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

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2
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0answers
18 views

Commutator subgroup of rank-2 free group is not finitely generated.

I'm having trouble with this exercise: Let $G$ be the free group generated by $a$ and $b$. Prove that the commutator subgroup $G'$ is not finitely generated. I found a suggestion that says to ...
-1
votes
0answers
20 views

Let G be a group and let x be a fixed element of G. Define Γ(x) = {g ∈ G : gx = xg} [on hold]

(a) Prove that Γ(x) is a subgroup of G. (b) Let G = A4, let x = (1 3)(2 4) and let y = (2 4 3). Find (i) Γ(x) and (ii) Γ(y).
1
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0answers
15 views

Permutation modules and their vector space dimensions

I'm given a field $k$, a finite group $G$ and a set $S$ which $G$ acts on transitively. I'm then told to consider the permutation module $M = kS$. My first problem is understanding what the ...
0
votes
0answers
11 views

Stochastic processes on group-valued variables

I have had this question in my head for a long time, and if I don't find out the answer I may explode. So I'm familiar with a basic Ito process, let's say: $dX_t = \mu d t + \sigma d Z_t$. There ...
1
vote
0answers
16 views

Exponentials of Representations of Lie Algebras

Assume G is a lie group and g is its lie algebra. Consider a representation of G : D:G->End(V). Then there is a corresponding representation of g : d:g->End(V). My question is, when you can express ...
0
votes
2answers
16 views

Part of simple proof of nontrivial center in p-group

I'm trying to understand the proof of a Burnside theorem (as stated in Beachy's Abstract Algebra p. 328): Let $p$ be prime number. The center of any $p$-group is nontrivial. Now, In the proof they ...
0
votes
0answers
15 views

Showing that f restricts to a group homomorphism

I have two abelian groups C and C' with corresponding homomorphisms $d:C→C$ such that $d^2=0$ and $d':C'→C'$ such that $(d')^2=0$. Then let $f:C→C'$ be a group homomorphism such that $fd=d'f$. I need ...
0
votes
1answer
10 views

Multiplying Cosets

1) Let $ah$ be a coset of the subgroup $H$. Suppose there are two elements $ah_1\in aH$ and $ah_2\in aH$ such that $(ah_1)(ah_2)\in aH.$ Show that this implies that $a \in H$ and so $aH=H$. 2) ...
1
vote
2answers
19 views

How to define binary operation on arbitrary set in order to create a group structure.

Is it (and if yes how?) possible to define an an binary operation $*$ for an arbitrary set $M$ such that $(M,*)$ is a group? If $M$ is finite or countable infinite this is trivial, but is it also ...
0
votes
1answer
17 views

Elements of $\operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^k$

Let $p > 2$ be a prime number and $n\ge 1$ an integer, and consider the group $G = \operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^n(p^{2n} - 1)$. Let us denote by $\operatorname{Inn}(G)$ (the ...
0
votes
0answers
33 views

Sufficient conditions for $G\cong N\times G/N$ [duplicate]

Given a normal subgroup $N$ of a group $G$, do there exist sufficient conditions that allow us to conclude that we have an isomorphism $$ G\cong N\times G/N?$$
1
vote
1answer
32 views

Finding all homomorphisms between $\mathbb{Z}/m\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$

I want to find all group homomorphisms $\varphi: \mathbb{Z}/ m \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$, with $m$ and $n$ natural numbers. Clearly $\varphi(0)=0$ since the identity in ...
3
votes
2answers
59 views

Showing that the image of a homomorphism $d$, with $d^2=0$, is contained in its kernel

Suppose I have an abelian group $C$, with a group homomorphism $d\colon C\to C$ such that $d^2=0$. I need to show that the image of $d$ is contained in the kernel of $d$. My original attempt was to ...
1
vote
1answer
23 views

Order of a permutation divides n in Sn

Let $\theta \in S_n$, and for any $k \in \mathbb{N}$, either $\theta^k = I_{I(n)}$ or $\theta^k$ has no fixed elements. Show that $o(\theta) | n$. $I_{I(n)}$ denotes the identity. I'm completely ...
0
votes
1answer
27 views

Subgroups of $\mathbb F_{p^n}$

Is it possible to give a discription of the possible subgroups (with respect to $+$) of the finite field $\mathbb F_{p^n}$ (obviously, $p$ is a prime number). Of course, if $n = 1$, $(\mathbb F_p,+)$ ...
0
votes
0answers
28 views

number of automorphisms of $ \mathbb Z_m \times \mathbb Z_n$

How to find the number of group automorphisms of $\mathbb Z_m \times \mathbb Z_n $ provided $m \& n $are not relatively prime?
0
votes
1answer
18 views

Does the product of elements being in a group imply the individual elements are in that group?

