Questions tagged [group-theory]
For questions about the study of algebraic structures consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility.
42,008
questions
1
vote
0answers
9 views
Great Orthogonality Theorem Identity (group theory)
Let G be a group of or order n with Γ matrix representations. We define the vector $\psi^{(\Gamma,i,k)}_g=\sqrt{\frac{d_{\Gamma}}{n}}D^{\Gamma}_{ik}(g)$ where $(Γ,i,k)$ is a label, with $i, k ∈
1, . . ...
0
votes
2answers
14 views
Words in the alphabet of minimal generators of group
Let us suppose that $A=\{a,b,\ldots z\}$ is a minimal set of generators for a finite group $G$. Then, does the sequence $a,ab,abc,\ldots, abc\ldots z, abc\ldots za, abc\ldots zab,\ldots\}$ produce all ...
0
votes
1answer
39 views
Why the following implies that $G \cong H$?
If I have the following sequence:
$$J \xrightarrow{v} H \xrightarrow{u} G \rightarrow 0 \qquad (1)$$
Where $v$ is the zero map.
Does that mean $G \cong H$ i.e., $G$ isomorphic to $H$?if so why?
Or, do ...
-3
votes
1answer
34 views
If $G$ is an infinite group, what can you say about the number of elements of order $n$ in the group? [closed]
I just started with group theory. I know that for any finite group $G$, the number of elements of order $n$ in group $G$ will be multiple of $\phi(n)$ where $\phi$ is the Euler phi function. But what ...
4
votes
0answers
51 views
In van Kampen’s theorem, what happens to the loops not in $\pi_1(U_\alpha \cap U_\beta)$?
I’ve just read the set up and the statement for van Kampen’s theorem. Here’s the version that we use in our class.
van Kampen's Theorem. Suppose we have an open cover $\left\{U_{\alpha}: \alpha \in A\...
6
votes
1answer
30 views
Transfer and fusion in a centralizer
Suppose $G$ is a finite group of order divisible by $8$, with an element $\tau$ of order 2 whose centralizer $C_G(\tau)$ is elementary abelian of order 4. I suspect $G/[G,G]$ must have even order, but ...
0
votes
0answers
37 views
What is the amalgamated free product $\mathbb{Z} \ast_\mathbb{Z} \mathbb{Z}$? [duplicate]
This is a small exercise to help with my understanding of amalgamated free product of groups. I think I got it partially, but not sure if I can/should further simplify it. The theoretical construction ...
-1
votes
0answers
46 views
cyclic group under multiplication
𝑁=𝑁(𝑍𝑛) for set of all nilpotent elements in Zn and 1+𝑁={ 1 + z | z ∈ N}
I am to prove that 1+𝑁(Z(27)) is a cyclic group under multiplication.
I am aware that a group is cyclic if there exits ...
1
vote
1answer
33 views
Is $H$ is a subgroup of $D_4?$ Yes/No [duplicate]
Is $H=\{ x\in D_{4} \mid x^2=1\}$ is a subgroup of $D_4?$
My attempt : I think not
Take the elements $s$ and $rs$ of $D_4$
Here $s^2=1$ and $( rs)^2 =rsr^{-1}s=s^2=1$
But $s(rs)=rs^2=r \neq 1$
...
1
vote
0answers
24 views
Decomposition of representation that has multiple copies of isomorphic irreducible representations, of a finite group.
To better show where the weightlifting point is, I'll use $D_{3d}$ group as an example. The problem is that a 24 dimensional reducible representation of $D_{3d}$ is given, which is a matrix form $r_{\...
1
vote
0answers
25 views
Minimal normal subgroup contained in a normal Sylow $p$-subgroup is abelian
I am studying the classification of sharply k-transitive groups, and I am reading a proof that sharply $2$-transitive groups are contained in an affine group: Aff($GF(p)$).
One of the steps it makes ...
0
votes
0answers
40 views
Semidirect product as a morphism
let $G$ be a group and $H$ be a normal subgroup of $G$. Also, let $\alpha:H \to G$ the inclusion and $\varphi:G \to\frac{G}{H}$ the quotient morphism.
