Questions tagged [group-theory]

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 is the study of groups.

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inverse of isomorphism from $\mathbb{C}[G]$ to $\bigoplus_V \text{End}(V)$

I understand that for any finite group $G$, there is an isomorphism from the group algebra $\mathbb{C}[G]$ to the direct sum of endomorphisms in which each irreducible representation of $G$ appears ...
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How can I learn about the Monster group?

There are several questions about the Monster group on this site, but none really answer the question in the title. While reading about groups in a first year algebra course, I was told about the ...
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Normal subgroups orbits

Let $G$ be a topological group acting transitively and effectively on the space $X$ and let $J,K$ be two normal subgroups of $G$ such that $G=J\cdot K$ and $J\cap K\not =\{e\}$. Let $Gx_0$ be the ...
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Maximal order of a torsion element in a hyperbolic group

Suppose that $G = \langle X \rangle$ is a $\delta$-hyperbolic group, i.e. all geodesic triangles are $\delta$-thin (the inverse image of a point under the projections onto a tripod has diameter ...
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Assistance with Wielandt's permutation group problem involving blocks containing pairs

EDITED to clarify and incorporate the comments and improve notation (sorry the original was poor). Thank you Derek Holt and Mesel. Let $G$ be a finite group acting on a set $\Omega$. A block is a ...
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Could every finite simple group be related to a pair of Lie Groups?

In terms of the classification of simple groups, it is known that every finite simple group is either: Cyclic Alternating Of Lie type One of the 26 Sporadic groups On the other hand there are 5 ...
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239 views

Explanation of a proof of the Second Sylow Theorem (Conjugation of Sylow p-subgroups)

I am an undergrad Mathematics student and I've been reading some additional literature for my lectures and came upon a quite short and seemingly elegant proof of the Second Sylow Theorem. Though, I ...
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Determination of the unitary irreps of $\mathbb{R}$ using Stone's theorem

I've tried to find the unitary irreducible representations of the additive group $(\mathbb{R},+)$ and came up with a pair of results, which I want to verify if are correct. They are: Theorem: Let the ...
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52 views

Complete reducibility for the Poincaré group

If $(\rho,V)$ is a unitary representation of a group $G$ which is finite dimensional, then complete reducibility is kind of easy to prove. Indeed, if $V$ is not irreducible, then it has one proper ...
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51 views

Isomorphism preserving operations

Questions : What are group modification processes ( group transformations ) which preserve the isomorphism property. One operation is abelianization of group does not preserve isomorphism. I mean to ...
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36 views

Groups with cyclic radicals

Let $G$ be a torsion-free group. For an element $g \in G \setminus \{1_G\}$ we define the radical of $g$ in $G$ as $$ \operatorname{Rad}_G(g) = \left\{r \in G \mid r^a \in \langle g \rangle \mbox{ ...
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45 views

Help me understand these topological properties of linear groups.

I'm studying Linear groups and already bamboozled after proving that there is a bijective correspondence from $SU_2$to $S^3$ and $SU_2$ can be thought of as the set of unit vectors in the quaternion ...
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1answer
182 views

Is it wrong to define $-1 \circ -1 = -1$

Problem: Quaternions are a set of objects that are an extension of imaginary numbers except that there are three of them $i$, $j$ and $k$, with the relations \begin{align*} i^{2} = j^{2} = k^{...
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Prove that $\langle F,\bullet \rangle$ is a group where $F=\{f\mid f:X→G\}$ and $\langle G,\ast \rangle$ is a group

Let $\langle G,\ast \rangle$ be a group and $X$ be a set. We define $F$ to be the set of all the functions from $X$ to $G$, meaning $F=\{ f\mid f:X \rightarrow G\}$. We define the operation $\bullet$...
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Existence of Complementary Subtorus

Suppose that $T$ is a closed subtorus (compact, connected, abelian subgroup) of $\mathbb{T}^d:=\mathbb{R}^d/\mathbb{Z}^d$. Is there a closed subtorus $T^\perp$ of $\mathbb{T}^d$ such that $\mathbb{T}^...
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68 views

Some nice basis for a subgroup of $Z^k$

I am currently reading the book "Algebra" by Hungerford. I saw some interesting theorem: If $F$ is a free abelian group of finite rank $n$ and $G$ is a nonzero subgroup of $F$, then there exists a ...
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84 views

Formal definition of subgroup / quotient group duality

Recently I read that a quotient group is the dual of a subgroup. This resource describes a lot of their properties. Wondering if: One could formally define both a subgroup and a quotient group ...
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Group action $GL(m,\mathbb{R}) \times GL(n,\mathbb{R})$

