The study of symmetry: groups, subgroups, homomorphisms, group actions.
0
votes
0answers
5 views
Are the quotient groups in a composition sequence necessarily subgroups?
Does there exist a finite group G and a normal subgroup N of G so that G/N is a simple group and G/N is not isomorphic to any subgroup of G ?
0
votes
1answer
48 views
In a transitive action there is a bijection between the fixed points of a stabilizer of a and the lateral clases of the stabilizer in his normalizer
the question is the given above, specially in the case infinite:
If the action of G is transitive, then there is a bijection between the fixed points of the stabilizer of a element a and the lateral ...
2
votes
1answer
47 views
A group with six elements which are given partially by relations.
In a textbook I saw the following Group
$$
G = \{ 1, x, x^2, y, xy, x^2y \}
$$
and it was said that it is the $S_3$, surely the $S_3$ is a model of this group, but when I set $x^3 = y$ (in $S_3$ ...
3
votes
1answer
98 views
Question on groups of order $pq$
Let $G$ be a group of order $pq$, $p>q$ and $p$, $q$ are primes. Then prove that
If $q\mid p-1$ then there exists a non abelian group of order $pq$.
Any two non-abelian groups of ...
2
votes
1answer
36 views
For every monoid $M$ with zero is there a group $G$ such that $\mathrm{Grp}(G,G)\cong M$?
All monoids that I will consider here have identities. A monoid $M$ is said to have a zero iff $\exists z\in M \forall x\in M (zx=xz=z)$.
Let $M$ be any monoid with a zero. Must there exist a group ...
4
votes
1answer
85 views
On Group of order $30$ and $60$.
In this question on yahoo answers ,
the answer says ,
"with $t = 6$, then there are 6 * (5 - 1) = 24 elements of order $5$ "
my question is , how did " 6 * ( 5 - 1 ) " come from ?
Which ...
2
votes
2answers
401 views
Group of order $60$
Let $G$ be a group of order $60$, pick out the true statements:
a. $G$ is abelian
b. $G$ has a subgroup of order $30$.
c. $G$ has subgroups of order $2$, $3$, and $5$.
d. $G$ ...
-3
votes
1answer
31 views
elementary consequences of Lagrange's theorem
Let G be a finite group. if G has an element of order p and an element of order q, where p and q are distinct primes, then the order of G is multiple of pq
-2
votes
2answers
27 views
Counting Cosets
Describe the cosets of the subgroups described:
The subgroup $\langle\frac{1}{2}\rangle$ of $\mathbb{R}^*$, where $\mathbb{R}^*$ is the group of non-zero real numbers with multiplication.
The ...
-2
votes
0answers
22 views
equivalence relations on group
Let $G$ be a group. A relation on $G$ is defined: if $H$ is a group of $G$, let $a\sim b$ iff $a^{-1}b\in H$. Is this the same equivalence relation as if $H$ is a subgroup of $G$, let $a\sim b$ iff ...
3
votes
1answer
68 views
Cocartesian squares in the category of abelian groups.
Recently, I've been doing a recap of (basic) category theory and found an old exercise I seem to be unable to solve. The question is as follows.
Let $A, B$ be abelian groups, $A'<A$ and $B'<B$ ...
3
votes
1answer
28 views
Infinite imprimitive non abelian group?
My new question is
Is there an infinite, imprimitive and non abelian group?
Thank you for the further answers.
4
votes
1answer
82 views
How to understand $\frac{d}{dt}\{(\exp(tX))_*(Y)\}|_{t=0}=[X,Y]$?
Let $G$ be a Lie group on which $X$ and $Y$ are two vector fields. Let $G\xrightarrow{\exp(tX)} G$ be the (Lie theory) exponential map corresponding to $X$. Then of fundamental importance is ...
0
votes
2answers
48 views
Representation Theory. Why does $a^{r}b^{s}a^{t}b^{u} = a^{i}b^{j}$?
I want to know that why we have $a^{r}b^{s}a^{t}b^{u} = a^{i}b^{j}$. Please let me know.
0
votes
1answer
110 views
Is $\{a + bi: a, b \in \mathbb{F}_3\}$ a field?
Is $P = \{a + bi: a, b \in \mathbb{F}_3\}$ a field? This is the question I am having.
First, I list the elements in $\mathbb{F}_3$, which consists of $\overline{0}$, $\overline{1}$ and ...
0
votes
1answer
51 views
Suppose that $G$ is nonabelian. Must $|\mathrm{Out}(G)| = |\mathrm{Aut}(G)|/|\mathrm{Inn}(G)|$?
I would say not. Suppose that $G = D_4$. By the article I found, $D_4$ has 4 outer automorphisms. I understand how reflection comes to play, but $|\mathrm{Out}(G)| \neq ...
