A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

learn more… | top users | synonyms (2)

2
votes
0answers
10 views

Why is every subgroup of a finitely generated nilpotent group closed in the profinite topology?

This should be a well known claim, but what is the proof? Why is every subgroup of a finitely generated nilpotent group closed in the profinite topology?
4
votes
0answers
22 views

Basic application of Weyl-Character-Formula

(I did not find a solution of my problem in any forum so far. Sorry if it exists...) I am new to Lie-Algebras and representations and actually do not need the mathematical background... I need only ...
1
vote
0answers
12 views

Hypercenter is the intersection of normalizers of Sylow subgroups.

I'm trying to prove that the intersection of the normalizers of the Sylow subgroups of a [finite] group $G$ is equal to its hypercenter, i.e., $$Z_\infty(G)=\bigcap\limits_{S\in ...
0
votes
0answers
49 views

Any hint on : Every $A_{n}$ elemnt is $n$-cycles product. [on hold]

[Added explanation] I found this exercise as follows in Hungerford : Abstract algebra (3rd edition) page 236, exercise number 40. Stated as follows : C.40. Prove that every element of $A_{n}$ is ...
1
vote
2answers
34 views

Generators in group $Z^*_{p}$

show that $g=2$ is a generator of group $Z^*_{19}$ Can anyone explain me how i can show in this example and generally that an element is a generator in a group?
0
votes
1answer
42 views

Hint to find the order of the group of $2\times 2$ matrices under multiplication [duplicate]

Let $G$ be the group of all $2\times 2$ matrices \begin{bmatrix}a&b\\c&d\end{bmatrix} where $a,b,c,d$ are integers modulo $p$ for $p$ prime such that $ad-bc\not =0$.$G$ forms a group relative ...
2
votes
1answer
34 views

$G$ be a group of order $p^n$ and $H$ be any subgroup of $G$; then does there exist $x \in G\setminus H$ , such that $xH=Hx$?

Let $G$ be a group of order $p^n$ , where $p$ is a prime and $n \in\mathbb N$ and $H$ be any subgroup of $G$; then does there exist $x \in G\setminus H$ , such that $xH=Hx$ ?
6
votes
0answers
36 views

index 2 subgroups of the infinite product of Z/2Z

is it possible to describe all the index 2 subgroups of the group $G = \prod_{i\in \mathbb{N}}\; \mathbb{Z}/2\mathbb{Z}$? For example, one can take the kernel of the $i$-th projection map ...
1
vote
2answers
25 views

Permutations: Interpreting Cycle Notation

I have a problem in interpreting permutation. I think the definition and my interpretation of it don't match each other. Let $\sigma=(1\ 2\ 4\ 3)$, and $\tau=(1\ 3\ 2\ 4)$ in one-line notation. I ...
2
votes
1answer
21 views

Can $\mathrm{PGL}_2$ be viewed as an affine algebraic group?

I was just wondering whether or not it is possible to view $\mathrm{PGL}_2$ as an affine algebraic group.
2
votes
1answer
32 views

On the number of Sylow subgroups in Symmetric Group

If $G$ is a finite group, and $P$ is a Sylow-$p$ subgroup of $G$, then the number of Sylow-$p$ subgroups in $G$ is at most $|G|/|P|$. In the Symmetric group $S_n$, the bound is attained only for ...
1
vote
0answers
10 views

Relation between torsion in torsion free of covariant derivative and torsion free group

Is there a relationship between "torsion free" of covariant derivatives and the torsion free group? Or is this just coincidence that people use the term "torsion free" here? It is in general required ...
1
vote
1answer
34 views

Presentation of the symmetric group of 5 symbols.

I am trying to write the presentation of the symmetric group $S_{5}$. We know that $S_{5}$ is generated by $a=(1,2)$ and $b=(1,2,3,4,5)$. Using this I am trying to write presentation of $S_{5}$. My ...
3
votes
2answers
34 views

For any $A, B \in SL(2, F)$, does knowing $\operatorname{tr}A$, $\operatorname{tr}B$, and $\operatorname{tr}AB$ specify $A$ and $B$?

In title, $F$ denotes a field. Does knowing the trace of two matrices and their product specify those two matrices? Up to some equivalence, perhaps?
2
votes
2answers
22 views

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle$.

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle $. I would like someone to check my solution. First of i will prove that $G$ ...
1
vote
2answers
44 views

The point of writting this isomorphism theorem like this?

In group theory there is this isomorphism theorem that doesn't seem to give any special information the way it is written. Let $T\unlhd G$ and let $S\leq G$ then $\frac{S}{S\cap T}\cong ...
2
votes
1answer
39 views

Determine the galois group of $x^5+sx^3+t$

im trying to show that the galois group of $x^5+sx^3+t$ over $\mathbb{Q(s,t)}$ is $S_5$. By just looking at the discriminant, it has to be $S_5$ or $F_{20}$. I know i could distinguish between those 2 ...
1
vote
1answer
60 views

Necessity of being well-defined in Group Homomorphism?

