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

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At what position do we insert the new number in the weights for extended Dynkin Diagrams?

Given a set of weights of a representation and the corresponding extended Dynkin diagram for some Lie algebra, we can delete a node, which yields the maximal subalgebra. I know how to draw the ...
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0answers
9 views

Homomorphisms in $Q_8$ [duplicate]

Prove directly that the 2-dimensional irreducible representation $\rho$ of $Q_8$ is not realisable over $\mathbb{R}$. Suppose $\rho: Q_8 \rightarrow GL_2(\mathbb{R})$ is a representation with ...
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2answers
17 views

Action of $G$ on the left cosets of $H$ giving a non-trivial homomorphism

If $H < A_5$ is a subgroup of index $3$, the action of $G$ on the left cosets of $H$ gives a non-trivial homomorphism $$\underset{order \ 60}G \rightarrow \underset{order \ 6}{S_3}$$ which ...
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4answers
27 views

In a group $G$, $a$ is the only element of order $n$, for some $n\in \mathbb N$. Prove that $a\in Z(G)$.

If In a group $G$, the element $a\in G$ is the only element of order $n$, i.e., $a^n=e$ for some positive integer $n$. Then we have to show that $a\in Z(G)=\{x\in G : xg=gx, \forall g\in G\}$. How ...
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24 views

What is the Generator of GL(2,Z) [on hold]

I am wondering what is the Generating set of GL(2,Z)? I was told it's 1 1 0 1 -1 0 0 1 1 0 0 1 How to show it?
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1answer
27 views

Galois Extension whose Galois Group is $\mathbb{Z}_2\oplus\mathbb{Z}_4$

The book I am using for my Abstract Algebra course is Contemporary Abstract Algebra by Joseph A. Gallian. Let $E/F$ be a Galois extension with Galois group isomorphic to ...
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1answer
25 views

H is normal subgroup having index $3$, why is every $a^3$ in $H$? [on hold]

Let $H$ be normal subgroup of a finite group $G$ such that $H$ has index $3$. Show that $a^3$ is in $H$ for every $a$ in $G$. Anyone can help me?
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3answers
48 views

What would be an effective way to learn group theory on my own?

I've read the basics of this branch and I found it extremely interesing, and I would really love to learn more about it. I want to study as much as I can on my own, as my course doesn't have group ...
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3answers
87 views

A normal subgroup that is not a characteristic

In the book I'm study is written: A normal subgroup of a group need not be characteristic. And as an exercise I'm supposed to find an example, it also said that is pretty hard to find one. After ...
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1answer
22 views

Considering transitive $G$-set

Question. Suppose that $X$ is a transitive $G$-set of size greater than $1$ and let $\pi$ be the associated permutation representation with the character $\chi$. Show that some element $g \in G$ ...
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1answer
24 views

Primitive root problem

Let $p>3$ be prime number and $a$ primitive root modulo $p^2$. Prove that $x^{p-1}\equiv 1 \pmod{p^2}$ solutions are $\overline{a}^p,\overline{a}^{2p},\ldots ,\overline{a}^{(p-1)p}$. Any ideas on ...
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33 views

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ [on hold]

A group that can be written as $\frac{G}{Z(G)}$ for some group $G$ is called capable. I want to know which one of the following groups is capable? $(\mathbb{Z}_2\times D_8)\rtimes \mathbb{Z}_2$, ...
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1answer
21 views

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers.

Prove that every non-abelian group of order $8$ has a $2$-dimensional irreducible character all of whose values are integers. I get that if $G$ is non abelian it must have an irreducible ...
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1answer
33 views

Is there a classification of f.g. infinite abelian-by-finite groups?

Let $G$ be a f.g. infinite abelian-by-finite group, i.e. there exists a f.g. infinite abelian group $N$ which is normal in $G$ and such that the quotient $G/N = Q$ is finite. The problem of ...
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1answer
27 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 ...
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4answers
32 views

Show that A*B and B*A have the same order [duplicate]

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 ...
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0answers
32 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 ...
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15 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
32 views

how to find subgroup generated by elements [on hold]

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 help me with this? I have a confusion on $\langle ...
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1answer
42 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 ...
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2answers
46 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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27 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 ...
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3answers
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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 ...
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0answers
13 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 ...
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1answer
48 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?
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0answers
18 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 ...
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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 ...
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2answers
36 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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12 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 ...
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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 ...
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2answers
53 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 ...
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26 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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25 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 ?
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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 ...
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1answer
413 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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1answer
48 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
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1answer
33 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 ...
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0answers
33 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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21 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 ...
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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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32 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 ...
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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 ...
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8answers
916 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 $p^2$?
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1answer
57 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
17 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 ...
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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
33 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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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 ...