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.

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Every Cayley Digraph of $\mathbb{Z}_{n}$ is isomorphic to a Metacirculant digraph

Let $m,n$ be positive integers and $\alpha\in \mathbb{Z}_n^*$. Define $\rho, \tau:\mathbb{Z}_m\times \mathbb{Z}_n \to \mathbb{Z}_m\times \mathbb{Z}_n$ by $$\rho(i,j) = (i,j+1)$$ $$\tau(i,j) = (i+1,\...
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2 votes
1 answer
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show that $H\cap Z(P)>\{1_{G}\}$

I working on exrcise that state: Let G be a group, and subgroups $P,H\leq G$. Also, P is a p-subgroup(prime p), H abelian, $H\trianglelefteq G$. Assume, $H\cap P >\{1_{G}\}$. Show that $H\cap Z(P)&...
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action of the extra-special group

I'm reading a paper which has this line: A direct computation shows that P$\Omega_8$($\mathbb K$) has an elementary abelian subgroup $X = 2^2$ such that $C_{P\Omega_8$(\mathbb K)}(X) = T_4.2^{1+4}_+$. ...
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3 votes
2 answers
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Is this inverse image nontrivial?

I would like to prove that if a non-comutative group $G$ has a subgroup $H$ whose index is equal to $3$ or $4$, then $G$ is not simple. In order to do, I took a group action $$\phi : G \times G/H \ni (...
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Closure of the commutator group of Lie solvable group

Help me please to understand the following. Let $G$ be a connected Lie solvable group. Let $G':=\overline{[G,G]}$ be a closer of the commutator group $[G,G]$. Is it possible that $G'=G$? If so what if ...
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-1 votes
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A subgroup is equal to a group if it has elements of the conjugacy classes [duplicate]

Let $G$ be a group and $H$ be a subgroup of $G$. If $H$ contains at least one element of each conjugacy classes of $G$, then it holds that $H = G$? What would be the proof if this is true?
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2 votes
3 answers
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Intersection of two subgroups with order 39 and 65 is cyclic

Following is Question 9 of Chapter 10 of Abstract Algebra by Dan Saracino: Suppose $G$ is a group and $H$ and $K$ are subgroups of $G$ such that $|H|=39$ and $|K|=65$. Prove that $H \cap K$ is cyclic. ...
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  • 1,087
3 votes
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Groups and Cosets

I am studying the proof of the third Sylow's Theorem and I dont get this: Let $G$ be a finite group, $H$ a $p$-Sylow subgroup of $G$, $N(H)= \{g \in G : gH=Hg \}$. Note that $G = \cup_{g \in G}AgA$ (...
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Let $\phi : G\to G'$ be a homomorphism and let $S'\subseteq G'$. Is $\phi^{-1}(\langle S'\rangle ) = \langle \phi^{-1}(S')\rangle$?

I've shown that $\phi^{-1}(\langle S'\rangle ) \supseteq \langle \phi^{-1}(S')\rangle$ and was wondering whether the other inclusion holds, although I've not been able to prove it nor find a ...
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Difficulty with converting Yoneda's natural isomorphy into a group isomorphism in the proof of Cayley's theorem

$\newcommand{\A}{\mathscr{A}}\newcommand{\Gc}{\mathscr{G}}\newcommand{\G}{\mathcal{G}}\newcommand{\s}{\mathsf{Set}}\newcommand{\op}{^{\mathsf{op}}}\newcommand{\sym}{\mathsf{Sym}}$I am having ...
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1 vote
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Subgroups of $\mathbb{Z}\rtimes\mathbb{Z}$.