In Group Theory, homomorphism is isomorphism when we no longer restrict to bijective map; do we still need that map to be well defined in homomorphism (like in isomorphism) or homomorphism can be ...
1
vote
0answers
24 views

How many permutations cover alternating/reverse alternating permutations?

Given integers $1$ through $2n$, let $S$ be set of ordering of integers that respect even alternating or reverse alternating permutations (https://en.wikipedia.org/wiki/Alternating_permutation) up to ...
2
votes
2answers
103 views

How to prove the group $G$ is abelian?

Question: Assume $G$ is a group of order $pq$, where $p$ and $q$ are primes (not necessarily distinct) with $p\leqslant q$. If $p$ does not divide $q-1$, then $G$ is Abelian. I know that if the ...
1
vote
1answer
46 views

Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.

I am teaching myself information about groups, but don't really understand how to work through this problem. Here is what I have been thinking so far (please note that I do not need to work through ...
0
votes
0answers
28 views

Are permutation group block only defined in the context of finite sets?

From Dummit and Foote (emphasis mine): Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ s.t. $\forall \sigma\in G:$ either $\sigma(B)\cap B ...
0
votes
1answer
31 views

A question about normal subgroups of nilpotent group

Assume G is a nilpotent group and to any n dividing $|G|$, if there is always a normal subgroup of G with order n?
5
votes
2answers
61 views

Show that $f$ is a homomorphism.

There is a group $G$ of order $p^3$, where $p>2$. Show that $f:G\rightarrow Z(G) $ with $f(x)=x^p$ is a homomorphism. My attempt: Case a): Suppose $|Z(G)|=p^3$. Then $G=Z(G)$, so $G$ is abelian, ...
2
votes
1answer
28 views

Where do I use that $G$ is a permuation group?

This is about question $4.1.7$ from Dummit and Foote, and also related to my previous question. The question is (summarised a bit): Let $G$ be a transitive permutation group on a finite set $A$. ...
2
votes
1answer
34 views

How to read $[G:N]$?

For a group $G$ and a normal subgroup $N$ of $G$, the quotient group of $N$ in $G$ is written $G/N$. I could find from this link how to read $G/N$ ("$G$ modulo $N$" or "$G$ mod $N$"), but I couldn't ...
-3
votes
0answers
32 views

How to prove the theorem on group algebra of permutation group [on hold]

How to prove the following theorem: If $t$ is a vector in group algebra of permutation group, then $\cal{y}t\cal{y}=\lambda_t \cal{y}$, where $\cal{}y$ is the Young operator of permutation group and ...
3
votes
1answer
93 views

Subring of $M_7(\mathbb{Z}_2)$ isomorphic to $\mathbb{F}_{128}$?

Let $A \subset M_7(\mathbb{Z}_2)$ be a subring such that no proper nonzero subgroup $V \subset \mathbb{Z}_2^7$ is invariant under all matrices in $A$. I suspect that $A \cong \mathbb{F}_{128}$, but ...
4
votes
2answers
84 views

Algebraic proof that a set generated by irrational rotations is dense in $S^1$.

This is exercise 1.9 in Lie Groups, Lie Algebras and Representations - Hall. Suppose $a$ is an irrational real number. Show that the set $E_a$ of the numbers of the form $e^{2\pi i n a}$, $n \in ...
2
votes
3answers
61 views

Defining a coproduct in $\mathsf{Grp}$ using group presentations

I've encountered this exercise in Aluffi's Algebra: Chapter 0. It might be helpful to say that the book doesn't introduce functors at this stage,, and that the book defines a presentation of a group ...
0
votes
1answer
44 views

Prove that $(\mathbb{Z}_n , +)$, the integers (mod $n$) under addition, is a group.

Prove that $(\mathbb{Z}_n , +)$, the integers (mod $n$) under addition, is a group. To show that this is a group, I know I need to show three things (in our text, we do not need to show that addition ...
3
votes
1answer
37 views

Difference between “$G$ acts on $A$” and “G is a permutation group on $A$ (i.e. $G\leq S_A$)”

This question is inspired by questions $4.1.1$ and $4.1.2$ of Dummit and Foote. The hypothesis for the first question is formulated as: "Let $G$ act on the set $A$", and the hypothesis for the second ...
4
votes
0answers
69 views

First appearance of modern definition of a group [migrated]

What is the first appearance in print of the modern definition of an abstract group? To qualify, it should be a formal definition, contain the word "elements" (so Burnside's 1897 restriction to ...
0
votes
0answers
33 views

Commutator subgroup of general linear group [on hold]

Let $G$ and $S$ be the group of all invertible $n\times n$ matrices and invertible matrices with determinant $1$ of the same order respectively over the field of real numbers. Prove that $S$ is ...
-2
votes
1answer
47 views

topic between algebra and geometry [on hold]

I have to do an exam on Differential Geometry and my teacher wants that I prepare a choosen topic, outside lectures program, that I will talk about at the oral part of the exam. I am interested in ...
1
vote
0answers
38 views

Number of conjugacy classes of nonabelian group of order $pq$.

