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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19 views

Difficulty understanding this definition of Galois extensions

I'm having some trouble understanding the definition of a Galois group given in class. It says that a field extension $E/F$ is a Galois extension if its group of automorphisms is such that $Fix(Aut(E/...
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25 views

Let $S_1 \leq S_2 \leq..$ be a strictly increasing chain of simple subgroups of a group $G$. Show that $\cup_i S_i$ is a simple subgroup of $G$

Let $S_1 \leq S_2 \leq..$ be a strictly increasing chain of simple subgroups of a group $G$. Show that $\cup_i S_i$ is a simple subgroup of $G$ So I understand that how to show that $\cup_i S_i \leq ...
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36 views

If $G = \langle x,y,z\rangle$ and $x \in \Phi(G)$, the intersection of all maximum subgroups of $G$, then $G = \langle y,z\rangle$

If $G = \langle x,y,z\rangle $ and $x \in \Phi(G)$, the intersection of all maximum subgroups of $G$, then $G = \langle y,z\rangle $. I have already proved that $\Phi(G) \triangleleft G$. So i'm ...
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1answer
24 views

1-ended Cayley graphs

I have been trying to understand the famous theorem for Cayley graphs, which states that, essentially, most infinite groups have a $1$-ended Cayley graphs (by characterizing precisely the cases in ...
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1answer
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Notation of a Unit Circle: Does $S^1$ only mean a unit circle? Or does it have something to do with symmetric groups?

I am currently dealing with some group theory problems. My algebra textbook denotes the unit circle on the complex plane by $S^1$. I am pretty confused because when I searched the term circle group ...
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19 views

subgroup of additive reals of index 2 [duplicate]

Is there a subgroup of the real numbers under addition of index 2? (and if so, can we classify them somehow?) [I am trying to solve the functional equation $f(x)f(y)=f(x-y)$ for all real $x,y$. If $f$...
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1answer
26 views

If $[\mathbb Z_p:H]=n$ then $n\mathbb Z_p< H$

I'm trying to prove that If $H$ is a subgroup of $\mathbb Z_p$ of a finite index, then $H = p^i\mathbb Z_p$ for some $i\in\mathbb N$. Here, $\mathbb Z_p$ is the p-adic integers. My problem is showing ...
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1answer
24 views

Calculate $|GL(n,q)|$ in which $n$ be a positive integer number [on hold]

Problem: Let $q$ be a power of a prime number. Calculate $|GL(n,q)|$ in which $n$ be a positive integer number. Here is the solution of Martin Isaac in his book "Finite group theory" I didn't ...
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2answers
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If ord(a) = n where n is odd, then ord(a^2) = n

$\DeclareMathOperator{\ord}{ord}$ Exercise 10.D.4 from Pinter says: Let $a$ be any element of finite order of a group $G$. Prove the following: If $\ord(a) = n$ where $n$ is odd, then $\ord(a^2) =...
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38 views

Should I care if some system forms a group?

I am currently studying a discrete dynamical system where the function composition on the set of periodic points forms a finite group. From a dynamical systems perspective, is there any reason to ...
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1answer
31 views

Validate my proof of for $\forall g \in G$, $|g|$ divides $|G|$ where $(G, \cdot)$ is a finite group.

First i want to talk about my motivation for this theorem and how i came up with this proof. In modular arithmetic, we have the group $U(n)$ closed under modular multiplication and by Euler's Theorem ...
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40 views

Determine whether $F^\times$ has a cyclic subgroup of order $10$ where $F=\mathbb{R}((t)),\mathbb{C}((t))$

I wanted to verify my attempt: Determine whether $F^\times=\langle F\setminus \{0\},\cdot, 1\rangle$ has a cyclic subgroup of order $10$ where: $F=\mathbb{C}((t))$. $F=\mathbb{R}((t))$. ...
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Control of fusion in GAP

Suppose that $G$ is a finite group, $p$ is a prime dividing $|G|$ and $P \in \operatorname{Syl}_p(G)$. A subgroup $N$ of $G$ which contains $P$ is said to control $G$-fusion in $P$ if whenever two ...
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1answer
41 views

Let G be a group of order $pqr$,where $p<q<r$ are primes. Prove that the Sylow r-group is normal.

I need to show that there exist unique r-Sylow subgroup. I know that the number of Sylow r-groups denoted by $n_r$ is congruent to $1mod \ r$. Also $n_r$ divides $pq$. So $n_r$ can be $1,p,q,pq$. But ...
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Proof Verification for Lagrange's Theorem [duplicate]

I am reading I.N.Herstein-Topics in Algebra Question: For $G$ a finite group. Let $|G|=n$, then $O(a)|n \enspace \forall a\in G$. Note: I have already proved that if $G$ is a finite group, then $...
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2answers
29 views

Find all elements of $D_4$ such that $s$ commute with them.

