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

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7
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2answers
89 views

Does every homogeneous space allow a group structure?

Let $(X,\tau)$ be a homogeneous space, that is for all $x,y \in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Is there a group operation $*:X\times X\to X$ such that ...
0
votes
1answer
23 views

Is it true that all proper normal subgroups of $D_{24}$ abelian?

Is it true that all proper normal subgroups of $D_{24}$ abelian ? If Yes, is it true only for $D_{4n}$ groups, or for all $D_{2n}$. I was trying to list all proper normal subgroups of $D_{24}$, Using ...
0
votes
0answers
40 views

How can I have a copy of this old paper? [on hold]

How can I have a copy of this old paper (or a translation of it)? Frobenius, G. (1902). Uber primitive Gruppen des Grades n und der Klasse n - 1. S. B. Akad. Berlin 1902, 455-459.
5
votes
1answer
26 views

Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$

Let $p$ be the smallest prime dividing $|G|$, and suppose that some $P \in \mathsf{Syl}_p(G)$ is cyclic. Prove that $N_G(P) = C_G(P)$. So I let $G=p^\alpha m$ $p$ does not divide $m$. P is cyclic, ...
1
vote
1answer
16 views

Clarification on proof that all groups of order $< 60$ are solvable

I've manged to prove that all groups of order $< 60$ are solvable, using Burnside's theorem. However, I found an alternate proof here Question about solvable groups It states that: "Note that ...
0
votes
1answer
19 views

Number of mutually non isomorphic Abelian groups

Let p and q be distinct primes. How many mutually non-isomorphic Abelian groups are there of order p^2q^4. I think there are 6 of them: p^2q^4 q, qp, q^2p q^2, q^2p^2 p, pq^3 pq, pq^3 q, q^3p^2 in ...
4
votes
1answer
32 views

The behavior of quotient groups under homomorphisms

We're learning normal subgroups, kernels, homomorphisms and isomorphisms in abstract algebra right now. I'm trying to tie the ends together: I know that if $G$ is a group, $N$ a normal subgroup of ...
2
votes
1answer
34 views

Groups of order $2\cdot 31\cdot 61$.

What are all groups (up to isomorphism) of order $2\cdot 31\cdot 61$? Letting $n_p$ be the number of Sylow $p$-subgroups of such a group, $G$, you can show $n_{31}=1$ using the Sylow theorems ...
2
votes
2answers
28 views

Epimorphism that is not surjective in the category of Torsion Free Abelian Groups

In reading about cokernels (relating to a homework question I have) I came across the following: https://www.dpmms.cam.ac.uk/~jg352/pdf/CTSheet4-2013.pdf I specifically wondered about question 5a. ...
0
votes
0answers
17 views

$n_p$ - the largest power of the prime $p$ which divides $n$

I was reading this article called "On A Theorem of Frobenius: Solutions to $x^n=1$ in Finite Groups" by I.M. Isaacs and G.R. Robinson (www.jstor.org/stable/2324902). In the third para of the first ...
1
vote
0answers
35 views

$\phi: G\to G$ is a homomorphism. Denote $Ker \phi := K$ and $ Im\phi := H$. If $K \cap H =\{e\}$, can we say that $G=KH$?

While solving a problem, I came across the following question : Let $G$ be an abelian group. Suppose $\phi: G\to G$ be a homomorphism. Denote $Ker \phi := K$ and $ Im\phi := H$. If $K \cap H ...
0
votes
1answer
20 views

Prove $f(x) = x * a$ is bijective (preferably using inverse)

I have this question that I am stuck at. Let $(G; * ; I)$ be a group and let 'a' in $G$. Let $f : G \rightarrow G$ be the function defined by $f(x) = x * a$ for all $x$ in $G$. Prove that $f$ is ...
0
votes
1answer
28 views

understanding a quotient group

Let $G=\mathbb{Z}\times \mathbb{Z}$ . Let $K$ be the subgroup of $G$ generated by $(3,6)$ and $(3,1)$. Describe the rank and invariant factors of the abelian groups $K$ and $G/K$. My Try: Since $\phi ...
3
votes
1answer
29 views

Fermat's Little Theorem: group and multiplication modulo

$p$ is a prime number. $G$ is a group of integers $\{1,2,\dots,p-1\}$ under multiplication mod $p$. $d$ is a divisor of $(p-1)$ Is it possible to prove that the number of elements $a$ in $G$ such ...
1
vote
0answers
42 views

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 ...
1
vote
0answers
22 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!
0
votes
1answer
25 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 ...
3
votes
1answer
32 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 ...
0
votes
2answers
24 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 ...
2
votes
2answers
112 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 $H=G$ ? $$ $$ ...
2
votes
1answer
56 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 ...
5
votes
2answers
324 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 ...
0
votes
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 ...
3
votes
3answers
48 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
25 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
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 ...
0
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0answers
19 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
22 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
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 ...
0
votes
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) ...
1
vote
2answers
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 ...
2
votes
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 ...
0
votes
0answers
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 $$ G\cong N\times G/N?$$
1
vote
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 ...
3
votes
2answers
70 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
30 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
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?
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
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 ...
5
votes
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 ...
0
votes
0answers
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?
3
votes
1answer
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 ...
2
votes
1answer
33 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, ...
-4
votes
1answer
29 views

When is the permutation group cyclic? [on hold]

When is the permutation group cyclic and when not? Thanks in advance.
1
vote
2answers
15 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
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 = ...
0
votes
1answer
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 :)