0
votes
1answer
107 views
Show that for any two circlines, $X, Y$, there is a Möbius transformation such that $M(X) = Y$
I understand how to show this if both $X, Y$ are lines or are circles, but I'm stuck on what to do if say $X$ is a circle and $Y$ is a line.
First, using the transformations that you would if $X, Y$ ...
3
votes
0answers
43 views
Intuitive understanding of the Reidemeister-Schreier Theorem
I am reading Combinatorial Group Theory by Lyndon and Schupp, and I'm having some trouble getting through the proof of the Reidemeister-Schreier theorem (page 103 in that book) - you can read that ...
8
votes
2answers
70 views
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
I have tried to using the definition but failed.can someone help me ...
14
votes
2answers
104 views
Schur's Lemma in Group Theory
The analogue of well celebrated Schur's Lemma in group theory will be
"If $G$ is a finite simple group, and $\phi$ is a non-identity homomorphism from $G$ to $G$, then $\phi$ is an isomorphism".
...
1
vote
1answer
55 views
If the group if abelian, must all automorphisms be outer automorphisms?
I am having some thought about this question.
Clearly, if the group $G$ is abelian, then $\mathrm{Inn}(G) = {e}$. But what about $\mathrm{Aut}(G)$ and $\mathrm{Out}(G) = ...
3
votes
1answer
42 views
Showing that if $N \le G$ is finite minimal normal and every simple homomorphic image is abelian, then N is elementary abelian
This is part 2 of this question. Unsurprisingly, I'm having some difficulty with it. Hints are much appreciated. Here's the setup again:
"Let $N$ be a finite minimal normal subgroup of a group ...
2
votes
1answer
52 views
About commutators and center o a certain group
Let $G$ be a group and $H, K\unlhd\ G$ (normal subgroups), with $K\leq H$ such that $\left[H, Aut(G)\right]\leq K$. Why does this imply that $H/K\leq Z(G/K)$? $Z(G)$ denotes the center of $G$.
3
votes
0answers
47 views
Discrete subgroups of SU(n) and SO(n).
Thank you very much for your concern. I am in physics background, any simpler but complete explanation would be helpful.
I would like to know whether there is a complete understanding of discrete ...
3
votes
4answers
66 views
How to find all elements of $\mathbb{Z}_{4} \times\mathbb{Z}_{4}/\langle(1,1)\rangle$?
I am studying factor groups, and I saw an example that says
Find all the elements of the factor group $\mathbb{Z}_{4} \times \mathbb{Z}_{4}/\langle(1,1)\rangle$.
I know that the order of ...
5
votes
5answers
368 views
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
Prove that if $G$ is abelian then the set $H$ of all elements of $G$ that are their own inverses is a subgroup of $G$.
Naturally in an abelian group, $ab = ba$ for $a, b \in G$, however I'm not ...
3
votes
1answer
43 views
Action of $S_7$ on the set of $3$-subsets of $\Omega$
Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem:
Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
2
votes
1answer
47 views
Subgroups of $\mathbb{Z}^k$ of finite index $n$
I want to describe all subgroups in $\mathbb{Z}^k$ of finite index $n$.
I have solved it for the case $k=2$. In $\mathbb{Z}^2$, each subgroup of index $n$ corresponds to a matrix $\left( ...
2
votes
1answer
27 views
Irreducible representation of tensor product
Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$.
What are the irreducible representaions of $\varphi_2 \otimes \varphi_3$?
1
vote
2answers
49 views
Suppose a finite group $G$ has an element $a$ which…
I was solving past exam papers and was stuck on the following one:
Suppose a finite group $G$ has an element $a$ which is not the identity such that $a^{20}$
is the identity. Which of the following ...
2
votes
0answers
42 views
The intersection of all $gA$ containing $x$,where $g\in G$ , is a block.
Excuse me for the previous problem in my post,i have a problem with my computer the question is:
Let $G$ transitive in $ X$, $x\in X$ and $\emptyset\neq A\subset X$. Then
$$\bigcap_{x\in ...
7
votes
3answers
240 views
Presentation of Rubik's Cube group
The Rubik's Cube group is the group of permutations of the 20 cubes at the edges and vertices of a Rubik's group (taking into account their specific rotation) which are attainable by succesive ...
13
votes
2answers
115 views
The smallest nontrivial conjugacy class in $S_n$
Find the smallest nontrivial conjugacy class in $S_n$.
For small $n$, the answer is not hard to find:
$$\begin{array}{cc}
n & \text{smallest nontrivial class(es)} \\
1 & \text{none} \\
2 ...
4
votes
0answers
61 views
What do linearly ordered abelian groups look like?
Recently a post on MathOverflow connected a theorem about ordered abelian groups to the axiom of choice, and it made me want to try and play with these objects a bit.
But it appears to me that I ...
