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

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6 views

If $G_{ab}$ is cyclic then $G$ is cyclic

In my notes I have the following theorem: Let $G$ be a (nilpotent?) group. Suppose that $G_{ab}$ is cyclic. Then $G$ is cyclic. Actually I don't know if the hypothesis that $G$ is nilpotent is ...
0
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4answers
25 views

Show that A*B and B*A have the same order

How can I show that the elements A * B and B * A have the same order? where A, B belong to a finite group G How can I prove that 2 elements have the same order? I was thinking of showing that ...
0
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0answers
22 views

Group Theory: Finding Homomorphisms From a Cyclic Group to an Automorphism Group.

I have to find all the homomorphisms, $$h:C_{5}\to Aut(C_{31})$$ Given that there are thirty elements in $Aut(C_{31})$, do I have to find the order of each of the elements and then see which of them ...
0
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0answers
14 views

Inducing $A_4$ from $\langle (123) \rangle$

Let $G=A_4$ and $H=\langle (123) \rangle < G$. Compute $Ind_{H}^G \chi$ for every irreducible $\chi$ of $H$. Choose the right transversal of $H$ in $G$ as $V_4=\{ 1, (12)(34),(13)(24),(14)(23) ...
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2answers
20 views

how to find subgroup generated by elements

Let $G=\mathbb{Z}_6$ the cyclic group of order $6$ then $\langle 2\rangle=\{0,2,4\}$ but $\langle 2,3\rangle=\mathbb{Z}_6$ Can someone solve this? I have a confusion on $\langle 2,3\rangle$
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1answer
35 views

Is the direct product $\mathbb{Z} \times \mathbb{Z}$ a cyclic group, with the operation $(x, z) + (y, a) := (x + y, z + a)$? [on hold]

Is the direct product $\mathbb{Z} \times \mathbb{Z}$ a cyclic group, with the operation $(x, z) + (y, a) := (x + y, z + a)$? I really do not know how to this question, any hints or a hopeful step by ...
1
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2answers
42 views

Is there an isomorphism of additive groups when $\mathbb{Q/Z}$ isomorphic to $\mathbb{Q}$? [duplicate]

I know that I have to study the order of every element in $\mathbb{Q/Z}$. But what do I do? I've been struggling of what to do for this question
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0answers
23 views

Some notes on $D_n, S_n$ and $A_n$

http://www.stat.uchicago.edu/~lekheng/courses/repth/sol2.pdf In these solutions it refers to See "Some notes on $D_n, S_n$ and $A_n$". Does anyone know where these notes can be found? They sound ...
7
votes
3answers
42 views

Proving that $D_{12}\cong S_3 \times C_2$

Prove that $D_{12}\cong S_3 \times C_2$. I really dont know how I should start this question. My gut feeling says in some way I have to consider normal subgroups of $D_{12}$ but I cannot see how ...
0
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0answers
12 views

Finding #Groupoid like subsets

Given $S=\{x \in \mathbb{R}: 1 \leq |x| \leq 100\}$, find all subsets $M$ of $S$ such that for all $x$, $y$ in $M$, their product $xy$ is also in $M$. My attempt: If any number with magnitude ...
1
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1answer
44 views

There exists a homomorphism $f : G \to H$ with $|G| = 20$ and $|im f | = 6$

There exists a homomorphism $f : G \to H$ with $|G| = 20$ and $|im f | = 6$? Is this true? I know that I have to use the first isomorphism theorem but I don't know what to do next?
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0answers
28 views

normalisers grow in nilpotent groups [on hold]

What are some intuitive examples of normalisers grow in nilpotent groups?
-3
votes
0answers
17 views

Subgroups, intersection, unions [on hold]

The one question I couldn't get on our final review sheet. Suppose that $H$ and $K$ are subgroups of a group $G$. 1. Show that $H\cap K$ is a subgroup of $G$. 2. Give an example to show that ...
3
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0answers
49 views

Sylow p-subgroups and set X not divisible by p

Let $P$ be a Sylow $p$-subgroup of $G$ and suppose that $P\subseteq Z(G)$. Show that the set $X$ of elements of $G$ with order not divisible by $p$ is a subgroup of $G$ and that $G=P\times X$. I ...
1
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2answers
31 views

prove the nomalizer $N(H)$ of the subgroup $H$ in $G$ is a group

I need some help on the following question. For an arbitrary subgroup $H$ of the group $G$, the normalizer of $H$ in $G$ is the set $N(H) = \{x \in G \mid xHx^{-1} = H\}.$ Any help??
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0answers
11 views

Help needed for statistical analysis of pitch class sets

Within Music Analysis, there is a quite mathematical type of analysis which looks at pitch class sets ($pcs$), not surprisingly known as pitch class set analysis. See ...
0
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1answer
33 views

Show that it is a homomorphism?