I've got some questions about an exercise. We consider $\mathcal{M}_{m \times n}(\mathbb{R})$ and $GL(m,\mathbb{R}) \times GL(n,\mathbb{R})$, and the action of group given by : $(P,Q).A = PAQ^{-1}$. ...
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Permutation Representations of an Interesting Family of Commutative Algebras

Consider a finitely generated (in fact finite) commutative algebra $A$ generated by projectors $P$ and (invertible) torsion elements $I$ with extra relations of the form $P*I = P$ for certain $P$ and ...
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49 views

Reconstructing groups from conjugacy class sizes

It is of interest to me to find if there is a way to reconstruct the group knowing the (multi)-set of its conjugacy class sizes or, in other words, given the indices of all centralizers is there a ...
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1answer
120 views

Sylow Counting Generalization of Hall Theorem

In a solvable group $G$, Hall's theorem (see, e.g., Th 9.3.1 in M. Hall's The Theory of Groups) implies that the number of Sylow-$p$ subgroups is a product of numbers each of which is congruent to $1$ ...
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189 views

Double Coset of Upper and Lower triangular matrices

This question is from Artin Algebra, Ch 2, Exercise M11(d): Most invertible matrices can be written as a product $A=LU$ of a lower triangular matrix $L$ and an upper triangular matrix $U$, where ...
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79 views

Set-theoretical inequality

Let $G$ be a group and suppose $E \subseteq G$ finite and symmetric (i.e. $g^{-1} \in E$, for all $g \in E$) . Define $E^n=\lbrace g_1g_2...g_n \mid g_i \in E \rbrace$, for all $n \in \mathbb{N}$. For ...
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1answer
158 views

Splitting field of $X^5-t$ in $\mathbb{Q}(t)[X]$

Given $K = \mathbb{Q}(t)$, $f(X) = X^5-t \in K[X]$ with $t$ trancedental over $\mathbb{Q}$. a) Is $f$ irreducible? b) Determine the degree of the splitting field of $f$ over $K$. c) ...
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Prove that $C_g(x) = gxg^{-1}$ is an isomorphism

Let $G$ be a group and $g \in G$. Define $C_g : G \to G$ as $C_g(x) = gxg^{-1}$ Prove that $C_g$ is an isomorphism. Homomorphism: $C_g(xy) = gxyg^{-1} = gx(gg^{-1})yg^{-1} = (gxg^{-1})(gyg^{-1}) = ...
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70 views

Generalization of the well-known commutator relation

Let $G$ be a finite group. Denote $$G^1 = G, G^2 = [G,G], G^3 = [G,G,G] = [[G,G],G], \ldots, G^k = [G^{k-1},G].$$ Suppose we consider a commutators of $n$ copies of $G$ that are not neccesarily left ...
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Practical application of enumerating groups

In learning a little about group theory I came across computational group theory which lead to this: https://groupprops.subwiki.org/wiki/Category:Groups_of_a_particular_order I also saw this: ...
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Difficulties with “old” definitions

I have a paper by Jacobson Nathan Jacobson.Structure of Rings, Volume 37, Part 1. American Math-ematical Soc., revised edition, 1956. Which really uses definitions that seem very complicated for ...
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186 views

A group where every element is its own inverse

Is it possible to find a group with more then say $30$ elements where $a*a=e$ for all $a \in G$ ? I'm having trouble thinking about this problem, I don't even know where to start in thinking of a ...
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1answer
63 views

Prove that for all elements $a$ of a group, $a^k a^l = a^{k+l}$ for all $k,l \in \mathbb{Z}$

This is my proof, using multiple induction. Base Case: $k = l = 1$ $$(a^1)^1 := a^1 := a$$ Assume the inductive hypothesis $a^k a^l = a^{k+l}$ For $k+1$ $$(a^{k+1})^l = (a^k a)^l = (a^k)^l a^l = ...
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What are the topological groups the points of which are separated by their finite-dimensional representations?

It is known that the continuous finite-dimensional irreducible unitary representations separate the points of Hausdorff compact topological groups (this is, in part, the Peter-Weyl theorem). These, ...
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Geometry and equivalence of orbit spaces

Consider the action of $G = {\rm SO}(2)$ by conjugation on real symmetric matrices, that is: $$g\cdot A = gAg^{-1} = gAg^{t},$$ for $A$ a symmetric real matrix $2 \times 2$. The author says that I can ...
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If $N\lhd G,g\in G$ show that $g\in N$

I have to show that: If $N\lhd G,g\in G$ and $[G/N]=n<+\infty$ with $gcd(m,n)=1,g^m\in N$ then $g\in N$ Attempt: I wrote $1=am+bn$ for some $a,b\in \mathbb{Z}$ so $g=g^{am+bn}=(g^{m})^{a}(g^n)^{...
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Show that $A_5$ has no subgroup of order $20$.

