The study of symmetry: groups, subgroups, homomorphisms, group actions.
0
votes
1answer
17 views
Understanding equivalent definitions of left cosets
I understand the canonical definition of left coset, but the one I do not understand (or see why it is advantageous) is the definition that follows:
Let $H\leq G$. Then a left coset of $H$ is a ...
2
votes
1answer
27 views
Group action on a manifold with finitely many orbits
I'm looking for a result along the lines of the following:
Let $G$ be a group acting on a set $X$. If the action partitions $X$ into finitely many $G$-orbits, then $\dim G \geq \dim X$.
For ...
5
votes
2answers
54 views
Using the orbit-stabilizer theorem to count graphs
Revising for a course in group and representation theory, I have come across the following interesting problem:
Use the orbit-stabilizer theorem to compute the number of isomorphism classes of ...
1
vote
2answers
30 views
Nitpicky Sylow Subgroup Question
Would we call the trivial subgroup of a finite group $G$ a Sylow-$p$ subgroup if $p \nmid |G|$? Or do we just only look at Sylow-$p$ subgroups as being at least the size $p$ (knowing that a Sylow-$p$ ...
2
votes
1answer
34 views
Order of products of elements in a finite Abelian group
We want to show that if $a,b\in G$ where G is a finite Abelian group, we have
LCM (|a|,|b|) = |ab| given that $ab \neq e$.
How I approached this question was by saying let LCM(|a|,|b|) = L. Then if ...
4
votes
2answers
69 views
Clarification on quotient groups
I've only recently started looking at quotient groups, so I don't know if this question will make sense...
In this wiki article, $G/H$ is defined as the set of left cosets of $H$ in $G$, without any ...
2
votes
1answer
73 views
What is a semi-join ($\ltimes$)?
From its subsection in Relational Algebra on Wikipedia, it is my understanding that:
$$R \ltimes S = \{t | t \in R,\ s \in S:\quad \bowtie(t \cup s)\}$$
Or to use exclusively relational algebra:
...
0
votes
0answers
14 views
How to decide whether a p-subgroup of some sporadic groups is cyclic?
Suppose that H is a subgroup of some sporadic groups (say convey groups Co1, Co2, etc.) and $|H|=p^k$ for some prime p and integer k>1, how to determine whether H is cyclic?
-1
votes
1answer
39 views
How to decide whether a p-subgroup is cyclic？ [closed]
Suppose that H is a subgroup of G and $|H|=p^k$ for some prime p and integer k>1, how to determine whether H is cyclic? Here, G are some sporadic groups, say Co1, Co2, J4, etc.
2
votes
1answer
77 views
A group $G$ is Abelian iff $(ab)^n = a^n b^n$ for all $a,b \in G$ and $n \in \Bbb Z$
Prove that $G$ is Abelian if and only if $(ab)^n =a^n b^n $ for all $a, b \in G$ and $ n \in \mathbb{Z} $.
I used proof by induction in the $ \rightarrow$ direction of the proof and I'm done with ...
3
votes
1answer
35 views
Are homomorphisms into PGL related to the Schur multiplier?
I've been trying to understand homomorphisms from a finite group $G$ into $\operatorname{PGL}(n,R)$ for $n$ a positive integer, and $R$ a commutative ring with 1, usually a field.
I had been under ...
0
votes
0answers
18 views
Supersolvable group and nilpotent maximal subgroup
$G$ is Supersolvable group. $|G|=pq^{b}$ (p,q are difrent prime number). $Q$ is q-Sylow subgroup of $G$.and $Q=\langle x\rangle$ and $Cor_{G}(Q)=\langle x^{q}\rangle$. If $M$ is maximal subgroup of ...
0
votes
1answer
38 views
first-order logic: Any finite set of first-order sentences true in all divisible abelian groups is true in some non-divisable one.
The question is the Subject. This comes from Barwise: Intro to First Order Logic, it is prop 2.1 and it is answered 2 pages later. I however don't understand the proof well.
As Barwise says, the ...
1
vote
0answers
22 views
No. of orbits of certain size
Let $U,V$ be subgroups of $G$ and $G/U = \{ g_1 U,...,g_e U \}$, with $U_i = g_i U g_i^{-1}$.
Assertion: The number of $V$-orbits of $G/U$ having cardinality $1 \leq j \leq e$ is $N_j/j$, where $N_j$ ...
2
votes
1answer
44 views
Abstract Algebra and Parallel Computing
I've recently been learning a bit about parallel computation. Two things I recently learned about are Reduce and Scan.
Where Reduce is defined as 2-Tuple of a Set of elements and a binary operation ...
6
votes
3answers
222 views
Conjugate subgroup strictly contained in the initial subgroup?
Probably a very stupid question:
Let $G$ be a group, $H\subset G$ a subgroup, $a\in G$ an element. Is it possible that $aHa^{-1} \subset H$, but $aHa^{-1} \neq H$? If $H$ has finite index or finite ...
