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

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2
votes
2answers
14 views

Right-angled Artin groups are residually finite

I know that residual finitness of RAAGs (Right-Angled Artin Groups) follows from linearity, but does there exist a more direct proof, maybe simpler?
0
votes
2answers
33 views

generators of groups from exact sequence

Suppose I have a middle term exact sequence of finitely generated abelian groups $G \longrightarrow H \longrightarrow K$. How do I get the generators of $H$ if I know the same for other two groups?
7
votes
4answers
256 views

What is a short exact sequence telling me?

Let's take a short exact sequence of groups $$1\rightarrow A\rightarrow B\rightarrow C\rightarrow 1$$ I understand what it says: the image of each homomorphism is the kernel of the next one, so the ...
-2
votes
1answer
40 views

Elements whose orders are multiple of $p$ [on hold]

Let $G$ be a non-solvable group, $N$ a cyclic subgroup of order $p$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. By these assumption we can conclude that $G$ has elements of orders $p, ...
3
votes
1answer
47 views

Computing values of centralizers in a non-solvable group with a given property

A finite group G satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
2
votes
1answer
28 views

The minimum size of generating set of the external direct product

I have seen the following theorem here: Suppose that $A$ and $B$ are finite groups whose orders are relatively prime to each other, and the minimum size of generating set (i.e., the smallest ...
0
votes
0answers
50 views

Direct product of $G'$ and $Z(G)$ with some conditions

Let $G=G'\times N$ be a non-solvable group, such that $G/N$ is a non-abelian simple group and $N=Z(G)<C_G(N)$ is an abelian normal minimal $p$-subgroup of $G$. What can we say about $G$? In a ...
2
votes
1answer
60 views

Subgroup contained in all other subgroups

This is Problem 2.13.10 from Herstein, Topics in Algebra: Let $G$ be a finite abelian group such that it contains a subgroup $H_0 \neq (e)$ which lies in every subgroup $H\neq (e)$. Prove that $G$ ...
1
vote
1answer
34 views

Elements of orders $2k$, for $k\geq 5$ in a semidirect product

Let $G$ be a non-solvable group, $N$ be an abelian 2-subgroup of $G$ such that $N=C_G(N)$ and $G/N\cong Sz(8)$. Does $G$ has elements of orders $2k$, for $k\geq 5$?
0
votes
0answers
9 views

Factorization of unipotent groups.

How to prove that $U = L_P U_P$? Here $U \subset P$, $P$ is a parabolic subgroup of an algebraic group $G$, $L_P$ is the Levi of $P$ and $U_P$ is the unipotent radical of $P$. Thank you very much.
3
votes
2answers
50 views

Groups reluctant to have infinite subgroup

Is there a group with only one infinite subgroup‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
2
votes
3answers
213 views

What is the meaning of “fix” in field theory?

What is the meaning of "fix" in field theory? Example: I found a definition of field automorphism, A field automorphism fixes the smallest field containing $1$, which is $\Bbb Q$, the rational ...
3
votes
1answer
51 views

Simple groups with cyclic odd Sylow subgroups

I know that every odd Sylow subgroups of $PSL(2,p)$ is cyclic. Is there any other simple group with cyclic odd Sylow subgroups. Thank you in advance
3
votes
0answers
33 views

Proving that a certain maximal subfield has countably infinite index

Let $K$ be a maximal subfield of $\mathbb C$ which does not contain $\sqrt{2}$. I've shown that $\mathbb C$ is an algebraic extension of $K$, and that the Galois group of any finite extension of $K$ ...
9
votes
4answers
147 views

Does every group act faithfully on some group?

Cayley's Theorem shows that every group acts faithfully on some set. In other words, one can find an injective group homomorphism $\sigma: G\to S_{A}$ where $S$ is the set of all bijections on some ...
2
votes
1answer
23 views

Normal subgroups of factor group

Suppose $G$ is a finite group and $N$ is a normal subgroup of $G$. Then subgroups of $G/N$ are of the form $A/G$ for $A\le G$. But how does the normal subgroups of $G/N$ look like? Is it true that $A ...
4
votes
2answers
104 views

If a group has no maximal subgroups then all elements are non-generators? Frattini subgroup characterization

This question is the last leg of an exercise I've been working on in which we characterize the intersection of all maximal subgroups as the subgroup of all non-generators. I've already shown that if a ...
3
votes
1answer
35 views

calculating signature and showing group homomorphism

I got this question in my group theory book. I think I understand the theory behind it but can't seem to use it to get a solution im happy with. Let $V = M_2(\mathbb F)$. For $x,y \in V$ define ...
3
votes
1answer
61 views