For any abelian group $G$ we have $e_n: G \to G, e_n(g) = g^n$. By convention $e_0(g) = 1$. For a Field $F$ we have the subgroup $\{1,-1\} \leq F^*$. When $F$ is of characteristic $2$, this is the ...
5
votes
2answers
50 views

Help to prove that a group is cyclic

As part of my study of Abstract Algebra I'm trying to prove that $U_p$ si cyclic for $p$ a prime number. It's a classical result, but I'm trying to prove it following 4 steps stated as problems in my ...
3
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0answers
25 views

Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...
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0answers
24 views

Cyclic subgroups of $\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$

$\mathbb{Z} /100\mathbb{Z} \oplus\mathbb{Z}/25\mathbb{Z}$ has 24 elements of order 10. Why each cyclic subgroup of order 10 has four elements of order 10 ?
2
votes
2answers
32 views

Group Theory and Lagrange's Theorem: coprime subgroups. [duplicate]

Let $G_1$ and $G_2$ be finite groups, and let $K≤G_1 \times G_2$. Let $H_1 = \{ g \in G_1 : (g,e) \in K\}$ and $H_2 = \{g \in G_2 : (e,g) \in K\}$ and suppose $|G_1|$ and $|G_2|$ are coprime. Then ...
15
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1answer
367 views

Can I represent groups geometrically?

I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How ...
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votes
1answer
47 views

Are ℚ/ℤ and ℚ isomorphic as (additive) groups? [on hold]

Is there an isomorphism $${\Bbb Q} / {\Bbb Z}\cong\Bbb Q$$ (of additive groups)? Justify your answer.
3
votes
1answer
27 views

Why can we write the weights of a representation in terms of the simple roots?

I'm currently trying to get my head around the fact that we can write the weights of any representation in terms of the simple roots of the algebra. Is there any, not too-technical, explanation? I ...
1
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0answers
32 views

A free group is residually nilpotent

How can I prove that a free group is residually nilpotent group. Definition- A group G is residually nilpotent if for every non-trivial element $g$ there is a homomorphism $h$ from G to a nilpotent ...
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0answers
13 views

group theory problem in m.a. [on hold]

if G is finite group of order n and G/z(G)=4 show that 8 divide n.
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0answers
20 views

Compatibility of direct product and quotient in group theory

This question came to me when I tried comparing direct product and quotients of groups with products and quotients of natural numbers. When we divide a number by another and multiply the result with ...
0
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0answers
26 views

centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$

I need to find the centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$ in order to find the action of $H$ on $X$ which will help me find the orbits of $X$ I Know that the centralizers of $M_2(F_p)$ ...
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0answers
31 views

Need an example

Let $p$ be a prime number. I need an example of finite group $G$ generated by the elements of order $p^n$ ($n\in \mathbb N$) , which contains a normal subgroup $H$ that is not generated by the ...
3
votes
2answers
56 views

Intuition behind quotient groups?

I am having a hard time seeing the intuition behind quotient groups or rings. Intuitively, for a group, say Z/nZ would the quotient groups be the different sub groups of order 0 to n-1? Or how would ...
18
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8answers
896 views

What does it mean for something to hold “up to isomorphism”?

For example, to say that there are 2 such groups up to isomorphism such that the order of G is equal to $p^2$?
2
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1answer
56 views

Why do we have a basis?