Here comes the new deal: Let's suppose that there ...
0
votes
0answers
23 views
About regular representation and the irreducible representation
I am now learning group theory using Sadri Hassani's book "mathematical physics."
In the book, the author introduces the orthogonality condition for the unitary representation, which reads ...
0
votes
1answer
24 views
What determines the number of real parameters needed in the exponential map from Lie algebra to Lie group?
Here, https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#Examples, we can see a number of examples:
The unit circle centered at $0$ in the complex plane is a Lie group (called the circle ...
2
votes
1answer
49 views
Artin Proposition 2.4.2.
I'm trying to understand Proposition 2.4.2 in Artin's algebra textbook.
Let $\langle x \rangle$ be the cyclic subgroup of a group $G$ generated by $x$, and let $S$ denote the set of integers $k$ such ...
1
vote
0answers
21 views
Haar measure of the orthogonal group and Lie algebra
I am looking to find the expression of the Haar measure of the $SO(3)$ group as a function of the Lie algebra basis
$$
R(x,y,z) = \text{exp}\left( x L_x + yL_y + zL_z \right)
$$
where $L_x, L_y, L_z$ ...
0
votes
1answer
34 views
Every element of $S_n$ can be written as a product of transpositions of the form $(1 i )$, for various $i$ [duplicate]
Let $n \geq 2$. Show that every element of $S_n$ can be written as a product of
transpositions of the form $(1 i )$, for various $i$.
I have proved by induction that if $n \geq 2$, then every ...
6
votes
1answer
53 views
How to use a character table
This is quite an open-ended question. I have learned representation theory of finite groups and have an understanding of how to obtain the character table of a finite group. I see how the character ...
0
votes
0answers
40 views
How to implement a direct group product with matrices?
Is there a matrix product that sends two matrices (each element of a group $G_1$ and $G_2$, respectively) to a third matrix (element of $G_1\times G_2$)?
Say, I have a matrix element of SU(2) :
$$
s(...
-2
votes
0answers
36 views
Let $G$ be cyclic of order $n$. Let $d$ be an integer such that $n\nmid d$. Then, there exists $y\in G$ such that $y^d\neq e$. [closed]
Let $G$ be a cyclic group of order $n$.
Let $d$ be an integer such that $n$ does not divide $d$.
Then, I would like to prove there exists $y\in G$ such that $y^d\neq e$.
My attempt :
If we assume all $...
2
votes
0answers
40 views
Computing a subgroup of the Galois group of $\Bbb Q(\sqrt3,\sqrt[3]2,\sqrt[4]2,i)$
I'm trying to compute the subgroup $H$ of the Galois group of $K = \Bbb Q(\sqrt3,\sqrt[3]2,\sqrt[4]2,i)$ corresponding to the subfield $E = \Bbb Q(\sqrt[3]2,\sqrt[4]2)$.
So far I've computed $[K:\Bbb ...
4
votes
1answer
68 views
J. J. Rotman's proof that two free groups are isomorphic iff they have the same rank
Rotman's "An Introduction to the Theory of Groups" contains the above result as Theorem 11.3. However, I failed to pickup a step of the proof. It goes something like this:
($F \simeq G \...
-2
votes
0answers
21 views
A problem about the orders of a finite group [duplicate]
Can I prove that a finite group with more than one element has element with prime order, like this,
for some element "$a$" let the order of it be $n$ ($a$ is not the identity)
let $n=pk$, ...
1
vote
0answers
25 views
Co-cyclic group and abelian monolithic group
A $\textbf{cocyclic group}$ is a group $G$ with $\displaystyle\bigcap_{\{1\}\neq H\le G}H \neq\{1\}$.
By this referement: https://encyclopediaofmath.org/wiki/Cocyclic_group
result equivalent this ...
0
votes
1answer
44 views
Exercise 1.1.7 of Dummit and foote . Is $G$ a group?
This question is from "Abstract Algebra" by Dummit and Foote , third edition , page 21 , exercise 1.1.7 .
Let $G = \{ x \in \mathbb R | 0 \leq x < 1 \}$ and for $x,y \in G$ let $x*y$ be ...
