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.
50,575 questions
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Quaternionic groups $GL(n, \mathbb H)$ and $Sp(p,q)$
Let $\mathbb H$ be the quaternions. The general linear group $GL(n, \mathbb H)$ is defined as
$$GL(n, \mathbb H) = \{A \in GL(2n, \mathbb C) \mid JA = \bar{A}J \}$$
where
$$J = \begin{pmatrix} 0 & ...
- 305
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41
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Is There A Better Deterministic Discrete Logarithm Decision Algorithm?
I recently wrote a fairly simple discrete log calculator (here https://github.com/FastAsChuff/Discrete-Log) which works pretty efficiently for small numbers, in my case moduli below $2^{32}$. It uses ...
- 324
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1
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Normality of the lower central series
The lower central series of a group $G$ is defined as follows:
$$
\gamma_1(G) = G, \quad \gamma_{i+1}(G) = [\gamma_i(G), G],
$$
where $[\gamma_i(G), G]$ denotes the subgroup generated by all ...
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27
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How to calculate Borel subgroups of an algebraic group over an algebraically closed field?
Let $G$ be an algebraic group defined over an algebraically closed field $k$. A Borel subgroup of $G$ is by definition a connected solvable subgroup of $G$, which is maximal with respect to the ...
- 1,649
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32
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The number of the double cosets $K\backslash G/H$ equals $ \langle \pi_{G/K} , \pi_{G/H} \rangle$
$H$ and $K$ are subgroups of $G$. $\pi_{G/K}$ is the permutation character of the set $G/K$ (the left cosets).
My thoughts
I think we are finding the number of the $K$-orbit on the set of left cosets $...
- 55
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86
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Is there a ring whose unit group is the $p$-Prüfer group?
My professor said “not every group is actually the unit group of some ring” even though some are.
He gave us examples, like $(\mathbb Z/2, +)$ being the unit group of $(\mathbb Z/4 , +, ·)$, while he ...
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40
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Cataloging the Core-free Subgroups of the Permutation Group
A subgroup H of G is called core-free if for every normal subgroup $N\subset G$ we have that $N\subset H$ implies $N=\{1\}$. That is, H is a core-free subgroup of G iff $\{1\}$ is the only normal ...
- 503
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1
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56
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Showing that a group of order $231$ is a semi direct product
I am having a real hard time with this problem. My understanding is that to show that $G$ is a semi direct product, I must find 2 subgroups of $G$ such that one is normal, they have trivial ...
4
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94
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Characterization of transpositions in symmetric group?
For a symmetric group on an arbitrary (indeed potentially infinite) set, is it true that $x$ is a transposition if and only if $x^2=1$, $x \not= 1$, and $y^2=1$ and $C(x) \subseteq C(y)$ implies $y=x$ ...
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Canonical Identification of Tangent Vector of Matrix Lie Group
Question Setup
(See edits made for comments below)
In an abstract sense, one of the defintions that we can use to define a tangent vector at a point $p\in M$, where $M$ is a differential manifold $(M,\...
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$H = \mathbb{Z}_6\times\mathbb{Z}_{15}/ \left<(0,3)\right>$ is isomorphism to which group? [duplicate]
If $ H = \mathbb{Z}_6\times\mathbb{Z}_{15}/\left<(0, 3)\right>$ , then $H$ is isomorphic to which group?
I know the order of $H$ is $18$.
Every element of $H$ can be represented as $(a,b)+(0,3k)$...
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32
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A finite subgroup of $P\subseteq GL(n,\mathbb{C})$ has the determinant of the sum of its matrices in the integers [duplicate]
Let M be the sum of all matrices of this finite subgroup, prove that $det(M) \in\mathbb{Z}$
I know that I must use the fact that M is invariant under multiplication by elements of P, which can be ...
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universal locally finitely presentable groups [closed]
In the snippet below, I think that there is a typo:
$C_P$ should read $C_X$, right ?
I'm attaching a snippet instead of re-typing the paragraph just because of the reproducing the error.It is taken ...
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Understanding the case when $P_3$ is normal and $P_2 \cong \mathbb Z _2 \times \mathbb Z_2$ is not normal to classify a group of order $12$
I am studying the classification of groups, when the Klein 4-group arises as a subgroup. The first example comes from Artin about groups of order $12$. He uses proof by cases, and the only case that ...
- 1,314
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How to find the $\mathbb Q$ conjugates?
This question is in the textbook: Abstract Algebra, A Comprehensive Introduction by John W. Lawrence and Frank A. Zorzitto.
Find the $\mathbb{Q}$-conjugates of $\alpha = 7^{1/3}+2i$.
I am trying to ...
- 301
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2
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62
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On $\mathrm{Gal}(\mathbb{Q}(e^{\frac{2i\pi}{n}}/Q)$
I want to show that $\mathrm{Gal}(\mathbb{Q}(e^{\frac{2i\pi}{n}}/Q)$ is abelian but without using or proving the fact that $\mathrm{Gal}(\mathbb{Q}(e^{\frac{2i\pi}{n}}/Q) \cong (\mathbb{Z}/n\mathbb{...