Let $N$ and $K$ be groups and let $x\in N \cap K$ and $k\in K$. If $kx=x'k$, for some $x'\in N$, does $kx \in N \cap K$ imply that $x' \in K$?
1
vote
1answer
31 views

can we have $gHg^{-1}\subsetneq H$? [duplicate]

It is well known that the following three definitions of a normal subgroup are equivalent: $gNg^{-1}\supseteq N$ for all $g\in G$ $gNg^{-1}\subseteq N$ for all $g\in G$ $gNg^{-1} =N$ for all $g\in ...
5
votes
2answers
36 views

$x^2+1=0$ in $\mathbb{Z}_7$

$x^2+1=0$ in $\mathbb{Z}_7$ By trying each number, I see that there is no solution, is this correct? And could you help me with a more direct solution, since this method is not going to work for ...
0
votes
0answers
34 views

How can I determine all the subgroups of order 8 in $S_4$

Is there any way to get all subgroups of order $8$ of the symmetric group $S_4$? In general, how can I find a subgroup of specific order?
3
votes
1answer
59 views

Infinite group not isomorphic to proper subgroup

We know that any finite group can't be isomorphic to any of its proper subgroups. Some countably infinite groups, like $\mathbb{Z}$, do have this property of course, as $\mathbb{Z} \cong ...
2
votes
1answer
31 views

Terminology of “G over H”

I am trying to find the definition of G/H (which is read as "G over H", "G modulo H", or "G mod H"). I believe that, in this case, G is a group and H is a subgroup of G.
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0answers
15 views

Automorphisms of direct and semidirect products

Let G be a group and let H, K be subgroups in G. Let G = H $\times$ K be. If H and K are both characteristic in G, then it holds: Aut(H $\times$ K) $\cong$ Aut(H) $\times$ Aut(K). (i) What could I ...
5
votes
2answers
38 views

Covering finite groups by unions of proper subgroups

A noncycic finite group $G$ may be expressed as a union of some of its proper subgroups. (Say the subgroups "cover" $G$ in this case.) A relatively simple exercise in some introductory algebra texts, ...
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votes
1answer
28 views

When is the permutation group cyclic? [on hold]

When is the permutation group cyclic and when not? Thanks in advance.
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0answers
46 views

Studying specific abstract algebra topics [on hold]

I have heard that the topics of direct products, homomorphisms and factor groups are the most intensive parts of abstract algebra. My instructor loves to give questions that may seem tricky when you ...
1
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2answers
14 views

non-zero elements in $\mathbb Z_3[i]$ form an abelian group

How shall I show that all non-zero element of $\mathbb Z_3[i]$ form an abelian group of group of order $8$ under multiplication... Please any hint how shall I show this result?
0
votes
2answers
37 views

Multiplication of subgroups

Let $H$ and $K$ be subgroups of a finite group $G$. Define $HK = \{hk\mid h \in H, k \in K\}$ and $KH = \{kh\mid k \in K, h \in H\}$. a) Show that in general $HK \ne KH$. (For example, consider $G = ...
0
votes
1answer
23 views

A finite and stable part of a group is a subgroup

How to prove that a a finite and stable part H of a group G is necessarily a subgroup ? This is equivalent to proving that every element x of H has its inverse in H too :)
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1answer
31 views

groups of order 4 [on hold]

How many different groups (non isomorphic) of order 4 are there ? Prove it.
2
votes
2answers
36 views

Find the kernel of the group homomorphism $G\to\text{Bij}(G/H),\;a\mapsto(b\mapsto abH)$

Let $G$ be a group and $H\subset G$ be a subgroup. Find the kernel of the group homomorphism $$G\to\text{Bij}(G/H),\;a\mapsto(b\mapsto abH)$$
1
vote
1answer
35 views

irreducible representation contained in regular rep

Why is every irreducible representation contained in the regular representation? Suppose $W$ is a irreducible representation. ( i.e. a vector space over $\mathbb{C}$ which $G$ acts on with no ...
1
vote
1answer
16 views

Trying to find the equivalence class of an equivalence relation

Let $G$ be a group and $H$ a subgroup of $G$. For $a,\;b \in G$, let $a \sim b$ if $a^{-1}b \in H$ I've managed to show that this is an equivalence relation. Now I have to show that $$ ...
3
votes
0answers
21 views

Groups where there's always a “deformation retraction” homomorphism onto any subgroup.