2
votes
2answers
49 views
Let $|G|=5780$, prove $G$ has one and only subgroup of index $4$
Let $G$ be a group of order $5780$. prove that $G$ has one and only subgroup $H$ such that $[G:H]=4$, and by that conclude that $G$ has one and only 5-Sylow subgroup.
My Attempt:
$5780 = 2^2 \cdot 5 \...
0
votes
1answer
28 views
Sylow tower in a supersoluble finite group
Let a finite supersoluble group $G$ and let $p$ a prime number.
Let $\pi=\{q$ is prime number$: q\ge p\}$.
Then i want proof that $G$ has a normal Sylow $\pi$-subgroup (i.e. normal Hall $\pi$-subgroup)...
1
vote
0answers
44 views
Irreducible representations of wreath products of cyclic groups.
Let $G= C_m \wr C_n$ be a wreath product of cyclic groups $C_m$ and $C_n$.
I am interested to find all irreducible representation of $G$.
Update:
My thoughts: If I take $H:=\underbrace{C_m \times \...
1
vote
0answers
53 views
Find all $7$-Sylow subgroups of $S_{14}$
Recently, while studying for my group theory class, I tried to solve the following problem
Find all $7$-Sylow subgroups of $S_{14}$.
Since the order of $S_{14}$ is $14!$ I know that $P$, a $7$-Sylow ...
3
votes
2answers
74 views
If a normal subgroup $N$ of $A_n$ contains any $3$-cycle, then $N = A_n$
Let $n\geq 3$, if a normal subgroup $N$ of $A_n$ contains any $3$-cycle, then $N = A_n$.
What I have done
If $n\geq 5$ the result follows by these lemmas: (1) * Let $n\geq3$, then every element of $...
0
votes
1answer
79 views
Is this a mistake in my group theory textbook?
In "Algebra with galois theory" by Emil Artin.
The first exercise is to construct a 'mutiplication' table and show closure.
We are given $f_1= x; f_2= (1/x); f_3= (1-x); f_4= 1/(1-x); f_5= x/...
0
votes
1answer
27 views
Prove the subset of a linear code consisting of codewords with even weight is a subgroup.
"Let $C$ be a linear code over $Z_2$. Let $C^+$ be the subset of $C$ consisting of those elements of $C$ with even weight. Show that $C^+$ is an additive subgroup of $C$."
It's an exercise ...
2
votes
1answer
42 views
Artin's theorem 2.33: subgroups of $\mathbb{Z}$
Theorem 2.33 in Artin states:
Let $S$ be a subgroup of the additive group $\mathbb{Z}^+$. Either $S$ is the trivial subgroup $\{0\}$, or else it has the form $\mathbb{Z}a$, where $a$ is the smallest ...
0
votes
0answers
54 views
No nonzero integers > 1 or < -1 have a multiplicative inverse. How does one prove this formally?
I understand intuitively why this is true: no integers > 1 or < -1 have a multiplicative inverse. However, I'm not sure how to prove this formally. Without relying on intuition, how do you KNOW ...
4
votes
0answers
100 views
Are there any interesting theorems of first-order group theory
I've heard that theoremhood in first-order group theory is uncomputable, so there should be some theorems that are difficult to prove. However the few theorems of group theory I know are about ...
2
votes
1answer
59 views
Cohomology of Heisenberg Group
If we let $G :=$ the Heisenberg group over the integers, then I can show that the center of this group is isomorphic to the integers $\mathbb{Z}$ and that the abelianization of G is isomorphic to $\...
4
votes
1answer
48 views
If $G$ has an $\Omega$-composition series, prove that every $\Omega$-subgroup [of $G$ has] a composition series.
This is part of Exercise 3.1.4 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE.
The Details:
Since definitions vary, on page 15, ...
2
votes
1answer
51 views
Finitely generated image from commutative diagram
Let the following Diagram commute with abelian groups
If $\text{im}(C \rightarrow F)$ and $\text{im}(B \rightarrow E)$ are finitely generated and $C \rightarrow D \rightarrow E$ is an exact sequence ...