1
vote
2answers
29 views
Application of Cauchy theorem to prove normality of a subgroup
Let $G$ be a group $o(G)=pq$, where $p,q$ are both distinct prime numbers. Let $H<G$ be a subgroup of $G$ and $o(H)=p$. I want to show that $H$ is normal in $G$. My argument goes as follows. First ...
10
votes
6answers
209 views
What is the importance of examples in the study of group theory?
When I study topics in group theory (I am currently following Dummit and Foote) I don't care about examples so much. I read them, try to understand the applications of the theorems and corollaries on ...
0
votes
1answer
38 views
Symmetric group and cyclic permutations
Show that if $n$ is at least $4$, every element of $S_n$ can be written as a product of two permutations, each of which has order $2$. I have no idea to prove the above statement. Any suggestions?
2
votes
2answers
63 views
Group generated by a set of normal subgroups is normal
For an indexing set $I$, if $\{N_i:i \in I\}$ is a set of normal subgroups of a group $G$, then the smallest subgroup containing all the $N_i$ is given by $\langle N_i:i \in I\rangle = \bigcap_j H_j$, ...
4
votes
3answers
135 views
Are Clifford groups very *non-commutative*?
Clifford groups seem to be very non-commutative by the relation \begin{equation}
\gamma_{i}\gamma_{j}=-\gamma_{j}\gamma_{i}.
\end{equation} But is it really so? Can we put this degree of ...
0
votes
0answers
23 views
How many irreducible factors of grade $6$ there is in $\mathbb{F}_{2}\left[ x\right]$?
How many irreducible factors of grade $6$ there is in the polynomial ring $\mathbb{F}_{2}\left[ x\right]$?
I have solved this by using the fact that every irreducible polynomial of grad $i$ is a ...
0
votes
0answers
37 views
Generalizations of fitting subgroup
The fitting subgroup of a group G has two generalizations; one of them is F*(G) and the other is F^~(G).I haven't got any information about the last one! Could you please introduce some useful ...
6
votes
2answers
102 views
if $G$ is a group , $H$ is a subgroup of $G$ and $g\in G$ , Is it possible that $gHg^{-1} \subset H$? [duplicate]
if $G$ is a group , $H$ is a subgroup of $G$ and $g\in G$ , Is it possible that $gHg^{-1} \subset H$ ?
this means , $gHg^{-1}$ is Proper subgroup of $H$ ,
we know that , $H \cong gHg^{-1}$ , so if ...
6
votes
1answer
78 views
Applications of representation theory in physics
The notes of a lecture on basic group and representation theory I attended last semester begin with a bit of motivation for the argument. They give the following examples for applications in physics:
...
2
votes
0answers
46 views
Group of order $11^2\times 13^2$
I am trying to show that a group of order $11^2\times 13^2$ is abelian. I managed to show, by Sylow theorems, that if $G$ has order $11^2\times 13^2$, then $G = H.K$, with $|H|=11^2$, $|K|=13^2$ and ...
0
votes
1answer
45 views
What is wrong with my thinking, simple groups order $168$
How many elements of order $7$ are there in a simple group of order $168$?
I will work on this more but I have seen some solutions out there. My only question is regarding what is wrong what my ...
3
votes
1answer
88 views
Conjugacy of projective representations
Given characters of the Schur covering group of $G$ of the same degree, how does one tell if the projective representations (as homomorphisms from $G$ into $\operatorname{PGL}$) are conjugate in ...
1
vote
2answers
53 views
What powers of an $18$ cycle permutation are also $18$ cycles?
Suppose we are given a permutation $A = (1,2,...,18)$ which is an $18$ cycle in $S_{18}$. We want to find $i \in \mathbb{Z}$ such that $A^i$ is also an $18$ cycle. Now I know how to do this by trial ...
0
votes
1answer
26 views
Consequence of injectivity of projections from covering spaces
We have the theorem which says that the induced homomorphism $p_* : \pi_1(\tilde X,\tilde x_0)\rightarrow \pi_1(X,x_0)$ is injective (hence a monomorphism). Here $\tilde X$ is a covering space of $X$.
...
0
votes
1answer
27 views
Schmidt group and Permute 2-maximal subgroup with 3-maximal subgroup
$G$ is Schmidt group With abelian Sylow subgroup Then every $2$-maximal subgroup of $G$ permuts with all $3$-maximal subgroup of $G$.
3
votes
1answer
50 views
An integer $n$, such that $nx = 0$, where $x$ belongs to the quotient group $\Bbb Q/\Bbb Z$
Well, first, let $x$ be an element of the factor group $\mathbb Q/\mathbb Z$ ($\mathbb Q$ and $\mathbb Z$ are the additive groups, by the way).
Now, how do we find an integer n, such that, $nx = 0$, ...
2
votes
1answer
30 views
About free group and kernel of homomorphism
I'm now reading textbook in group theory but couldn't understand its briefy explanation below
"Let $G=<a,b\mid a^4=e,b^2=e,bab^{-1}=a^{-1}>, S=\{a,b\},F(S)$ be a free group and $N$ be the ...
