Questions tagged [group-theory]
For questions about the study of algebraic structures consisting of a set of elements together with a well-defined binary operation that satisfies three conditions: associativity, identity and invertibility.
47,166
questions
7
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Why is studying centralizers the/a key to classifying finite groups?
In this MO thread https://mathoverflow.net/questions/38161/heuristic-argument-that-finite-simple-groups-ought-to-be-classifiable, Borcherds says
One problem, as least with the current methods of ...
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0
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28
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What is the point when the profinite topology of a group $G$ induces the full profinite topology on a subgroup?
I have a question. I don't know if is trivial or even it makes sense, but is something I could not understand. For example, consider the following exercise of Ribes-Zalesskii:
Exercise 9.1.1 Let $G = ...
- 450
1
vote
2
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46
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If the cardinality of the quotient set of a subgroup is $1$ does that mean that the subgroup is equal to the whole group?
Let $(G, \cdot)$ be a group and let $H \leq G$ (i.e. $H$ is a subgroup of $G$) such that $\mid G:H \mid = 1$ (i.e. the index of $H$ in $G$ is $1$). Does this imply that $G=H$?
In the case that $G$ is ...
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0
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34
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About the number of disjoint cycles in a product of permutations [duplicate]
I am being troubled by a quick doubt on products of permutations. Concretely suppose we have the permutation
$$(123\dots n)^k,\quad\text{for some}\quad k=1,\dots,n$$
I wish to find the number of ...
- 1,060
2
votes
1
answer
31
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Understanding rearrangement and simplification of products of adjacent transpositions.
Whilst working with permutations I have attempted to write several as products of disjoint cycles and adjacent transpositions. Below is an example:
$\sigma =
\begin{pmatrix}
1 & 2 & 3 & ...
- 23
0
votes
1
answer
47
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kernel of subgroup homomorphism
Let $f:A \longrightarrow B$ be a group homomorphism, and note $C$ a subgroup of $A$ and $D$ a subgroup of $B$. Can we find a link between the kernel of $f$ and the kernel of the group homomorphism $g: ...
- 27
0
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0
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53
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Non-$p$-torsion difference of $p$-torsion elements in a $p$-group
I have a relatively simple question. For $p$ odd, does there exist a non-abelian $p$-group such that for every pair of non-commuting $p$-torsion elements $g$,$h$ their difference $g^{-1}h$ is not $p$-...
- 476
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1
answer
17
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I am trying to understand how to rearrange and simplify adjacent transpositions of permutations
Whilst working with permutations I have attempted to write several as products of disjoint cycles and adjacent transpositions. Below is an example:
$\sigma =
\begin{pmatrix}
1 & 2 & 3 & ...
- 23
-1
votes
0
answers
56
views
Number of involutions in $S_n$? [duplicate]
I was having some fun with the number of involutions (didn't know they were called that) in the symmetric group $\Psi(S_n$), and tried to come up with a simple formula for it, I'm like, so close to ...
- 21
1
vote
1
answer
54
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Which wallpaper groups do these images belong to?
The three images below are 'pieces' of infinite wallpaper patterns. The first one corresponds to an infinite checkerboard pattern with alternating white and black squares. The second image corresponds ...
- 25
4
votes
1
answer
58
views
The Iwasawa decomposition of $\text{GL}(2,\mathbf R)$
The Iwasawa decomposition of $\text{SL}(2,\mathbf R)$ is
$$
\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} r & 0 \\ 0 & 1/r \end{...
- 2,331
-1
votes
0
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26
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Does the exponential of multivectors map to the double cover of $GL^+(R)$? [closed]
The exponential of bivectors maps to the Spin(n) group which is the double cover of SO;
$$
\exp \mathbf{f} \to {\rm Spin}(n) \cong \widetilde{{\rm SO}(n)}
$$
Is it safe to assume that the exponential ...
- 2,331
1
vote
0
answers
13
views
Representing $G=\text{GL}^+(2,\mathbf R)$ as the matrix product $G=TH$. If $H=\text{SO}(2)$, what is $T$?