1
vote
0answers
38 views
Embedding of countable abelian group into a finitely generated group
Prove/ Disprove/ or object
Let, $G=\langle p,q\rangle$ be a group. Then there exists an abelian group $A$ with $\left| A\right| \leq\left|{\mathbb{N}}\right|$ and such that for every group ...
14
votes
2answers
179 views
Structure of the Brainball group
This is a Brainball:
It consists of $13$ numbered pieces arranged in a ring and a core; each piece has one side white and one side yellow. Part of the core, the blue caps in the picture above, can ...
2
votes
0answers
51 views
Prove that this 'generator' function $\boldsymbol{\Psi}$ for cyclic groups is equal to $0$ if $G$ is not cyclic
I came across this question in my Abstract Algebra class, and am having difficulty in proving it.
Let $\boldsymbol{\mathcal G}$ be the class of all finite groups. We have a function $\boldsymbol{\Psi}...
2
votes
2answers
65 views
Can elements of automorphism groups be expressed as some form of matrix?
I was reading on automorphism groups and Galois theory and this idea came to mind:
Since by definition, automorphisms are isomorphisms, we have $\phi(a*b)=\phi(a)\cdot\phi(b)$ where $\phi$ is an ...
0
votes
1answer
43 views
Prove or disprove $\langle a,b,c\rangle$ is free
Suppose that $\langle a,b\rangle$, $\langle b,c\rangle$, and $\langle a,c\rangle$ are free, is it true then that $\langle a,b,c\rangle$ is a free group of rank 3? We assume that $a,b,c$ are distinct ...
0
votes
3answers
50 views
Let $G$ be a group and $H\unlhd G$ such that $[G:H]=20$ and $|H|=7$. Suppose $x\in G$ and $x^7=e$. Show that $x\in H$.
I'm currently on an exercise problem from Dan Saracino Abstract Algebra, Exercise 11.20. It says the following:
Let $G$ be a group and $H$ be a normal subgroup of $G$ such that $[G:H]=20$ and $|H|=7$. ...
2
votes
2answers
69 views
Let $G$ be a finite group and $Z(G):=\{x\in G : gx=xg \ \ \forall g\in G\}.$ Prove that $[G:Z(G)]$ is not a prime number. [duplicate]
Let $G$ be a finite group and $Z(G):=\{x\in G : gx=xg \ \ \forall g\in G\}.$ Prove that $[G:Z(G)]$ is not a prime number.
I think that I should use the Lagrange Theorem, i.e.,
If $Z(G)$ is a subgroup ...
0
votes
0answers
33 views
On the proportion of 2-groups.
Suppose $T(n)$ denotes the number of non-isomorphic 2-groups (groups of order a power of 2) of order at most $n$, and $G(n)$ the number of non-isomorphic finite groups of order at most $n$. It is well ...
0
votes
0answers
14 views
$(\mathbb{Z}^n/H)/p(\mathbb{Z}^n/H)\cong\mathbb{Z}^n/(H+p\mathbb{Z}^n)$ where $p$ is a prime
I was going through the answer of the following question:
Let $H$ is the subspace of $\mathbb{Z}^n$ that is generated by
$(a_1,a_2,...,a_n)$, determine the rank and the torsion subgroup of
$\mathbb{Z}...
1
vote
1answer
24 views
Conditions for Pollard's rho algorithm for logarithms
I am trying to implement Pollard's rho algorithm for logarithms, but it looks like I am missing some initial condition which has to be met, because in my case it never works.
Example:
$203^{\text{7}}=...
0
votes
0answers
66 views
$(G/N)/\ker\pi \cong G/H\to\ker\pi＝H/N$
Let $G$ be a group and $N$ and $H$ are normal subgroup of $G$ and $N$ is normal subgroup of $H$.
$\pi:G/N\to G/H$ be natural projection, that is, $x\pmod{N}\to x\pmod{H}$.
Then, I would like to ...
5
votes
1answer
71 views
How to determine the number of double cosets from table of marks
I remember reading a while back that a lot of information about double cosets of a group can be extracted from the group's table of marks. I can't recall the source. The group is an arbitrary finite ...