Textbooks on graph theory

I've read the textbook Groups and Their Graphs by Grossman, and I'm interested in learning more about graphs. I know about O. Ore's book in the same series (Graphs and Their Uses), but I'm interested ...
0
votes
1answer
20 views

torsion and torsion-free groups

I have the following statement: Every finitely-generated abelian group $G$ is isomorphic to $T\bigoplus F$, where T and F are torsion and free groups. As an example is given that all abelian groups ...
-2
votes
0answers
38 views

Order of the elements of a right coset [on hold]

What can we say about order of the elements of a right coset in a finite group.
3
votes
1answer
49 views

Linear representation of a free group

I need to prove the following: "Prove that the free group of rank 2 is linear." So to my best understanding, and please correct me if I'm wrong: I actually need to show a homomorphism from the free ...
7
votes
1answer
126 views

A question about Sylow subgroups

Let $G$ be a finite group and $P\neq\{e\}$ be a Sylow $p$-subgroup of $G$ and $P^g\neq P$ be its conjugate in $G$. If we know that $P\cap P^g\neq \{e\}$, can we conclude that $Z(P)\cap Z(P^g)\neq ...
0
votes
0answers
35 views

Finite generated abelian group $G$ and $H<G$. What is the rank of $(G/H)/(G/H)_t$?

I saw another question about this problem here. However there are quite different answers from my expectation. Anyway, here are my trials. Trial 1 : By structure theorem, $G\cong G_t\oplus F_1$ ...
3
votes
1answer
38 views

Computing the order of the first cohomology group $|H^1(S_n, \mathbb F_p^n)|$

Assume $n\geq 3$, $p$ is a prime, and that $S_n$ acts on $V=\mathbb F_p^n$ by permuting the basis vectors $v_1,\ldots, v_n$. I want to compute the order of the first cohomology group of this action. ...
2
votes
0answers
52 views

Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...
3
votes
1answer
31 views

Proving the Thompson Transfer Lemma

Let $G$ be a finite group of even order $n=2^kr$, $T$ a Sylow-$2$ subgroup of $G$, and $M$ an index $2$ subgroup of $T$. I want to show that if $G$ has no subgroup of index $2$, then every element $x$ ...
0
votes
0answers
28 views

An error in least square optimization problem in Matlab

I am new to MATLAB and I want to formulate the following lease square expression in Matlab. I have some codes that I am typing here. But the optimization problem solution seems not to be correct. Does ...
0
votes
2answers
30 views

Isomorphism between $G$ and $S$ Implies that $_0(G)=_0(S)$? [on hold]

If two finite groups $G$ and $S$ are isomorphic then is it neccesary that $_0(G)=_0(S)$?
0
votes
0answers
26 views

Part of Zassenhaus’s butterfly lemma [on hold]

Let $Y,B$ be subgroups of $G$ , $X$ be a normal subgroup of $Y$ and $A$ be a normal subgroup of $B$ , then how do we prove that $(Y \cap A)X$ is normal in $(Y \cap B)X$ ? Please help
0
votes
0answers
28 views

Must the index $k=|G:HC_G(x)|$ be finite?

I want to solve the following Exercise from Dummit & Foote's Abstract Algebra text: Assume $H$ is a normal subgroup of $G$, $\mathcal{K}$ is a conjugacy class of $G$ contained in $H$ and $x ...
2
votes
1answer
46 views

If direct limits of matrices are isomorphic, is the direct limit of the transpose matrices also isomorphic?

On the one hand, the following conjecture seems reasonable, but on the other hand it doesn't seem natural because some objects are being dualised while others are not. I would appreciate if anyone ...
2
votes
2answers
42 views

I.N. Herstein, “Topics in algebra” group theory section 2.8 example 2.8.1

I.N. Herstein, "Topics in algebra" group theory section 2.8 example 2.8.1 it is written that Let $G$ be a finite cyclic group of order $r$, $G=(a)$, $a^r=e$. Suppose $T$ is an automorphism of $G$. If ...
6
votes
1answer
45 views

Definition of the normalizer of a subgroup

Let $G$ be a group and $H$ a subgroup of $G$. Is there any counterexample to the assertion $N_G(H):=\{g\in G\mid gHg^{-1}=H\}=\{g\in G\mid gHg^{-1}\subset H\}$? Thanks!
1
vote
1answer
40 views

Help with semidirect product

I need help with this problem, i am trying to understand the semidirect product, so if anyine could help or give me some ideas Let $G$ be the group generated by $<a,b>$ and the relations ...
0
votes
3answers
44 views

Subgroups of Sufficiently Large Symmetric Groups / Cayley's Theorem explanation

Here's the question: is every finite group a subgroup of a symmetric group of sufficiently large order? More specifically, if a group $G$ has order $n$, then is it true that $G \le S_{n}$? For ...
1
vote
0answers
22 views

For a discrete abelian group $G$, is the Gelfand Representation of $\ell^1(G)$ injective?

