \rangle \cap \langle t ...
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1answer
22 views

Eigenvalues of operator on $S_n$'s group algebra

Take the group algebra of the symmetric group $S_n$ (or equivalently consider $S_n$'s regular representation) - I guess over $\mathbb{C}$. If $e_{i,j} \in S_n$ denotes the element which swaps only ...
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1answer
23 views

Primitive Roots Modulo $2^n$ for $n\geq3$

Question: (a) Prove that there is no primitive root modulo $2^n$ for any $n\geq3$, where $\bar{a}\in(\mathbb{Z}/2^n\mathbb{Z})^\ast$ is a primitive root modulo $2^n$ if the order of $\bar{a}$ is ...
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23 views

Specific Subgroups of an Abelian Group

I am looking for an elementary proof of the following result: If G is a finite abelian group and H is a subgroup of G, then G contains a subgroup isomorphic to G/H. This can be proved rather easily ...
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35 views

A Problem from I Martin Isaacs Algebra: A graduate course [duplicate]

Suppose G = H U K U L where H,K, and L are proper subgroups of G. Prove that [G:H]=[G:K]=[G:L]=2. I have trouble believing this, let alone proving it. Thanks for any pointers.... or a proof! Gary
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2answers
44 views

Do there exist pro-$p$ groups with finite quotients of non $p$ power order?

We define a pro-$p$ group to be a projective (i.e. inverse) limit of $p$-groups. My question is exactly as stated in the title: If a subgroup $H$ of a pro-$p$ group $G$ has finite quotient, must ...
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1answer
23 views

Inequivalent representations of a finite group

I'm looking for this result: A finite group has only finitely many inequivalent representations of given degree over a field of characteristic $0$. Do someone know where I can find a proof of ...
2
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1answer
29 views

Generators of $\Gamma_0(N)$

Let $\textbf{T}:=\bigl(\begin{smallmatrix} 1&1\\ 0&1 \end{smallmatrix} \bigr)$, $\textbf{S}:=\bigl(\begin{smallmatrix} 0&1/\sqrt{N}\\ -\sqrt{N}&0 \end{smallmatrix} \bigr)$ and $H$ the ...
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1answer
47 views

Homology group $H_1(G;\mathbb{R})$ is a vector space?

I am reading a paper which is asking me to view the homology group $H_1(G;\mathbb{R})$ of a (presentation of a) group as a vector space. Now, my knowledge of homology is basically non-existent, but I ...
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1answer
41 views

A normal intermediate subgroup in L30 lattice with an additional index condition?

This post is a sequel of: A normal intermediate subgroup in L30 lattice? Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups ...
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11 views

A relationship between central-by-finite groups and FC-groups II [on hold]

Just now asked a question with the same title. Now I would like to improve my question with the following. Let $G$ be a locally finite group. Suppose that $G$ is a FC-group. Let $x$ be a element of ...
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1answer
16 views

A relationship between central-by-finite groups and FC-groups

A group is said FC-group if for all $x\in G$ is true that the set $x^G$ is finite. Equivalently, $G$ is a FC-group if $|G:C_G(x)|$ is finite for all $x \in G$. A group is said a central-by-finite if ...
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15 views

Group theory and centres [duplicate]

If $G$ is a $p$-group and $H$ is a non trivial normal subgroup of $G$, how do I show that the size of $H\cap Z(G)$ (where $Z(G)$ is the centre of $G$) is $\ge p$? A hint is given to consider the ...
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42 views

True False Statements (Group Theory )

Let $G_1$ be an Abelian group of order $6$ and $G_2$ = $S_3$. a. Both $G_1$ and $G_2$ have unique subgroup of order $2$. b. Neither $G_1$ and $G_2$ have unique subgroup of order $2$. c. $G_1$ ...
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32 views

Intuition behind the link between coding theory and group theory

I am trying to find an easy link between group theory and coding theory. The usual path that most of the texts follow is that they present introductory material on groups, fields, rings, etc., and ...