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For which $m \leq n$ does an arbitrary $n$-cycle and $m$-cycle in $S_n$ not powers of each other generate $S_n$ or $A_n$? [closed]
Let $n \in \mathbb N$. For which $m \leq n$ does an arbitrary $n$-cycle and $m$-cycle in $S_n$ not powers of each other generate $A_n$ or $S_n$?
I'd be willing to assume that $n$ is prime.
Context: ...
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Visual intuition for normal subgroups
I've read this discussion on Intuition behind normal subgroups, but I'm still unable to create a mental picture of what is going on behind the scenes. I'm trying to build an alternative visualization.
...
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Proving that $B_3/\langle (ab)^3\rangle\cong SL_2(\mathbb Z)/\langle -I\rangle$ where $B_3$ is the braid group of three strands
Let $B_2=\langle a,b:aba=bab\rangle$ be the braid group of three strands. It is known that $Z(B_3)=\langle(ab)^3\rangle$. For the sake of simplicity, set $Z=Z(B_3)$. Let $\varphi:B_3/Z\to SL_2(\mathbb ...
- 5,036
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In finite fields of large characteristics,what does prevent shrinking the field size down to their larger order in order to solve discrete logarithms? [closed]
In the recent years, several algorithms were proposed to leverage elliptic curves for lowering the degree of a finite field and thus allow to solve discrete logarithm modulo their largest suborder/...
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52
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Understanding Invariant $p$-forms
I'm currently reading through Bryant and Salamon's 1989 paper, 'On the construction of some complete metrics with exceptional holonomy', and trying to understand the construction of the subgroups of $...
0
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27
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Algebraic Notation for Tessellations
Is there a numerical or algebraic way to describe the nature of tessellating certain objects? For example, take a simple 2D object like a square. Obviously this shape can tessellate space as you can ...
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48
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Is the group $\langle x,y:x^5=y^2=1\rangle$ a quotient of some subgroup of $GL_2(\mathbb F)$?
The projective special linear group $PSL_2(\mathbb Z)$ is defined as $SL_2(\mathbb Z)/\langle -I\rangle$, where $SL_2(\mathbb Z)$ is the special linear group. It is well known that $PSL_2(\mathbb Z)$ ...
- 5,036
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Turing degree of group isomorphism problem
The group isomorphism problem of a finitely presentd group is undecidable. But where is the problem embedded in the arithmetical hierarchy, i. e. what is the Turing degree?
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Determining if a field extension of a function field is a splitting field
Here is a textbook question from a textbook: Abstract Algebra: A Comprehensive Introduction by John W. Lawrence and Frank A. Zorzitto. I am trying to solve this question in preparation for my final ...
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2
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Clarification about the definition of word (group theory)
I'm a Bachelor student (third year) and I'm having trouble with the definition of word in group theory.
From Wikipedia (our professor also gave us a similar definition):
Let G be a group, and let S ...
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Prove that general polynomial of degree n>_5 is not solvable by redicals. [closed]
Prove that general polynomial of degree n>_ is not solvable by redicals
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Exercise 2.6 in Katok's book "Fuchsian Groups"
Exercise 2.6 in Svetlana Katok's book Fuchsian Groups asserts the following
Let $G$ be a subgroup of isometries of a proper metric space $X$. Then the following are equivalent:
(i)/(ii) The action is ...
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A confusion related to concrete category in Hungerford Book
I am reading the T.W. Hungerford's Algebra book. In Chapter 1, section 7, categories: Products, coproducts, and free objects are included. I am the first learner of Category Theory. Here an example of ...
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Interpreting group in normal abelian subgroup of finite index?
I am self-studying Hodges' Model Theory, and exercise 5.3.6 asks:
Show that for $G$ a group with normal abelian subgroup $A$ where $G/A$ is finite, $G$ is interpretable (potentially with parameters) ...
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$Aut(\mathbb{Z}_n)=\{\varphi:\mathbb{Z}_n\rightarrow\mathbb{Z}_n|\varphi$ is an automorphism$\}$ [duplicate]
Let $n\in \mathbb{N}$. For group of divisors $\mathbb{Z}_n$ we label $Aut(\mathbb{Z}_n)=\{\varphi:\mathbb{Z}_n\rightarrow\mathbb{Z}_n|\varphi$ is an automorphism$\}$.
$1.$ I was able to prove that $...
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$G$ is finite iff $G/N$ and $N$ are finite
Let $G$ be a group and let $N$ be a normal subgroup of $G$. Prove or disprove:
a) $G$ is finite iff $G/N$ and $N$ are finite.
b) if $[G:N]=n$ for some $n\in \mathbb{N}$, then for all $g\in G$ holds ...
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Subsets of groups that contain a (large) coset [duplicate]
Let $G$ be a group.
Suppose that finitely many translates of $A\subseteq G$ cover $G$.