The problem asks to show that a nonabelian group of order $pq$ has $p+\frac{q-1}{p}$ conjugacy classes. I have shown a. $p$ divides $q-1$, b. $|Z(G)| = 1$, Now I'm using the class equation to ...
-2
votes
1answer
31 views

H subgroup of G such that H=Inn(G)=Z(G) [on hold]

Let $G$ be a group such that the condition in the title is fulfilled. What can be said about $H$? Is it finite? Cyclic? Many thanks in advance.
1
vote
0answers
10 views

Weyl (anti-)invariant differential operators on spheres

The permutation group $S_n$ acts as via Weyl reflections of $A_{n-1}$ on $R^{n-1}$ und thus on the sphere $S^{n-2}$. On this sphere, we have a natural action of $SO(n-1)$ generated by the angular ...
0
votes
1answer
28 views

The set of nonneg integral powers of $2$ is a group under $\max(a,b)/\min(a,b)$ [on hold]

Let $X$ be the set of all nonnegative integeral powers of $2$. Prove or disprove: $X$ is a group under the operation $a . b = \max(a,b)/\min(a,b)$
3
votes
3answers
45 views

Why is $\phi : \operatorname{Hom}(\mathbb{Z},G) \to G$ given by $ f \mapsto f(1)$ surjective?

I was working on showing $\operatorname{Hom}(\mathbb{Z},G) \cong G$ for $G$ abelian. The proposed map given by evaluating a given $f \in \operatorname{Hom}(\mathbb{Z},G)$ at $1$ is easily seem to be a ...
2
votes
2answers
40 views

In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian

Let $G$ be a group. If for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then phow can I prove that $G$ is abelian?
4
votes
2answers
45 views

In a group $G$, prove the following result

Let $G$ be a group in which $a^5=e$ and $aba^{-1}=b^m$ for some positive integer $m$, and some $a,b\in G$. Then prove that $b^{m^5-1}=e$. Progress $$aba^{-1}=b^m\Rightarrow ab^ma^{-1}=b^{m^2}$$ ...
1
vote
0answers
29 views

Show that in a finite abelian group $G=\{a_1,\dots, a_n\}$, and $x=a_1\dots a_n$, then $x^2=e$ [duplicate]

Let $G=\{a_1, a_2, \dots, a_n\}$ be a finite abelian group and $x=a_1a_2\dots a_n$. Then show that $x^2=e$. Let $a_i\in G$, then $a^{-1}_i\in G$. Suppose $a_i^{-1}=a_j$. How can show the required ...
1
vote
0answers
32 views

Possible orders of the elements of the alternating group $A_n$

What are the possible orders of the elements of the alternating group $A_n$?
4
votes
3answers
92 views

Why Composition and Dihedral Group have reverse order of operation?

NOTE - I didn't receive any answer in here and I think because my first post is not clear, so I entirely made another example: $K={\{id,r^2,r^4,s,r^2s,r^4s}\}$ is a proper subgroup of the dihedral ...
0
votes
1answer
38 views

Prove that $H\times K \cong K\times H$

According to the book: Let $G$ be the internal direct product of subgroups $H$ and $K$. Then $G$ is isomorphic to $H\times K$. From that it results $H\times K \cong K\times H$. Is there any ...
0
votes
0answers
19 views

Discrete series representations of the group SO(2n,1)

I am interested in discrete series representation of the group SO(2n, 1). Can someone recommend me a paper or a book about it?
0
votes
0answers
26 views

On the definition of Internal Direct Products

Let $G$ be a group with subgroups $H$ and $K$ satisfying the following conditions. • $G=HK={\{hk:h∈H,k∈K}\}$; • $H∩K={\{e}\}$; • $hk=kh\ \text {for all}\ k∈K \text {and}\ h∈H$. My question is: ...
1
vote
1answer
75 views

Intuitively and Mathematically Understanding the Order of Actions in Permutation GP vs in Dihereal GP

I define $r$ to be one rotation clockwise, and s to be reflection on the 'horizontal' line (see the figure). So I can make these bijections: (in clockwise order) $$\begin{align*} 1,2,3,4,5,6 ...