A corollary that is in my book that I think is relevant to my question is: If E is an extension field of F, $\alpha \in E$ is algebraic over F, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ ...
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1answer
15 views

nilpotent algebraic groups in terms of extensions

Let $N$ be a nilpotent linear algebraic group over a field $k$. If $k = \mathbb{C}$ and $N$ is connected, one can write $N = U \times T$, where $U$ is a unipotent algebraic group and $T$ is a ...
3
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0answers
32 views

Inner automorphisms as the kernel of a homomorphism

By a straightforward computation, it is not hard to show that the set $\operatorname{Inn}(G)$ of the inner automorphisms of a group $G$ is a normal subgroup of $\operatorname{Aut}(G)$, see for example ...
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2answers
32 views

A prime order group must be cyclic [duplicate]

I have a question about prime order group. This answer by amWhy says that: It follows that any group of order 5 (and any group of prime order) must be generated by a single element and is hence, ...
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0answers
22 views

assume subgroup $H$ of $G$ such that $N$ is also a subgoup of $H$, then $ P_{G/N}(H/N) = P_{G}(H)/N$

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
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3answers
29 views

Why is it the smallest subfield containing F and $\alpha$?

Please take a look at the sentence in red: I understand that $\phi_\alpha[F[x]]$, is a subfield which contains $\alpha$, and F(we just need to evaluate $\phi_\alpha$ at the appropriate values). But ...
3
votes
1answer
57 views

Difference between definitions of $p$-subgroup and Sylow $p$-subgroup

I'm reading Abstract algebra by Dummit and Foote and the following definitions are made: $1$. A group of order $p^{\alpha}$ for some $\alpha\geq1$ is called a $p$-group. Subgroups of $G$ which are ...
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3answers
47 views

Does every finite field have a subfield $\mathbb{Z}_p$?

It seems that in the answers for my exercises in the book, the book uses that every finite field, has a subfield $\mathbb{Z}_p$. Is this true? They seem to use it in the answer for one exercise. But ...
-1
votes
1answer
28 views

Covering relation over functions

F is a group that includes all functions from N to N K is relation over F. For f,g ∈ F: (f,g) ∈ K iff ∀ n∈N, f(n)≤g(n). Obviously K is Partially ordered set and not Total Order. My problem is with ...
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0answers
19 views

Let $N$ be a minimal normal subgroup of $G$, then $G/N$ is supersoluble? [on hold]

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
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1answer
19 views

If $\sigma\in S_p$ with $|\sigma| = p$, why is $\sigma$ a $p-$cycle, and why is $|\sigma^r| = p$?

If $\sigma\in S_p$ with $|\sigma| = p$, why is $\sigma$ a $p-$cycle, and why is $|\sigma^r| = p$ for each $r$, $1\leq r < p$? ($p$ is a prime) I guess I am just having a hard time understanding ...
1
vote
1answer
27 views

Is the unreduced Burau representation completely reducible?

To be specific, my question is about specializations $\beta \colon B_n \to GL_\text{n}\left( \mathbb C \right)$ of the unreduced Burau representation given by \begin{array}{cr} \beta \left( \sigma_{i} ...
1
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0answers
24 views

How can I find the Weights of a Subalgebra

I'm currently trying to understand how we can derive the weights of a subalgebra of a given representation of a Lie group. For example, if we start with the 16-dimensional representation of ...
2
votes
1answer
40 views

Finite presentability of a group

Le $F$ be a free group of infinite rank (say countable rank) and consider a semidirect product $G = F \rtimes \mathbb Z$. Is it possible that $G$ is finitely presentable? If not, can we say something ...
9
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2answers
104 views

Prove that $x$ has order $5$.

let $ x \in G$ such that $(a^{-1})*(x^2)*(a) = x^3$ for some self inverse $a.$ Prove that $x$ has order $5.$ I don't know how to start this proof. Seems really difficult.
2
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2answers
58 views

Is every group of odd order isomorphic to a subgroup of $A_n$ for some $n$?

Is every group of odd order isomorphic to a subgroup of $A_n$ for some $n$? If not, what is a counterexample; if so, how can I prove it? Hints will be appreciated.
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0answers
42 views

Proving a group is $PSL(2,q)$ with $q>3$ odd.

How can I prove the following theorem If $G$ is a nonabelian simple group with Sylow $2$-subgroups being of order $4$ then $G=PSL(2,q) $ where $q>3$ is odd. with the help of this theorem: ...
2
votes
1answer
29 views

If $G$ is finite group that supersoluble then $G$ satisfy the maximal permutizer condition?

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...
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votes
0answers
24 views

What is the permutizer of the Sylow 3 subgroup in $S_4$ ? [on hold]

The permutizer of a subgroup $H$ of $G$ is defined to be the subgroup generated by all cyclic subgroups of $G$ that permute with $H$, i.e. $\langle x \in G | \langle x \rangle H = H \langle x \rangle ...