In this paper (Equation 2.6 and 2.7) the author seems to suggest that one can represent the $\text{GL}^+(4,\mathbf R)$ group using the product of two exponentials: $\exp (\epsilon \cdot T) \exp (u \...
- 2,331
5
votes
2
answers
56
views
Elements of the coset $G/H$, where $G=\text{GL}^+(2)$ and $H=\text{SO}(2)$
In this paper, in section $2$, a method to write the elements of the coset of $G/H$ is provided for $\text{GL}(4)$, but I am interested in $\text{GL}^+(2)$.
My matrix representation of $\mathfrak{gl}(...
- 2,331
0
votes
1
answer
51
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Proof that all Quotient Groups are Abelian- where is my error?
I'm having trouble finding the error in my proof:
Theorem: If H is a normal subgroup of G, then the quotient group G/H is abelian.
Proof: $\forall x, y \in G$, $xy(yx)^{-1} = xyy^{-1}x^{-1} = e \in H$....
1
vote
0
answers
29
views
Representing $GL(2)$ as dilations $\times$ shears $\times$ $SO(2)$
In this paper1 in section 2, it is stated that one can write the GL(4,R) using the generator of dilation (1 parameter), those of shears (9 parameters) and the Lorentz generator (6 parameters).
Here, I ...
- 2,331
2
votes
1
answer
58
views
Proof for order of composition for permutations on indices into a sequence
Suppose I have a sequence $s$ that contains some elements in a well-defined order. Furthermore, let $i$ be an integer (an index into the sequence) and $s[i]$ shall denote the $i$-th element from $s$.
...
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1
vote
0
answers
19
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Mellin transform defined for function on group $(\Bbb R^+,\times)$ but integration domain is over semigroup
I came across this question: Mellin transform defined for function on group $(\mathbb{R}^{+}, \times)$.
Let $\varphi_x: s\mapsto x^s $ be a group isomorphism from $(\Bbb R,+)$ to $(\Bbb R^+,\times).$
...
- 176
2
votes
2
answers
58
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Prove every Sylow Subgroup of $G$ is Cyclic
I'm working on the following problem. Let $G$ be a finite group such that for each $n \mid |G|$, we have that $G$ contains at most $1$ subgroup of order $n$. Prove that every Sylow $p$ subgroup of $...
- 1,099
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0
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19
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Showing a non-unimodular group operation is not associative
There is a binary operation defined by
$$(f*g)(x)= \int_G f(xy^{-1})g(y)dy$$
where G is not a unimodular group.
Show is operation is not associative.
Workings so far:
$$((f*g)*h)(x)=\int (f*g)(xy^{-1}...
- 155
1
vote
0
answers
55
views
Conditions for a representation of a group. (updated)
My lecturer wrote that a matrix representation of a group $G$ is a group homomorphism $\rho$ from $G$ to the set of invertible square matrices over a field. He proceeded to write that this is ...
- 113
0
votes
0
answers
31
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Using properties of bijection for quicker proof of conjugate subgroup symmetry
My textbook, Szekeres's "A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry", tacitly alludes to the following lemma, which I've tried to write out and ...
- 661
10
votes
3
answers
618
views
Axioms for finite cyclic groups
tl;dr: What is a minimal definition of a finite cyclic group, without resorting to the definition of a group?
Let $G$ be a finite set and $\star$ be a binary operation on $G$. My question is whether a ...
- 277
-2
votes
0
answers
14
views
show that Ĝ is non empty and an abelian group when equipped with pointwise operations. [closed]
Assuming G is a locally compact group ( not abelian). A character of G is a continuous group homomorphism x: G → 𝕋 , where 𝕋 is the circle group . Denote by Ĝ the set of characters of G.
Here how ...
0
votes
1
answer
84
views
Definition of Presentation of Group in Dummit’s Abstract Algebra [closed]
A subset $S$ of elements of a group $G$ with the property that every element of $G$ can be written as a (finite) product of elements of $S$ and their inverses is called a set of generators of $G$. We ...
- 3,180
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0
answers
40
views
On coset equalities
Let $H \le \textrm{Sym}(n)$, where $n \in \mathbb{Z}$ and Sym(n) stands for the symmetric group over $n$ elements. Furthermore let $g \in \textrm{Sym}(m)$ where $m \in \mathbb{Z} \land m \ge n$. This -...