Consider the Dihedral Group $D_4$. Find all elements of $D_4$ such that $s$ commute with them. We know that $Z(D_4)=\{1,r^2\}$ Hence these elements commute with $s,sr$. Now I found that $s$ ...
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1answer
41 views

When the derived (commutator) subgroup is in the center

I have the following problem in group theory from a previous exam in a college course which stumped me: Let $ G $ be a group such that $ G/Z(G) $ is abelian. We are asked to show that for any fixed ...
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2answers
54 views

Order of a^k is a divisor of the order of a

Exercise 10.D.2 from Pinter says: Let a be any element of finite order of a group G. Prove the following: The order of <...
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1answer
39 views

Group generated by groups.

Given a group $G$ is a external product denoted by $G = \displaystyle \prod_{ i = 0 }^n G_i $ and $H_i$ be the image of the inclusion homomorphism $f_i : G_i → G$, show that $G = ⟨H_1,...,H_n⟩$. I ...
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1answer
47 views

generation of a symmetric group

Here is the question: My attempt: We know that any permutation can be expressed as a product of transpositions. Any kind of transposition $(a_{k}, a_{l})$ can be expressed as $(a_{k}, a_{k+1})(a_{k+1}...
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1answer
22 views

Left transversal of finite group that is also right transversal

I am working on proving the following fact: Let $G$ be a finite group and $H$ be a subgroup. Then there exists a complete set of left coset representatives for $G/ H$ which also form a complete set ...
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48 views

Is there a name for the function that maps to the identity of a group?

Let $G$ be a group with identity $e$. Let $f:G\to G$ such that $\forall x\in G:f(x)=e$ Does this function have a name?
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Quotients of the Cayley graph of $F_2$

Given the Cayley graph of the free group on two generators, I know that the quotient by $\{a^n b a^{-n}| n \in \mathbb{Z}\}$ is the real line with a loop at integer points. What about if we quotient ...
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How do I indicate the numbers not factorisable by some primes in the multiplicative group generated by all primes? [on hold]

How do I indicate the numbers not factorisable by some primes in the multiplicative group generated by all prime natural numbers? Let $\langle P\rangle$ be the free abelian group generated by all ...
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1answer
57 views

Why can't $S_3$ be a coproduct which contains $C_2$?

I am doing Aluffi's problem II.6.11 which states that: Since direct sums are coproducts in Ab, the classifications theorem for abelian groups mentioned in the text says that every finitely ...
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51 views

Elaboration of the proof that “a group is free iff it is isomorphic to $F[\mathscr{A}]$ for some $\mathscr{A}$”.

The proof is given below: But I do not understand the proof starting from the eighth line starting from "If now $G$ is ...." till the end of the proof, could anyone explain it to me in a ...
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1answer
36 views

Preimage of a cyclic group under the canonical epimorphism is cyclic

$\pi:G\twoheadrightarrow G/X$, $X=\langle x\rangle \subset G$, $A=\langle aX\rangle \subset G/X$, $x,a\in G$, $x\neq e$, $a\neq e.$ $\pi:g\mapsto gX$. $G$ is finite, abelian. I want to show that $\...
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1answer
78 views

Showing there's no $2$-group with specific number of elements of each order

Are there any theorems that I can use to show that there are no $2$-groups with a specific number of elements of each order aside from the fact that the number of elements of order $2$ is always odd, ...
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1answer
65 views

Dixmier's lemma as a generalisation of Schur's first lemma

What mathematicians call Schur's lemma is known to physicists as Schur's second lemma: An intertwiner of two irreducible representations of a group is either zero or isomorphism. It is valid for all ...
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$[A:B] = [\overline{A}: \overline{B}]$ quotient groups correspondence.

Consider $N \unlhd G$ and write $\overline{A}:= A/N$ for a subgroup $A \leq G$ containing $N$. I will write an element of the quotient group $G/N$ as coset $gN$. I want to prove that for $A \leq B$ ...
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A statement reciprocal to Schur's First Lemma

What the mathematicians call Schur's lemma is known to physicists as Schur's second lemma: An intertwiner of two irreducible representations of a group is either zero or isomorphism. It is valid for ...
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1answer
46 views

The Isomorphism for Smith Normal Form

Let $G$ be an abelian group that is generated by the elements $a$, $b$ and $c$ in it such that the relations $2a -4b =0$, $2b - 4c$ and $4a - 2c = 0$ generate all of the relations on $a$, $b$ and $c$. ...
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1answer
50 views

Is there a cleaner way to express the quotient $\frac{\langle(1,0),(0,1)\rangle}{\langle(2,-8)\rangle}$?

Is there a cleaner way to express the quotient $\frac{\langle(1,0),(0,1)\rangle}{\langle(2,-8)\rangle} \cong \frac{\mathbb{Z} \oplus \mathbb{Z}}{\langle(2,-8)\rangle}$? To give the question some ...
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A group that is also a category

I have a group with a category structure (i.e. category whose objects form a group), such that the left multiplication with any fixed element is a category automorphism. The same is true for right ...
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Is there an action of the abelianization of $SL(2,\mathbb{Z})$ on lattices?