Does $A$ contain the coset of a subgroup with finite index?
[If the answer is no, I'd appreciate references to ...
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Is there a medial quandle for which the relation $\sim$ defined by $a\sim b$ iff $a\ast b=a$ is not an equivalence relation?
Definitions:
A quandle is a set $Q$ with a binary operation $\ast$ satisfying three axioms, for all $a,b,c$ in $Q$:
$a\ast a=a$.
There exist a unique $x$ such that $x\ast a=b$.
$(a\ast b)\ast c=(a\...
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What is the continuous product
Two
Many
Continuous
+
$\sum$
$\int$
*
$\prod$
What goes here?
What goes in the missing box, and what do I search to learn more.
If there is no single symbol, a larger formula will work instead (...
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The center of $\langle x,y:x^3=y^2=1\rangle$ [duplicate]
Let $G$ be the group given by the following presentation
$$
G=\langle x,y:x^3=y^2=1\rangle
$$
I want to find $Z(G)$. I notice that $\langle x\rangle\cap\langle y\rangle=\{1\}$, Does that means that $Z(...
- 5,036
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The Grothendieck group, Serge Lang and simplified version?
I've across this post on how Serge Lang develops Grothendieck group by the free abelian group generated by monoid $M$. Problem statement:
Let $M$ be commutative monoid. We can construct a ...
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85
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Intuition cosets of $S_n$ under product of "subgroups"
Given group $S_n$ and subgroups $H, G$ that are identified as $S_2$ and $S_{n-2}$ (i.e. the subgroups that permute the first two and the last $n-2$ elements and are as such isomorphic to $S_2$ and $S_{...
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Orientation for groups
I am significantly rewriting this question, as, I hope, I can formulate it now better from a different angle.
Is there such a concept as an orientation for a group? (For some classes of groups?)
Here ...
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Real character implies isotypic subspace has real basis?
Let $ \rho: G \to GL(n,\mathbb{C}) $ be an irreducible complex representation of $ G $. Let $ \chi_\rho $ be the character of $ \rho $. Let $ \rho^{\otimes k} $ be the $ k $ tensor power of the ...
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Basic groups and linear equations [closed]
I am in 9th grade right now and joined a research program, the basic outline is solving problems using group theory. We started off with the basics (dihedral, abelian etc) and now reached 'weights' ...
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25
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ordering of conjugacy classes in MAGMA
I have a family of finite groups, and I want to use the compute algebra system MAGMA to evaluate certain characters of these groups at specific conjugacy classes. The CharacterTable commmand returns ...
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finding presentation for a group quotient modulo normal cyclic subgroup
Let $G$ be the group given by the following presentation
$$
G=\langle x,y:x^5=y^2\rangle
$$
Clearly, $x^5$ commutes with both $x$ and $y$, so $x^5\in Z(G)$. Hence $N=\langle x^5\rangle$ is normal ...
- 5,036
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Let G be a group of order 45. Explain why the centre of G cannot have order 9 or 15 [duplicate]
I am not sure how to
I tried to use the class equation to prove that the centre $Z(G)$ of a group $ G$ of order 45 cannot have order 9 or 15, but I’m struggling with how to proceed, especially since ( ...
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What is the smallest finite group $G$ with a proper nontrivial noncentral subgroup $H$ to have a non-abelian centralizer $C_G(H) \neq G$?
What is the smallest finite group G with a proper nontrivial noncentral subgroup H to have a non-abelian centralizer C(H)≠G in G?
I found one: the alternating group A6 of order 360, non-abelian simple,...
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1
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0
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G has a quaternionic irrep iff G has a central element of order 2?
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$I just learned an interesting ...
- 9,194
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0
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Is the group of isometries on $R^2$ the Euclidean group, $E(2)$? [duplicate]
I‘m looking for a proof that the group of isometries on $R^2$ is the Euclidean group, E(2). I know elements of the Euclidean group E(n) can be represented by pairs (A,x) where A is in the orthogonal ...
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1
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0
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38
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Preserving finite index via profinite completion
Let $G$ be a finitely generated residually finite group and let $H\leq G$ of finite index. Denote by $\hat{G}$ the profinite completion of $G$ and by $\bar{H}$ the topological closure of $H$ in $\hat{...
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2
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1
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50
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Inverse image of a cyclic normal tower is also cyclic and normal via canonical homomorphism
Let $G / H = M_0 \supseteq ... \supseteq M_n$ be a normal tower of $G / H$ with $M_{i} / M_{i + 1}$ being cyclic. Then, define the cannonical homomorphism:
$$\Phi: G \rightarrow G / H;\quad \Phi: g \...
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3
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Same conjugacy implies same isotropy?
I'm struggling at the class formula in Serge Lang's algebra. The formula is a consequence of
$$|S| = \sum_{i} |Gs_i| = \sum_{i} (G : G_{s_i})$$
Here, $G$ is some group that acts on $S$. Since the set ...
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