- 249
1
vote
1
answer
106
views
Why is Cayley's theorem stated for finite groups?
My textbook writes Cayley's theorem on groups as follows:
Every finite group $G$ of order $n$ is isomorphic to a permutation group.
By permutation group is meant a subgroup of the group of ...
- 661
0
votes
0
answers
61
views
Prove: $\mathbb{C}^* \cong U_1 \times \mathbb{R}^*$ (isomorph to circle group)
Define the circle group $U_1 = \{z \in \mathbb{C} \mid |z| =1 \} \subset \mathbb{C}^*$. Prove: $\mathbb{C}^* \cong U_1 \times \mathbb{R}_{>0}^*$.
I would like to know if I did it correct. I ...
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votes
1
answer
56
views
Is this proof that a homomorphism preserves identities correct (sufficient)?
My textbook writes the following proof for the statement in the title:
For a homomorphism $\varphi:G \to G'$ and any $g\in G$, $\varphi(g) = \varphi(ge) = \varphi(g)\varphi(e)$. Multiplying both ...
- 661
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votes
0
answers
31
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Is there a name for a group whose elements are Hurwitz?
I am not familiar much with group theory except for some basics. I know that a group is defined by some axioms for addition and multiplications. I have heard of general linear group where invertible ...
- 55
2
votes
1
answer
58
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A non-trivial homomorphism from $SL(2, 7)$ to $GL(4, 11)$.
Let $G=GL(4, 11)$ be the general linear group of $4\times 4$ matrices over the field $\mathbb{F}_{11}$. Let $H=SL(2, 7)$ be a special linear group of $2\times 2$ matrices over the field $\mathbb{F}_{...
- 4,196
1
vote
2
answers
117
views
Why isn’t every subgroup of a Finite Group Cyclic?
I’m trying to prove or disprove wether every subgroup of a finite group is cyclic or not. I came up with this proof:
$G$ is a finite group of order $k$, with elements $g_1,…,g_k$, then every $g_i\in G$...
3
votes
1
answer
106
views
Does the direct sum have a universal property in the category of groups?
In other categories, like modules, the direct sum is the coproduct, giving it a neat category theory description. The direct product of groups, which is very similar to it, is the product in the ...
- 2,644
4
votes
3
answers
120
views
A ternary relation on a group
Let $G$ be a group. Consider the ternary relation $R \subset G^3$ defined by
$$R(x,y,z) \Leftrightarrow x y^{-1} z x^{-1} y z^{-1} = 1 $$
Show that $R$ is a symmetric relation, that is if $R(x,y,z)$, ...
- 50.4k
1
vote
1
answer
37
views
Classification of torsion-free nilpotent groups of class 2
Some background
Let $G$ be a torsion-free nilpotent group of class $2$ and rank $2$ (i.e., generated by two elements). Then, $G$ has to be isomorphic to the Heisenberg group.
This is relatively easy ...
- 1,945
-1
votes
1
answer
27
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Help showing associativity when multiplying group element by vector.
Consider a representation $\rho=(\rho_{ij})$ of a group $G$ of degree $d$ and a $d$-dimensional vector space. I want to turn $V$ into a $G$-module by defining the action of $x\in G$ on each of the ...
- 615
2
votes
1
answer
48
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How can we correctly understand the definition stating that the union of conjugacy classes forms a normal subgroup?
I am learning group theory and trying to practice the concepts I am learning. In particular, I learned that $H$ is a normal subgroup of $G$ if and only if $ghg^{-1}\in H$ for all $g\in G$ and for a ...
- 233
2
votes
0
answers
24
views
Is the group-like element functor strong monoidal?
Let $K$ be a field of characteristic $0$, $\mathcal{C}$ a subcategory of the category $\widehat{\mathbf{Hopf}}$ of complete Hopf algebras over $K$,
\begin{align}\mathbb{G}&: \widehat{\mathbf{Hopf}}...
- 51
1
vote
0
answers
28
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Show explicitly the rational function field over the fixed field is not Galois.