2
votes
2answers
105 views
All of the dihedral groups are factor groups of the infinite dihedral group.
Show that $\operatorname{Aut}(\Bbb{Z}) \cong \{\pm 1 \}$ and write $\alpha : \mathbb Z_2 \rightarrow \operatorname{Aut}(\Bbb{Z})$ for the nontrivial homomorphism. The semidirect product $\Bbb{Z} ...
7
votes
0answers
67 views
Ulm and Frattini Subgroups
Let $A$ be an abelian group. We define $U(A)=\cap (nA), n\in \mathbb N$ be the Ulm subgroup of $A$. The Frattini subgroup of $A$ is $\Phi(A)=\cap(pA)$ ($p\in \mathbb P$). I was trying to show that ...
0
votes
0answers
21 views
Conjugacy classes for su(2)
I am wondering how to calculate the conjugacy classes of the Lie algebra su(2). My guess is that they can be easily evaluated under the similarity transformations but I am not sure it that is all to ...
0
votes
2answers
48 views
General Linear Groups with Homomorphisms [closed]
Let $G=\mathrm{GL}_n(\mathbb R)$ and $H=\mathbb R^*$. Let $\phi : G=\mathrm{GL}_n(\mathbb R) \rightarrow \mathbb R^*$ be the map defined by $\phi(A)=\det(A)$. Show that $\phi$ is a group ...
4
votes
3answers
225 views
Show a certain group is contained in a Sylow p-group.
Statement: Let G be a group and p a prime that divides $|G|$. Prove that if $K\le G$ such that $|K|$ is a power of p, K is contained in at least one Sylow p-group.
I just started studying Sylow ...
2
votes
1answer
43 views
Show that the SU(2) group is a Lie group
How can I prove that the SU(2) is a Lie group with the Pauli matrices as generators?
2
votes
1answer
35 views
Group extension of $\mathbb Z_4$ by $\mathbb Z_2$
Let $f : G →\mathbb Z_2$ be an extension of $\mathbb Z_4$ by $\mathbb Z_2$. Suppose that the induced action $α_f :\mathbb Z_2 →\mathbb Z^{\times}_4$ carries the generator of $\mathbb Z_2$ to $−1$. ...
2
votes
1answer
44 views
Function spaces and transitive group actions
Note: this question is really a subquestion of this one, but I decided to ask it separately since it seems it might be attacked first.
Let $B$ be a topological space and $G$ a topological group ...
1
vote
2answers
55 views
When does the isomorphism $G\simeq ker(\phi)\times im(\phi)$? hold?
Suppose you have a group isomorphism given by the first isomorphism theorem:
$G/ker(\phi) \simeq im(\phi)$
What can we say about the group $ker(\phi)\times im(\phi)$? In particular, when does the ...
3
votes
1answer
41 views
Reference request for ordered groups
I've been reading Pete Clark's notes on commutative algebra, and I especially liked section 17 on valuation rings, and ordered groups in particular.
I'm looking for more introductory material ...
0
votes
1answer
55 views
Let $G$ be transitive.Then $\beta\in \operatorname{fix}(G_\alpha)$ implies $G_\alpha = G_\beta$
i am new in this forum.
My question is about group actions
We have a transitive action of $G$ and $\beta$ a element in the fixed points of the stabilizer of another element $\alpha$. Then $\alpha$ ...
2
votes
0answers
41 views
Universal coefficient theorem maps
Let $G$ be a group and $A$ a trivial $G$-module. The Universal Coefficient Theorem yields a split exact sequence
$$0\longrightarrow {\rm Ext}(H_{n-1}(G),A)\longrightarrow H^n(G,A)\longrightarrow {\rm ...
6
votes
0answers
81 views
Orders of elements and homomorphisms.
Corollary 4.6.8 There is a group $G$ of order $n^3$ given by $G= \{b^ic^ja^k \mid 0 ≤ i, j, k < n\}$, where $a$, $b$, and $c$ all have order $n$, and $b$ commutes with $c$, $a$ commutes with ...
0
votes
1answer
58 views
Is this $\mathbb{Z}_2^n$?
What group is formed by binary strings of a fixed length, $n$, and the XOR operation (^)?
For example, we have:
For $n=1$:
A^B = B^A = B
A^A = B^B = A
For ...
0
votes
1answer
35 views
the general linear groups - solvable
I'm doing some experimental mathematics and I'm in the situation where I need an answer to the question:
Consider the general linear groups $\operatorname{GL}(2,10)$, $\operatorname{GL}(4,10)$ (the ...
3
votes
0answers
71 views
How to recover the integral group ring?
I would like to solve the following exercise:
Suppose $R$ is a commutative semisimple ring of characteristic $p^t, t\geq1$, and we have two finite groups $G_1=H_1 \times A_1$ and $G_2=H_2 \times ...