I don't know whether it is an easy or a hard question, but for the proof of a stronger statement I need to know what the subgroups of $\mathbb{Z}\rtimes\mathbb{Z}=\langle x,y\mid x^{-1}yx=y^{-1}\...
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Determining if a finite matrix group is irreducible in the Adjoint representation

Let $ G $ be a finite group of matrices. In particular suppose that $ G $ is a finite subgroup of $ SU_n $. What would be the best way to use GAP to figure out if the adjoint representation of $ G $ ...
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2 votes
1 answer
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Automorphism group of $\mathbb{Q}/\mathbb{Z}$

Consider the group of all complex roots of unity, $\mathbb{Q}/\mathbb{Z}$ (where both groups are additive groups). I was wondering what its automorphism group is ?
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2 votes
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Identification of a small commutative group

Given the binary operator $*$ of a finite (even small) commutative group, literally or as a table, how can I proceed to identify the name a mathematician knowing group classification would call it (...
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Continuous group action and $\sigma(f(ω))=f(\sigma(ω))$

Let $G$ be a topological group(For example, absolute galois group ${\rm Gal}(\overline{L}/L)$), and $G$ acts continuously on topological field $L$, via $G\times L\to L$, $(\sigma,a)\to \sigma(a)$. ...
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  • 398
3 votes
1 answer
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Induced representations in GAP

Background. In a previous question I asked how one defines a (modular!) representation in GAP, and I came to learn that one just has to provide GAP with the entire group homomorphism using ...
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  • 233
3 votes
1 answer
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Haar measure on isomorphic subgroups

Let $G$ be an abelian locally compact Hausdorff group endowed with a chosen Haar measure $\mu$. Moreover, let $H,N$ be subgroups of $G$ where $N$ is discrete. Then we have the isomorphic quotient ...
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  • 53
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2 answers
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Let $H = \{ \sigma\in S_5\mid\sigma(5)=5\}$. Let $K$ be a subgroup of $S_5$. Prove that $HK=S_5$ iff 5 divides $|K|$.

Let $H = \{ \sigma\in S_5\mid\sigma(5)=5\}$. Let $K$ be a subgroup of $S_5$. Prove that $HK=S_5$ iff 5 divides $|K|$. Attempt: I found $|H|=24$. I know that $|HK|=\frac{|H||K|}{|H \cap K|}$. Suppose $...
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1 answer
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How to prove the restricted Lorentz Group is connected?

Many textbooks claim that $SO^{+}\left(1,n\right)$ is the identity component of $O\left(1,n\right)$. But how do we know that $SO^{+}\left(1,n\right)$ is connected itself?
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-4 votes
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Proof the follow exercise of group theory [closed]

let $a \in G$ where $G$ is finite. If $a^n$ has $m$ conjugates and $a$ has $k$ conjugates, the m|k.
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4 votes
1 answer
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Gorenstein's proof of the classification of solvable CN-groups

I am reading Gorenstein's Finite Groups. Chapter 14 is about CN-groups, a (finite) group where the centralizer of every non-identity element is nilpotent. Theorem 14.1.5 gives the classification of ...
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Show that if $G$ is finite and $G$ acts on $X$ and it is transitive, then $1=\dfrac{1}{|G|}\sum_{g\in G}|\operatorname{Fix}(g)|$ [closed]

I know you have to relate the $\sum_{g\in G} |\operatorname{Fix}(g)|$ with $\sum_{x\in X} |\operatorname{Stab}(g)|$ and use the orbit theorem and stabilizer but I get very confused.
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3 votes
2 answers
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Proving a different version of the Third Isomorphism Theorem for groups

Theorem: let $\phi_1:G_1\to G_2$ and $\phi_2:G_2\to G_3$ be surjective homomorphisms, then $$\frac{G_1/\ker \phi_1}{\ker(\phi_2\circ\phi_1)/\ker(\phi_1)}\cong \frac{G_1}{\ker(\phi_2\circ\phi_1)}.$$ ...
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-1 votes
0 answers
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Extension of a representation

My question is related with this Representation abelian subgroup of an abelian group. I want to show that X extends to a representation of G, which I think it's obvious from G being defined by its ...
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2 votes
0 answers
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Class of finite groups where all noncentral conjugacy classes have the same size but three different character dimensions

I'm looking for a class of groups where there are only two possible conjugacy class sizes, say $1$ and $k$, but at least three distinct dimensions of the irreducible characters. I know the conjugacy ...
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5 votes
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Is $\text{GL}_2(\mathbb{R})/\mathbb{R}^{\times}$ isomorphic to $\text{SL}_2(\mathbb{R})$?