For complex number field $\mathbb{C}$, we can consider the automorphism group $\mathrm{Aut}_{\mathbb{C}}(\mathbb{C}(x))$. It's not hard to show $\mathrm{Aut}_{\mathbb{C}}(\mathbb{C}(x))\simeq \mathrm{...
- 11
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votes
0
answers
72
views
$\phi:G\to \bar G$ is an isomorphism. Then the equation $x^k=b$ has the same number of solutions in $G$ as the equation $x^k=\phi(b)$ in $\bar G$
This is my first post here, so forgive for any formatting errors or lack of information; please let me know if I can rectify any issue with my post.
I've been stuck on this problem for quite a bit, ...
- 21
0
votes
0
answers
48
views
Moebius transformation on circles
I'm working on a program that draws the limit sets of Schottky groups defined by pairs of circles. If circle $C$ with radius $r$ is centered at $P$, then to map that circle to some other circle $C'$ ...
- 343
-1
votes
0
answers
27
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Sylow 2-subgroup of $GL(2,\mathbb{Z}_{2^n})$ [closed]
The structure of Sylow 2-subgroups of $GL(2,q)$ (or even of $GL(n,q)$) is well known, but I couldn't find any information about Sylow 2-subgroup of $GL(2,\mathbb{Z}_{2^n})$, so could anyone help me to ...
- 369
2
votes
0
answers
34
views
Defining $G$-module from representation.
Consider a group $G$ and a representation $\rho=(\rho_{ij})$ of degree $d$ as well as a $d$-dimensional vector space $V$. In the book Algebra by P. M. Cohn he states that we can turn $V$ into a $G$-...
- 615
2
votes
1
answer
28
views
If $\varphi: X\to Y$ is a $G$-equivariant map and the action on $Y$ is transitive, then $|Y|$ divides $|X|$
I came up with the following proof, which I would like help checking and/or cleaning up:
Theorem: Let $X$ and $Y$ be $G$-sets, and let $\varphi:X\to Y$ be a $G$-equivariant map, i.e. $\varphi(gx) = g\...
- 43
0
votes
1
answer
46
views
Orthogonal group of Lorentzian lattice $I_{1, n}$ is infinite for $n\geq 2$
I am looking for a reference or an elementary proof of the following fact:
The orthogonal group of the lattice $I_{1,2} = I^+ \oplus 2 I^-$ is ifinite.
Here the Lorentzian lattice $I_{1,n}$ is given ...
- 4,937
4
votes
1
answer
66
views
If $a^k\equiv a\mod p$ for all $a$, show $(p - 1)\mid (k-1)$
Problem statement. Suppose that $k$ is a positive integer and $p$ a prime such that $a^k\equiv a\mod p$ for all positive integers $a$.
Show that $(p - 1)$ divides $(k-1)$.
The proof is simple if we ...
- 5,146
2
votes
1
answer
68
views
$K \unlhd A \rightarrow \Phi(A)$ is isomorphism
Let $\Phi: G_1 \rightarrow G_2$ be a surjective homomorphism. Prove that there is 1-1 correspondence between subgroups that contain $K=\text{Ker}(\Phi)$ and subgroups in $G_2$
I tried proving ...
- 156
-4
votes
1
answer
67
views
Groups, Algorithms and Programming (GAP 4) lack of functions
I am a newbie in using of GAP. To learn it I downloaded a book "Groups, Algorithms and Programming" (subtitle GAP 3.4.4 of 20 Dec. 1995, gap3-jm of 19 Feb. 2018) and also Reference Manual of ...
- 29
0
votes
1
answer
34
views
Show that $H = \{a \in G \ | \ ax=xa \ \forall x \in G\}$ is an abelian subgroup of G. [duplicate]
Problem Statement: Let $G$ be a group. Show that $H = \{a \in G \ | \ ax=xa \ \forall x \in G\}$ is an abelian subgroup of G.
It is easy to show that H is a subgroup, but I couldn't show that it is ...
user1159783
-2
votes
1
answer
144
views
Abelian subgroups of $S_n$? [closed]
I was playing around with the group $S_4$ and it’s subsets, and I came to this conclusion and wanted to write my first paper on it. (my question is so stupid it might not be valued though)
The set $\...
- 21