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

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An example of a nilpotent group

Is there an example of a nilpotent group such that $G/G'$ is (non-trivial) torsion-free while $G$ is not? I cannot think of any example of this kind and I think that it is not proved any result like ...
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0answers
18 views

I need example to satisfy in this theorem (Hall Subgroup)

I need example to satisfy in this theorem: let $H$ be a subgroup of $G$ such that $\mid G : H \mid$ is a $\Pi$-number.If there is a nilpotent subgroup $K$ of $G$ such that $G=HK$ then $G=HK_{\Pi}$, ...
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2answers
35 views

Geometric Interpretation of S3

My impression was that the symmetric group $S_3$ acts on the vertices of a labeled triangle. However, I am not sure this is the case anymore, because of the following. (The triangle is labeled as ...
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1answer
23 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 ...
0
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0answers
18 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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0answers
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
22 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 ...
2
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1answer
27 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 ...
1
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3answers
59 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, ...
3
votes
2answers
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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0answers
35 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 ...
2
votes
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
22 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 ...
-1
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1answer
25 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 ?
7
votes
1answer
101 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 ...
2
votes
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 ...
0
votes
2answers
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 ...
0
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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\\ ...
3
votes
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 ...
2
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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
votes
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 ...
5
votes
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 ...
0
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0answers
24 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 ...
2
votes
1answer
51 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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0answers
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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0answers
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 ...
2
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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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votes
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) ?
3
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2answers
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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3answers
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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votes
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
31 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 ...
2
votes
1answer
23 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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0answers
24 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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0answers
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
45 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 ...
2
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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
votes
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
48 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 ...
2
votes
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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0answers
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 ...
2
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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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0answers
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 ...