Let $G$ be a finite group with the property that, for every subgroup $H$, there exists a homomorphism $f: G\to H$ such that $f(h)=h$ for all $h\in H$. What possible groups can $G$ be? If $P$ is a ...
5
votes
4answers
219 views

Understanding homomorphism and kernels

Let $G$ be a group and $\phi$ a Homomorphism $$ \phi:G\to G' $$ Now I know that the size of the kernel tells you how many elements in $G$ map to the same element in $G'$ I couldn't find this in my ...
1
vote
1answer
32 views

Show that the element $z=i \cos \frac{\pi}{3}+\sin \frac{\pi}{3} = i( \cos \frac{\pi}{3} - i \sin \frac{\pi}{3})$ belongs to $U_{12}$

Show that the element $\displaystyle z=i \cos \frac{\pi}{3}+\sin \frac{\pi}{3} = i( \cos \frac{\pi}{3} - i \sin \frac{\pi}{3})$ belongs to U12 What I don't understand: In what way $i \cos ...
0
votes
1answer
30 views

Show that $a \star b=a \cdot b+a+b$ is binary operation for the group $\Bbb{ Q} - \{-1\}$

The group $\left(\Bbb{ Q} - \{-1\},\star\right)$ has as its underlying set the rational numbers different from $-1$ and the operation $\star$ is defined as $a \star b=a \cdot b+a+b$ where ...
2
votes
2answers
42 views

Show that $\mathbb Z_3\times V$ is isomorphic to $\mathbb Z_2\times\mathbb Z_6$

Klein's group is often referred to as the klein four group and denoted by $V$. Show that $\mathbb Z_3\times V$ is isomorphic to $\mathbb Z_2\times\mathbb Z_6$. I'm totally stuck.
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0answers
26 views

Could the equivalence classes in the construction of quotient group be the orbits of some group action?

Given a group $G$, $S\le G$, the $G/S$ is the collection of all left cosets, $gS$ for all $g\in G$. And These cosets partition $G$. Given that the orbits of any group action on $G$ also partition ...
3
votes
1answer
24 views

Permutations Isomorphic to $S_4$

Prove that the group generated by permutations $(0 2 6 4)(1 3 7 5)$ and $(4 2 1)(6 3 5)$ are isomorphic to the symmetric group $S_4$. I approached this problem by labeling the vertices of a cube. ...
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3answers
33 views

Find a non-trivial semidirect decomposition of the following groups

Find a non-trivial semidirect decomposition of the groups $S_n$, $n \geq 3$, $D_{2n}$, $n \geq 3$ and $A_4$. Prove that $A_n$, $n \geq 5$ and $Q_8$ have no non-trivial semidirect decompositions. How ...
0
votes
1answer
29 views

A Lagrangian group

I have a non-Lagrangian group $G$ of order $pq^3$, $Q$ a Sylow $q$-subgroup of G and a $H$ a subgroup of $Q$ with $|H|=q^2$. It is clear that $Q \subseteq N_G(H)$. I must prove that $G$ doesn't posses ...
1
vote
1answer
20 views

Homomorphism between 2 abelian groups sending one given element to another given element

Let $G$ and $G'$ be arbitrary abelian groups. Fix a $g \in G$ and $h \in G'$. Then does there exist a homomorphism $\phi$ such that $\phi(g) = h$?
0
votes
1answer
49 views

What is meant by a kernel and homomorphism in algebraic structures?

I have just started with discrete maths. I was doing some group theory and I stumbled upon kernel and homomorphism. I didn't understand what was written in the book. I googled it up and also looked ...
1
vote
1answer
48 views

Free group on two generators and commutators. Why it's enough to add the relation ab=ba?

I've looked through lots of question on this topics, but I cannot find what I want to prove: I've seen in a lots of exercises sheets that the abelianization of a free group with two generators (let's ...
0
votes
0answers
27 views

Normal Form of Elements in a Group

Suppose that there is a family of groups $A_n$ with $n\in\mathbb{N}$ and $A_1$ is the trivial group. If there is a split exact sequence $$0\to B_n\to A_n\to A_{n-1}\to 0,$$ where structure of the ...
0
votes
1answer
28 views

If the nilpotent class of $G$ is $k$, what's the nilpotent class of $G/C_{k-1}(G)$?

I read from a website, it said the nilpotent class will be $k-1$ at most. But why? (I know as the quotient group its class should be $k$ at most.)
0
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1answer
18 views

In $S_6$, write the result as a product of disjoint cycles and then in the 2-row form.

(a) $(1,2,4)(4,3,5)(2,4)(1,2,4)^{-1}$ In the solution for this question, my professor has the product of disjoint cycles written as (1,3,5)(4,1). How would this make sense when disjoint cycles are ...
2
votes
0answers
19 views

Getting U.C.P map on group operator algebras using Fell's absorbtion principle.

I'm struggling a bit with this theorem: Let $\Gamma$ be a discrete group and $\mathbb{C}\Gamma$ be the group ring of $\Gamma$ i.e. the set of formal sums $\sum_{t \in \Gamma} \alpha_t t$. Furthermore ...
2
votes
1answer
31 views

If I have that all groups $\mid G \mid < 100 \ncong A_5$ are not simple, does this imply solvable?

If I have that all non-abelian groups $\mid G \mid < 100 \ncong A_5$ are not simple, does this imply that all groups $\mid G \mid < 100 \ncong A_5$ are solvable in a few steps? In other words, ...