Given a discrete group $G$, we can consider the Banach $*$-algebra $\ell^1(G)$, with convolution product $(\xi*\eta)(g)=\sum_{h\in G}\xi(h)\eta(h^{-1}g)$ and involution ...
0
votes
1answer
68 views

a potential application of the ping-pong lemma?

From my understanding, a simple result of the ping-pong lemma would state that if we have a set of linear transformations (matrices) $A_1,\ldots,A_n$ all of the same dimension, then if ...
0
votes
0answers
28 views

Can a compact topological group have the trivial topology?

I want to show that the following are equivalent for a compact topological group $G$: $G$ is the inverse limit of finite groups $G_i$. There's a family $\left\{N_i\right\}$ of open normal subgroups ...
3
votes
0answers
64 views

The order of the representative elements of conjugacy classes

Suppose $A$ is an arbitrary subset of group $G$ and $K_G(A)$ be the number of $G$-conjugacy classes contained in $A$. A finite group $G$ satisfies property $P_n$ if for every prime integer $p$, $G$ ...
1
vote
0answers
48 views

Order Preserving Isomorphism

If abelian group G has an archimedean order then there is an order preserving isomorphism $\phi$ of G onto a subgroup of $\mathbb{R}$. Here we can say that G is archimedean totally ordered abelian ...
2
votes
1answer
51 views

groups generated by two elements of order 3

I'd like to know whether it is possible to find a characterization (cardinality?) of the set of finite, non-abelian groups generated by two elements $a$ and $b$ whose order is $3$? Is it the same task ...
9
votes
2answers
96 views

Subgroups of $\mathrm{GL}(n,\mathbb{Z})$ which are not finitely generated

The group $\mathrm{GL}(n,\mathbb{Z})$ is finitely generated: take for example diagonal matrices, permutations and one elementary matrix (upper triangular). Are there some simple / nice examples of ...
1
vote
0answers
30 views

representation of extraspecial group

Let $H=\langle x,y\rangle$ be a non abelian group of order $p^3$ which has exponent equal to $p$. Also let $G$ be an extraspecial group of order $p^{2r+1}$ exponent $p$. We know that $G$ is isomorphic ...
7
votes
1answer
102 views
+100

The largest value of $k$ for $\Bbb{Z}^{k}$ to be embedded in $\mathcal{GL}(n,\Bbb{Z})$.

Reading my course on group theory, I asked my self the following question : Suppose that $\Bbb{Z}^{k}$ can be embedded in $\mathcal{GL}(n,\Bbb{Z})$. What is the largest value of $k$?
0
votes
0answers
39 views

Homology groups of $SL(2,\mathbb Z)$

I am reading Brown's book "Cohomology of Groups" and I can't solve exercise II.7.1.3.: "It's a classical fact that $SL_2(\mathbb Z) \cong \mathbb Z_6 *_{\mathbb Z_2}\mathbb Z_4.$ Use Mayer-Vietoris ...
0
votes
2answers
31 views

order of dihedral

I am learning abstract algebra, and I don't quite understand the order of the symmetry of dihedral. When you look at a squares, I agree that there will be 8 symmetry. But all the operations have cycle ...
3
votes
0answers
36 views

an elementary problem on wreath product groups with combinatorial flavor

Embarrassingly, I got stuck in solving the following elementary exercise. Let $G=H\wr \Gamma$ be a wreath product groups, $H,\Gamma$ are countable discrete groups, when $\xi\in\oplus_{\Gamma}H$, then ...
0
votes
3answers
28 views

How do you compute the inverse of the following permutation?

g = Row#1 (1 2 3 4 5 6) Row #2( 2 3 1 6 5 4) How do you compute g inverse and what is the identity of g?
0
votes
1answer
45 views

The relation between orders in a group

G is a group and N is a normal subgroup of G.what is the relation between the order of $x$ and $x.N$?