Since there is an action of the group $\text{SL}(2,\mathbb{Z})$ on the lattice $\mathbb{Z}^2$, and knowing that its abelianization is the group $\mathbb{Z}_{12}$, is there a corresponding quotient ...
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Embed a Spin group into an Orthogonal group

How do we embed a Spin group to an Orthogonal group? Say a Spin($n$) group is a Lie group with $\frac{n \cdot (n-1)}{2}=$ Lie algebra generators. Say a Spin(10) group is a Lie group with $\frac{10 \...
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1answer
62 views
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Explicit group decomposition

Let $E/F$ be a quadratic extension of non-archimedean fields. Let $p$ its maximal ideal and $O$ its ring of integers. I am interested in the subgroup $A$ of invertible matrices of the form $$ \left( \...
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415 views

What is the period of Langton's ant on a torus?

Langton's ant runs on an infinite white grid. At every white square, it turns right, flips the color of the square, and moves forward one square. At every black square, it turns left, flips the color ...
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1answer
37 views

Group generated by commuting elements of PSL(2, $\mathbb{R})$

I read a question that says given $f,g \in PSL(2,\mathbb{R})$, such that $f \circ g = g \circ f$, show $g(Fix(f)) = Fix(f)$ where $Fix(f) = \{x \in \mathbb{H} \cup \partial \mathbb{H}| f(x) =x\}$. ...
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Restricted Representation of the Quaternion Group

I want to determine the branching rules from the quaternion group to the group of the fourth roots of $1$. I have produced the character tables for the groups:       &...
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1answer
72 views

Can I get disjoint cycles decomposition of $\sigma \in S_n$ from the partition of $I_n$ into orbits under the action of $\langle \sigma \rangle$?

I'm aware of this similar post, whose answer, though, is too "implicit" for my understanding. Then, I reformulate as follows. Given a permutation $\sigma \in S_n$, let's consider the cyclic subgroup ...
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Identifying subgraphs relevant to cosets in Cayley graph of a semidirect product [on hold]

When considering the Cayley graph (say $X$) of $(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_3$, $S=\{s,u,t\}$, where $|s|=|u|=5,|t|=3$, I have drawn one of the Cayley graphs of a semidirect ...
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20 views

Explicit form of $\mathfrak{so}(3,1)$ as $4\times4$ matrix

Is there a short way of finding the explicit general, real $4\times4$ matrix in $\mathfrak{so}(3,1)$? I would have started with the form for the $SO(3,1)$ group via $A^tgA=g$ with $g= \begin{pmatrix} ...
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2answers
42 views

$D_3 \simeq S_3$ why can't we map elements to different orders from the element itself? [on hold]

We know that $D_3 =\{e,r,r^2,s,rs,r^2s\}$ and $S_3$ has the following elements: $(12),(23),(13),(123),(213),e$ Why is it that we can't map elements in $D_3$ to different orders in $S_3$, in ...
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2answers
44 views

For what $n$ is there an injective homomorphism from $\mathbb{Z_n} \to S_7$

So I came across this problem: For what $n$ is there an injective homomorphism from $\mathbb{Z_n} \to S_7$ And I have the solution but the solution doesn't make much sense to me. It said that $n=1,...
2
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1answer
38 views

Transitivity of $\Omega$ in subspace actions

Let $G=O(V, Q)$ be a finite orthogonal group acting naturally on a space $V\cong \mathbb{F}_q^n$ equipped with a quadratic form $Q$. Assume $n=\dim V\geq c$ for some large enough constant $c$ in order ...
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0answers
25 views

Two dimensional irreducible representation of SO(3)

It is known that the irreducible representations of SO(3) are of dimension 1, 3, 5, etc. Can anyone give a proof that there is no two-dimensional irreducible representation of SO(3)? It is known ...
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1answer
24 views

The inverse image of a cyclic tower

The main goal is to show that if $G$ is finite, abelian, then $G$ admits a cyclic tower. The proof I was given conducts an induction on the order of $G$. i.e., if $|G|=1$ then $G=<e>$ so it is ...
2
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1answer
50 views

Embed a Spin group to a special unitary group

How do we embed a Spin group to a Unitary group? Say a Spin(10) group is a Lie group with $\frac{10 \cdot 9}{2}=45$ Lie algebra generators. Say a special unitary group SU($n$) has a $n^2-1$ Lie ...
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31 views

GAP: how to obtain the Young Symmetrizer?

Given a partition $\lambda$ of $n$ and a standard Young Tableaux filled with numbers from $1$ to $n$ (e.g. increasing row by row), how does one obtain the corresponding Young Symmetrizer using GAP? ...