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

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Find this example

Let $H=\{e,(13)\}$ be a subgroup of $S_3$. Find element $a,b \in S_3$ where $bh_2ah_1 \in aH$ but $bH\ne H$. $h_1$ and $h_2$ are elements in $H$. My friend thinks that it is (123) and (132), but I ...
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10 views

Projective representaions of $(\mathbb{Z}/3\mathbb{Z})^2$

I have a very short question: is there a faithful projective representaion $\rho: \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\to {\rm PGL}(4,\mathbb R)$? Thanks!
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1answer
13 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 ...
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1answer
27 views

Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
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23 views

Question on order of elements in groups (subgroups)

I am a bit confused at the moment, but what can we say about the order of all elements in a finite (sub)group? Suppose we have a group $G$ such that $|G|=p^k$ for a prime $p$. Next let $H$ be a ...
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2answers
107 views

Group theory question for cyclic group

I met some problem during googling. The problem and its solution are next. and I'm wondering about 2nd YELLOW BOX $$ $$ $$ $$ Why $G$ has a unique element of order 2 in case of $H=G$ ? $$ $$ ...
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1answer
46 views

Number theory / Group theory: consecutive integers divisible by at least n prime numbers

Claim: There exist 15,251 successive positive integers $a_1, a_2\dots,a_{15251}$ such that each $a_i$ where ($1\le i\le 15251$) is divisible by at least 251 different prime numbers Is there a neat ...
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312 views

Group theory: subset of a finite group

Given $G$ be a finite group $X$ is a subset of group $G$ $|X| > \frac{|G|}{2}$ I noticed that any element in $G$ can be expressed as the product of 2 elements in $X$. Is there a valid way to ...
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1answer
19 views

Splitting a short exact sequence of orthogonal groups

How does one split the short exact sequence $$1 \rightarrow SO_n(\mathbb{R}) \rightarrow O_n(\mathbb{R}) \rightarrow \{\pm 1\} \rightarrow 1$$ ? I understand that there needs to be an injective ...
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3answers
44 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 ...
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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).
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22 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 ...
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18 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 ...
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20 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 ...
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2answers
17 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 ...
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1answer
14 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) ...
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23 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 ...
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1answer
23 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 ...
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35 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 $$ G\cong N\times G/N?$$
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1answer
38 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 ...
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2answers
65 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 ...
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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 ...
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29 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 $n = 1$, $(\mathbb F_p,+)$ ...
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30 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?
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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 $x' \in K$?
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1answer
33 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 ...
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2answers
38 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 ...
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35 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?
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62 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 ...
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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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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 ...
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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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1answer
29 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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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?
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38 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 = ...
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25 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.
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37 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 $$G\to\text{Bij}(G/H),\;a\mapsto(b\mapsto abH)$$
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38 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 ...
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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 $$ ...
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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 ...
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220 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 ...
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33 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 ...
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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 ...
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2answers
54 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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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 ...
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25 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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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 ...
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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 ...
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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 $\phi(g) = h$?