The study of symmetry: groups, subgroups, homomorphisms, group actions.
2
votes
1answer
41 views
prove that a non-abelian group of order $10$ must have a subgroup of order $5$.
prove that a non-abelian group of order $10$ must have a subgroup of order $5$.
using Cauchy's theorem proof is easy but how can I do this without using this?
0
votes
1answer
26 views
Simple meaning to Center of a group
Recently I was learning Center of groups and on referencing the group table, I observed is that all the rows that are also present as columns are the centers of any group.
So, I made a small program ...
0
votes
1answer
28 views
Invariant subspaces of tensor product of SU(2)
Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$.
I know that irreducible representations of $\varphi_2 \otimes \varphi_3 = \varphi_5 \oplus ...
1
vote
2answers
39 views
Give an example of the $a,b,c$ which satisfies conditions in the generating set
How to derive the specific case of the generating element of a group given its generating set. For example, when
$$G=\langle a,b,c|a^3=b^3=c^2=1,ab=ba,ca=a^2c,cb=b^2c\rangle$$
we can let $G\subset ...
1
vote
0answers
17 views
Dihedral and quaternion groups as subgroups of SO(n), SU(n), Spin(n), SO(n)$\times$SO(n), SU(n)$\times$SU(n)
This is a very simple question on whether these three discrete groups $D_4$,$Q_8$,$(\mathbb{Z}_2)^3$ are subgroups of certain Lie groups.
More precisely, given discrete groups below (a), (b), (c):
...
5
votes
3answers
47 views
Order of the largest cyclic subgroup of $\mathrm{Aut}(\mathbb{Z}_{720})$
Back to easier problems for a bit...
I have been told that it is possible to find the order of the largest cyclic subgroup of $\mathrm{Aut}(\mathbb{Z}_{720})$ without considering automorphisms of ...
1
vote
1answer
30 views
Computing the order, inverse, and parity of a permutation
How do you compute the order, inverse and parity of $\alpha=(12)(43)(13542)(15)(13)(23)$? Please explain all steps taken to get the answer.
I guess my thought process was to first put it into a ...
3
votes
1answer
54 views
Show that any nonabelian group $G$ of order $12$ which contains a normal subgroup of order $4$ must be isomorphic to $A_4$.
Show that any nonabelian group $G$ of order $12$ which contains a normal subgroup of order $4$ must be isomorphic to $A_4$.
Hint. Show that $G$ is a split extension of a group of order $4$ by ...
9
votes
1answer
70 views
a group is not the union of two proper subgroups - how to internalize this into other categories?
A well-known fact from group theory is that a group cannot be the union of two proper subgroups. I wonder, does this statement internalize into other categories than the category of sets? That is, is ...
4
votes
0answers
50 views
What is a simple axiomatisation of groups using division?
I recall from an old exercise I did as an undergrad that groups can be axiomatised using division rather than multiplication:
A group is a non-empty set equipped with a binary division operator / ...
5
votes
2answers
57 views
Examples of non-cyclic group with a cyclic automorphism group
In introduction to algebra we got the exercise:
Let $G$ be a group. Show that when $\operatorname{Aut}(G)$ is cyclic $G$ is abelian.
This doesn't make that much trouble. Denote the center (all ...
0
votes
0answers
21 views
Total differential of a map between SU(N) and its quotient group w.r.t. the diagonal subgroup
I am trying to figure out the following statement. Consider SU(N) and its normal subgroup T consisting of diagonal matrices. Then define the map
$\pi : SU(N)/T \times T \rightarrow SU(N) : (xT, t) ...
7
votes
1answer
42 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 ?
2
votes
1answer
52 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 $S_3$ ...
2
votes
1answer
46 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 ...
-1
votes
1answer
43 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
-1
votes
0answers
24 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 ...
-1
votes
2answers
30 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 ...
3
votes
1answer
34 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.
0
votes
2answers
51 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.
3
votes
1answer
70 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 $B'<B$ ...
1
vote
1answer
55 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 ...
8
votes
2answers
83 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 ...
1
vote
1answer
57 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
0answers
44 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 ...
2
votes
1answer
49 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( ...
3
votes
1answer
45 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
30 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
50 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
49 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 ...
4
votes
0answers
62 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 ...
14
votes
2answers
108 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".
...
3
votes
0answers
52 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 ...
0
votes
0answers
23 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
50 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 ...
7
votes
0answers
71 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 ...
2
votes
1answer
44 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 $−1$. ...
0
votes
1answer
52 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 ...
1
vote
2answers
56 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 ...
2
votes
1answer
47 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 ...
0
votes
1answer
58 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 $\alpha$ ...
6
votes
0answers
83 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 ...
4
votes
1answer
83 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 ...
2
votes
1answer
52 views
Proving that $X$ is a subgroup of $G$
If we're given that some $X\subset G $ such that $e\in X$, and $\forall g \in G$, cosets $gX$ partition $G$, is $X$ a subgroup of $G$?
I'm not quite sure what it is I have to do. If $X$ wasn't a ...
1
vote
0answers
52 views
Relationship between curls, gradients, and divergences; and the Isomorphism Theorem
I am trying to develop a geometric intuition for the relationship between the curl, the gradient, and the divergence based on the Isomorphism Theorem, where the Isomorphism Theorem says that "If ...
3
votes
2answers
61 views
Find $\mathrm{Aut}(G)$, $\mathrm{Inn}(G)$ and $\mathrm{Aut}(G)/\mathrm{Inn}(G)$ for $G = D_4$
Problem
Find $\mathrm{Aut}(G)$, $\mathrm{Inn}(G)$ and $\mathrm{Aut}(G)/\mathrm{Inn}(G)$ for $G = D_4$
My Attempt
I let $D_4 = \{e, x, y, y^2, y^3, xy, xy^2, xy^3\}$
I found that $\mathrm{Inn}(G)$ ...
2
votes
2answers
109 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} ...