Let $\text{GL}_2(\mathbb{R})$ be the set of real $2\times 2$ invertible matrices (where the operation is matrix multiplication). It has $\text{SL}_2(\mathbb{R})=\{ A \in \text{GL}_2(\mathbb{R}| \det A ...
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-1 votes
0 answers
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Discrete abelian group of finite rank [closed]

By definition of rank, a torsion-free abelian group of rank $k$ is isomorphic to a subgroup of $\mathbb{Q}^k$. So it is true that a discrete torsion-free abelian group of rank $k$ is isomorphic to a ...
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-1 votes
1 answer
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Sylow 2-subgroup of Suzuki Group $Sz(8)$

I need to find the isomorphism class containing the Sylow 2-subgroup of the Suzuki group $Sz(8)$. Can anyone give a reference?
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7 votes
1 answer
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Schönert & Seress Algorithm - Computing all block systems - blocks of imprimitivity

Atkinson as well as Schönert and Seress describe methods to compute the minimal block system; in particular in Permutation Group Algorithms by Ákos Seress, we find Theorem 5.5.1 Suppose that a set S ...
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3 votes
1 answer
155 views

What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$?

What is the general number of solutions of the equation $y^2=x^3-x\pmod{p}$, where $p$ is a prime number and $p>3$?. Calculations suggest that the number of solutions to this equation is $p$ if $p\...
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4 votes
4 answers
97 views

On local group rings

Let $G$ be a finite group and $k$ a field of characteristic $p$. Suppose that the ring $k[G]$ is local. How this implies that G is a p-group? Since all maximal ideals are prime, the nilradical $nil(k[...
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-1 votes
1 answer
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Prove that $S_n=⟨(12),(23),...,((n-1)n)⟩$.

Assume we know that $S_n=⟨(12),(13),...,(1n)⟩$, then prove $S_n=⟨(12),(23),...,((n-1)n)⟩$ I didn't understand this proof below: For $i=2$ at $(1 \ i)$ so $(1 \ i) \in S_n$, then assuming that $(1 \ i)$...
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0 votes
1 answer
26 views

Finitary Alternating Groups

If $X$ is an infinite set, then the finitary alternating group on $X$ can be defined in the following equivalent ways: the group of all even permutations on $X$ under composition the kernel of the ...
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-1 votes
1 answer
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Homomorphism and Isomorphism importance

From group theory, two groups $(G,\cdot)$ and $(S,*)$ are homomorphic if there is a map $f$ such that $f(a\cdot b)=f(a)*f(b)$. While these groups are isomorphic if the map $f$ is homomorphism and ...
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1 vote
0 answers
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Is the Galois group of $\mathbb{Q} \left(\sqrt{5}+\sqrt{7}+i \right)$ isomorphic to $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times\mathbb{Z}_{2} $?

First of all, I proved that $\mathbb{Q}(\sqrt{5}+\sqrt{7}+i)=\mathbb{Q}(\sqrt{5},\sqrt{7},i)$.Found that $|G|=|Gal(\mathbb{Q}(\sqrt{5},\sqrt{7},i)|=[\mathbb{Q}(\sqrt{5},\sqrt{7},i):\mathbb{Q}]=8$. ...
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0 votes
0 answers
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Group cohomology reference

I'm interested in studying group cohomology (for discrete groups). Are there accessible (lecture) notes that give a nice overview of the basics for group cohomology and develop the categorical ...
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1 vote
1 answer
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Naming convention: Lie groups with finitely generated discrete part

Consider a Lie group $G$ and let $G_0$ be the connected component of the identity. Then $G_{\mathrm{dis.}}:=G/G_0$ is a discrete group. We are writing a (physics) article in which finite-dim. Lie ...
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  • 107
2 votes
0 answers
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Representation abelian subgroup of an abelian group

Let G a non abelian group of order $p^{3}$ and exponent $p$, where $p$ is a prime. Let $w$ be a primitive complex p-th root of unity. So far, I have proved that: $Z(G)=[G:G]$ $|Z(G)|=p$ $G\cong \...
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3 votes
2 answers
145 views

Difference between products and coproducts.