8
votes
1answer
118 views
Generic properties of $p$-groups
I have the impression that most people in algebra believe the following statement to be true, but I have no reference for it.
Fix a natural number $n$. Consider for each prime $p$ the set of all ...
0
votes
1answer
40 views
Showing that $x^3+\omega x+1$ is irreducible over $\Bbb C(\omega)$
How to show that $f(x)=x^3+\omega x+1$ is irreducible over $\Bbb C(\omega)$?
This is what I have tried:
Suppose on the contrary that: ...
5
votes
2answers
63 views
A representation of a finite group which is not completely reducible
Maschke's theorem says that every finite-dimensional representation of a finite group is completely reducible. Is there a simple example of an infinite-dimensional representation of a finite group ...
0
votes
1answer
14 views
Irreducibility of a module of a cyclic group
Let $G = \langle x \rangle$ be a cyclic group of order $p$ ($p$ is a prime). Let $M$ be a vector space over $\mathbb Q$ with basis $\{m_0, m_1, \cdots, m_{p-1}\}$. Define $\rho(x)$ to be $$\rho(x)m_i ...
0
votes
0answers
32 views
Maximal subgroups of $G=Z_{3}\ltimes Q_{8}$
Let $G=Z_{3}\ltimes Q_{8}$. How can find Maximal subgroups of $G$ ? $$Q_{8}$$ is Quaternion group of order of 8 and $$Z_{3}$$ is cyclic group of order 3
2
votes
0answers
75 views
What is the intuition of conjugacy classes?
How can I fully understand what are conjugacy classes are in groups?
I know the definition, that $a$ and $b$ are conjugate if $gag^{-1}=b$ for some $g\in G$. But what is the intuition? Using a ...
2
votes
2answers
59 views
If $\chi\in\operatorname{Irr}(G)$, $N\unlhd G$, and $\langle\chi_{N},1_{N}\rangle\ne 0$, then $N\subset \operatorname{Ker}(\chi)$.
Let $N \unlhd G$ and $\chi \in \operatorname{Irr}(G)$. Suppose that $\langle\chi_{N},1_{N}\rangle\ne 0$. Show that $N\subset \operatorname{Ker}(\chi)$.
Hint: Use that, for any character ...
1
vote
0answers
23 views
Orthochronous Lorentz is time preserving and SL(2,R)
Let's consider the pseudosphere in $\mathbb{R}^{1,2}$ given by $x^2+y^2-z^2=-R^2.$
We know that the group $O(1,2)=\{ A \in \operatorname{Mat}(3,\mathbb{R}): A^tGA=G \}$ where ...
1
vote
1answer
79 views
What's the difference between Abstract Algebra and Group Theory?
I'm slowly beginning a student of certain higher mathematics.
I'm trying to see if I would prefer to study Group Theory or Abstract Algebra.
I know that Abstract Algebra seems to "come before" ...
2
votes
2answers
51 views
Presentation of a non-abelian group of order $pq$.
What is the presentation of the non-abelian group of order $pq$ where $p$ and $q$ are primes and $q\mid(p-1)$?
Thanks in advance.
1
vote
1answer
58 views
Show $G$ and $H$ are isomorphic
Let $G$ and $H$ be finite abelian groups of the same order $2^n$. If for each integer $m$, $$\left|\left\{x\in G\mid x^{\large 2^m}=1\right\}\right|=\left|\left\{x\in H\mid x^{\large ...
4
votes
1answer
13 views
Infinitesimal $SO(N)$ transformations
An infinitesimal $SO(N)$ transformation matrix can be written :
$$R_{ij} = \delta_{ij}+\theta_{ij}+O(\theta^2)$$
Now it has to be shown that $\theta_{ij}$ is real and anti-symmetric.
I've started ...
1
vote
1answer
135 views
Group of Isomorphisms of a Groupoid
Write a careful proof that every group is the group of isomorphisms of a groupoid. In particular, every group is the group of automorphisms of some object in some category.
First can someone tell me ...
3
votes
2answers
94 views
if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$? [duplicate]
If $A$ is abelian group and $B$ is a subgroup of $A$, $B$ is normal subgroup of $A$.
Is it true that $B \times A/B \cong A$?
I ask because I was watching an online lecture from a course in abstract ...
1
vote
1answer
27 views
How to choose a proper binary operation in a semigroup?
I am interested in generating a finite commutative semigroup which is not a group.
And by generating I mean choosing a number $n$ (number of elements in a semigroup) and then defining a $n \times n$ ...
8
votes
5answers
224 views
Does $(\{0,1\},*)$ form a group?
I am reading my first book on abstract algebra. I am not enrolled in a class on the subject.
Given $S = \{0,1\}$. Is $(S,\cdot)$ a group?
$S$ is closed under multiplication. ...
4
votes
2answers
57 views
bounds for number of groups with given order
The number of different groups with order n is usually called gnu(n).
I found a very good survey about the values. Up to 2047, gnu is
completely known and for many other values n too.
But in ...
