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)

1
vote
0answers
26 views

$|H|$ and $|$Aut$(N)|$ are relatively prime. Show that $H$ and $N$ commute.

Can anyone help me with this exercise? "Let $G$ be a group. Let $N$ be a normal subgroup and $H$ a subgroup of $G$. Assume that $|H|$ and $|$Aut$(N)|$ are relatively prime. Show that $H$ and $N$ ...
0
votes
1answer
64 views

Matrix Algebras: Generator

Problem Given the algebra $\mathcal{M}_\mathbb{C}(2)$. Denote the normals: $$\mathcal{N}:=\{N\in\mathcal{M}_\mathbb{C}(2):N^*N=NN^*\}$$ And their calculus: ...
2
votes
2answers
43 views

Permutations: Interpreting Image 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
20 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?
1
vote
0answers
27 views

Free groups: normal supplements of the commutator subgroup

Let $F$ be a free group and let $V$ be another verbal subgroup of $F$ such that $$ F = [F,F] V. $$ Is it true that $V=F?$ More generally, if $N$ is a normal (or even characteristic) subgroup of $F$ ...
7
votes
0answers
60 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 ...
4
votes
1answer
36 views

Cubic Planar Graphs have $2^m-1$ Hamilton Cycles

I looked at the symmetric difference of hamilton cycle (HC) in cubic planar graphs and found that, together with the empty graph, they build a subgroup of the abelian group $\Omega$ of symmetric ...
4
votes
1answer
54 views

Homomorphism from $\mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z}\rightarrow \mathbb{Z}\oplus \mathbb{Z}$ has non-trivial kernel: elementary argument

One can give an elementary arguments (avoiding "rank") to prove that any group homomorphism $f$ from $\mathbb{Z}\oplus \mathbb{Z}$ to $\mathbb{Z}$ has non-trivial kernel: Let $f:(1,0)\mapsto a$ and ...
0
votes
2answers
57 views

Automorphism of $D_6$

I need find an explicit way to express the group $Aut(D_6)$, and I have not idea how write this group, maybe this is an semidirec product of some groups but I don´t see this. thanks.
0
votes
0answers
41 views

Number of all possible groups of given order [duplicate]

Suppose $n=18$, then all possible groups of order $18$ is $5$. Among them $2$ are abelian and $3$ are non-abelian. Let $n$ be a natural number. How can I determine the number of all possible ...
2
votes
1answer
23 views

Homomorphism with intersection of all Sylow p-subgroups as kernel?

Does anyone know of a homomorphism from a group $G$ to another group with kernel as the intersection of all Sylow $p$-subgroups? I was trying to prove that the intersection of Sylow subgroups is ...
-4
votes
0answers
16 views

how to apply tietze transformation? [on hold]

how to prove this isomorphism? $$\langle a,b:a^bb^a=(b^{-1}a^2)^2=e \rangle\cong \langle x,y:x^2=y^3=e\rangle$$
4
votes
0answers
29 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 ...
6
votes
2answers
3k views

How to prove that if $G$ is a group with a subgroup $H$ of index $n$, then $G$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$

I would be very thankful if someone could give me a hint with proving next statement: If $G$ is a group with a subgroup $H$ of finite index $n$, then $G$ has a normal subgroup $K$ contained in $H$ ...
1
vote
0answers
16 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
52 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
39 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?
2
votes
1answer
23 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.
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
35 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 $xH=Hx$ ?
2
votes
1answer
268 views

Graph (or Group) in Astronomy

Is there an application of graph theory (or group theory) in astronomy. If there is, refer me some references.
4
votes
3answers
94 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 ...
2
votes
1answer
35 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 ...
2
votes
1answer
42 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
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 ...
1
vote
0answers
14 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
76 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 ...
1
vote
2answers
76 views

Proving complete reducibility of modular representations

Let $G$ = $S_{3}$ and consider the $3 \times 3 $ permutation representations. For example, we have $$ \psi (123) = \begin{pmatrix} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ ...
2
votes
2answers
24 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 $G$ ...
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?
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
3answers
63 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 ...
2
votes
1answer
84 views

Rank 3 permutation groups

Let $G \leq Sym(\Omega)$ be a finite permutation group of rank 3, $\alpha \in \Omega$ and $g,h \in G$ such that $x_1 := g(\alpha)$ and $x_2 := h(\alpha)$ are not equal. Now my question is: Is there ...
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
61 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
25 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 ...
1
vote
2answers
112 views

More Symmetric than the symmetric groups?

So I was considering the following question. Is there a group of size n! (or less), that contains a larger number of subgroups than $$S_n$$? In thinking about this problem one obvious contender that ...
5
votes
3answers
344 views

Can one construct a “Cayley diagram” that lacks only an inverse?

My group theory text asks for an example of a Cayley-like diagram that exhibits all the properties of a group except (only) that at least some elements lack an inverse. Is it possible to construct ...
6
votes
1answer
121 views

Extension of group with Ext$^{1} (A, B) = 0.$

Are there any infinite torsion free abelian groups $A$ and $B,$ with $A$ is not projective and $B$ is not divisible but $$\text{Ext} ^{1}(A, B) = 0.$$ Thanks
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
30 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
62 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 $A$. ...
-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 ...
4
votes
2answers
162 views

Definitions in a Theorem of Lang

I'm trying to understand the following theorem due to Serge Lang (Algebraic Groups over Finite Fields, 1956, Theorem 2): Let $p$ be a prime, and let $k$ be a finite field of $q=p^n$ elements. Let ...
0
votes
0answers
29 views

A question about primitive idempotent of group algebra [closed]

How to prove: $e_j$ is primitive idempotent of group algebra $\cal{L}$ iff $\forall\ t\in \cal{L}\ $, $e_j^2=e_j$ and $e_j t e_j=\lambda_te_j$. Or in which book can I find the proof.
3
votes
1answer
96 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 ...
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 ...