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

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2
votes
1answer
14 views

$p$-group acting on a finite set

Let $G$ be a $p$-group. Prove that if $G$ acts on a finite set $X$ and $p$ does not divide $|X|$, then $X$ contains some element that is fixed by every element in $G$. Any thoughts? I'm stumped ...
2
votes
0answers
21 views

Non-existence of $\mathfrak{S}_n \hookrightarrow \mathfrak{A}_{n+1}$.

Let $\mathfrak{S}_n$ be the symmetric group (permutations of $n$ items) and let $\mathfrak{A}_n$ be the alternate group. For $n \geq 5$, I have to show that there is no injective morphism ...
0
votes
1answer
16 views

Find all group such that there exists a surjective homomorphism

I have been asked to find all groups H (up to isomorphism) st. there is a surjective homomorphism from $D_{2p}$ ($p$ prime) onto $H$. The normal subgroups of $D_{2p}$ are: <$e$>, $<\rho>$ ...
0
votes
1answer
22 views

The stabiliser of $g.x$ is the subgroup $gGxg^{-1}$

Let $X$ a G-set and $x \in X$. Show that for any $g\in G$, the stabiliser of $g.x$ is the subgroup $gGxg^{-1}$. I found this question in the book A Course in Group Theory of J. Humphreys. I was ...
1
vote
1answer
21 views

Lifting representations, kernels and invariant subspaces

Let $G$ be a group, $N \triangleleft G$, $G/N$ the corresponding quotient group. Suppose $\rho : G/N \longrightarrow GL(\mathbb{C})$ is a representation of $G/N$. Then the composition ...
-1
votes
0answers
32 views

Prove they are not pairwise isomorphic [on hold]

Prove $\mathbb{Z}_8$, $G_s$(the group of symmetries of the square) and the quaternion group $Q$ are not pairwise isomorphic. Stuck on this question. Seems very difficult.
0
votes
0answers
10 views

At what position do we insert the new coeffiecent 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 ...
0
votes
0answers
12 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 ...
0
votes
2answers
20 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 ...
1
vote
4answers
30 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 ...
-2
votes
0answers
25 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?
4
votes
1answer
31 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 ...
0
votes
1answer
27 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?
5
votes
3answers
55 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 ...
2
votes
3answers
88 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 ...
0
votes
1answer
28 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 $g \in G$ ...
0
votes
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 ...
1
vote
0answers
38 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? $(\mathbb{Z}_2\times D_8)\rtimes \mathbb{Z}_2$, ...
3
votes
2answers
42 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 ...
1
vote
1answer
40 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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
0answers
16 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) ...
-2
votes
2answers
34 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 ...
-1
votes
1answer
44 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
vote
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
0
votes
0answers
32 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
47 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 ...
1
vote
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 ...
1
vote
1answer
50 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?
-3
votes
0answers
29 views

normalisers grow in nilpotent groups [on hold]

What are some intuitive examples of normalisers grow in nilpotent groups?
-3
votes
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 ...
3
votes
0answers
54 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 ...
2
votes
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??
0
votes
0answers
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 ...
0
votes
1answer
34 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
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 ...
3
votes
0answers
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 ...
-1
votes
0answers
26 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 ...
17
votes
1answer
432 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 ...
-5
votes
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
votes
1answer
35 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
vote
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 ...
-2
votes
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.
1
vote
0answers
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 ...
0
votes
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 $M_2(F_p)$ ...
0
votes
0answers
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 ...
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 ...