I am struggling with understanding the difference between products and coproducts in category theory. For the category of abelian groups what part of the universal property of coproduct implies the ...
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  • 437
0 votes
0 answers
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Discrepancy between order of the identity element & its inclusion in n-torsion sets

The order of the identity element of a group is usually considered to be one because $1 * e = e$ or $e^1 = e$. However, the text on torsion points (from Mathematical Cryptography by Silverman, Pipher ...
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  • 359
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0 answers
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How to conclude the order of $G$ is $p^{m+n}$.

I have this exact sequence $$ 0\to \mathbb{Z}_{p^m}\overset{f}\to G\overset{g}\to \mathbb{Z}_{p^n}{\to} 0 $$ where $G$ is finitely generated. I want to prove that $|G|=p^{m+n}$ and a friend told me to ...
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2 votes
0 answers
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Prove a nonempty set $G$ with an associative operation $\ast$ is a group iff the following equations are satisfied $y \ast g = h$ and $g \ast x = h$ [duplicate]

I'm currently working on an exercise and the body of the text for the exercise is as follows. I have a first draft of the proof but am missing some things and am unsure about some things as well so ...
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2 votes
0 answers
27 views

Prove that a normal subgroup $G$ of $S_4$ with $(12)\in S_4$ is equivalent to the entire group $S_4$ [duplicate]

Consider a normal subgroup $G$ of $S_4$. The simple transposition $(12)\in G$. Prove that $G=S_4$. I have already proved that the above case implies that $(12),(23),(34)\in G$. These are all the ...
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  • 319
2 votes
0 answers
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Show that $\langle a\rangle$ is a normal subgroup of $\langle a,b\rangle$ where $a$ and $b$ are permutations

I'm studying for an exam and in a previous year we had the question: Let $a=(12345678)$, and $b=(26)(48)$ and $G=\langle a,b \rangle$. Show that $\langle a \rangle$ is a normal subgroup of $G$ and by ...
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0 votes
0 answers
39 views

Does a $Z_N$ finite group approach a $U(1)$ in the limit of $N\rightarrow \infty$? [closed]

I was thinking that since a $Z_N$ symmetry can be represented as $\{e^{2\pi i\frac{k}{N}}\}$ for $k\in\{0,1,...,N-1\}$ and a continuous $U(1)$ symmetry can be represented as $e^{i\theta}$ for $\theta\...
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0 votes
1 answer
29 views

On nonsplit noncentral extension of finite simple groups

Let $G$ be a finite group and $N$ be its minimal normal subgroup such that $G/N$ is a finite simple group. It is well know that if $G=G'$ and $N=Z(G)$, then $N=\Phi(G)$. My Question is: Is it true ...
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  • 359
0 votes
0 answers
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show that $x^3 = (1234)$ in $S_7$ has three solutions (and find them?) [duplicate]

As title. This is question 52 in Chapter 5 of Gallian’s Abstract Algebra, 10th edition. My current line of logic is as follows. We have $x^3 = (1234)$, which gives us $|x^3| = 4$, implying that $|x| = ...
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10 votes
1 answer
131 views

Permutation Groups: Find $x$ such that $x^5 = (12345)$

I am wondering about how to solve question 35 from chapter 5 (Permutation Groups) from the 10th edition of Gallian’s Abstract Algebra. The full question is as follows: What is the smallest $n$ for ...
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1 vote
0 answers
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Construct subgroup of $\mathbb{Z_2} \times \mathbb{Z_2}$ where both groups involved in the direct product are not subgroups of $\mathbb{Z_2}$

I am tasked in an exercise to do the following: Construct an example of a subgroup of $\mathbb{Z_2} \times \mathbb{Z_2}$ which is not of the form $K \times J$ for some $K < \mathbb{Z_2}$ and $J <...
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2 votes
1 answer
35 views

Prove that the direct product of $2$ subgroups is a subgroup

I have an exercise where I am tasked to prove that for $2$ subgroups $K < G$ and $J < H$ of $2$ groups $G,H$ the following is a subgroup: $$K \times J \subset G \times H$$ I believe I have done ...
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