I have tried to solve it in the following way though I find little bit difficulty to reach at the desired conclusion.Here's my way. I take the action of $A_5$ on the set of all left cosets of $H$ in $...
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127 views

When Burnside's Lemma does not apply

I have a finite group G which acts on a set X. I want to establish the number of distinct members of a subset Y $\subset$ X, however Y is not closed under G (I think that the correct terminology here ...
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Order of $x$ in $\mathbb F_q[x]/(g(x))$

Let $\mathbb F_q$ be a finite field and $g(x)$ be a polynomial in $\mathbb F_q[x]$ with non zero constant term. Given the factorization of $g(x)=p_1(x)^{e_1}\dots p_k(x)^{e_k}$, give an estimate for ...
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Who discovered that normalizer of an abelian Sylow $p$-subgroup controls $p$-transfer?

Theorem: Let $G$ be a finite group and let $P\in Syl_p(G)$ and assume $P$ is abelian. Then $N_G(P)$ controls $p$-transfer. I wonder who discovered above theorem? Is it due to Burnside?
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Invariant factor decomposition of quotient group of two subgroups of $\mathbb{Z}^n$.

Determine the rank and the elementary divisors of the following group: $A/H$ with $A \subset \mathbb{Z}^5$ the group of all $5-$tuples with sum $0$ and $H = A \cap B(\mathbb{Z}^5)$ where $$ B=\begin{...
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Longest element of a subgroup

Say I have a finite Weyl group, $W$, and a set of generators $S:= \{s_1,...,s_k\}$ (making $W,S$ a coxeter system) and an automorphism $\theta: W\rightarrow W$ which permutes $S$. I know that the ...
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162 views

Right-to-left cycle multiplication in Wolfram alpha

Wolfram alpha carries out cycle multiplication from left-to-right. If the context dictates a right-to-left (function composition) operation, there probably is a way (short of rearranging the order of ...
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On constructing a group containing each subgroup once up to isomorphism

Let $\mathfrak{G}$ be the free product of all finite (say) groups. Then for every subgroup $K$ (finite or not), identify it with each subgroup of $\mathfrak{G}$ isomorphic to $K$ to obtain a quotient ...
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Examples of 1-cocycles on non-discrete groups

Let $G$ be a locally compact group. Let $\pi$ be an orthogonal representation of $G$ on a real Hilbert space $H$. A continuous mapping $b \colon G \to H$ such that $b(gh) = b(g) + \pi(g)b(h)$ for all $...
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Prove that $F(A \coprod B) = F(A) \ast F(B)$ in $\mathsf{Grp}$.

The problem statement above is from Algebra: Chapter 0 by Paolo Aluffi. As additional context, $F(A)$ was defined to be initial in the category whose objects consist of set-function, group pairs $(j, ...
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Proving a certain result regarding groups of order n where n is a cyclic number

I know the following result is true, but I haven't managed to find a proof to it: A cyclic number is a positive integer such that $\phi(n)$ is coprime to $n$, where $\phi(n)$ denotes Euler's totient ...
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142 views

Is a group which is equal to its derived subgroup necessarily semisimple?

let $G$ be either a (connected) Lie group or an (connected) algebraic group over a field (which is algebraically closed of characteristic zero). It is well known that if $G$ is semisimple then $G=G'$ ...
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stabilizer of an induced character

Suppose $K\trianglelefteq N\trianglelefteq G$ are finite groups (where $K$ isn't necessarily normal in $G$), and suppose $\chi\in\mathrm{Irr}(N)$ is induced from $\mu\in\mathrm{Irr}(K)$. Let $V=\...
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131 views

About normal closure a subgroup.

Let $H$ be a subgroup of $G$ where $\{H_i\mid i=1..,n\}$ is the set of all conjugate of $H$. Then $$H^G=H_1.H_2...H_n$$ where $H^G$ denotes the normal closure of $H$. This is an exercise from The ...
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435 views

Find all automorphisms of $\mathbb{Z}_{10}$

Since $\mathbb{Z}_{10}$ is a cyclic group (ex. generated by $\left<1\right>$), all automorphisms will be determined by finding $\phi(1)$. Since we want an isomorphism, we map $1$ to a generator, ...
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Proof verification that Aut$(A_4) \simeq S_4$

I've been watching a video proving $S_4 \simeq$ Aut$(A_4)$, but I was having trouble making sense of it. I wasn't quite sure I even understood exactly what I didn't understand, so I set out to prove ...
3
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1answer
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Action of $S_3$ on a set of subsets

Consider the group $S_3=\{1,x,x^2,y,xy,x^2y\}$ where $x=(123),\ y=(12)$. This group acts on the set $\mathcal P_3(S_3)$ of subsets of $S_3$ of cardinality 3 by left multiplication. Describe the ...