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.

3
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1answer
714 views

Group Which Isn't Solvable

For a recent qualifier problem, I was to show that if a group $G$ has two solvable subgroups $H,K$ such that $HK=G$ and $H$ is normal, then $G$ is solvable. This is simply a matter of showing that $G/...
6
votes
2answers
151 views

A question about normal subgroups in group theory

If $K$ is a normal subgroup of $H$, and $H$ is a normal subgroup of $G$, is it true that $g^{-1} K g$ is a normal subgroup of $H$ for all $g \in G$? I think I know the answer, but I just want to ...
4
votes
1answer
464 views

PSL(2,7) as subgroup of A_7

As $PSL(2,7)$ has a subgroup of index $7$, and $PSL(2,7)$ is simple, hence it can be embedded in $A_7$. How many copies of $PSL(2,7)$ are in $A_7$? (There should be at-least 15 copies: If $H\leq A_7$...
2
votes
1answer
241 views

Amalgamated Products

Let $H,G,G'$ be groups such that $H$ injects into both $G$ and $G'$. Then we may form the amalgamated product $G\ast_H G'$. Is the canonical map $G\rightarrow G\ast_H G'$ always ...
3
votes
2answers
99 views

Concerning the existence of a group isomorphism

Let $G = \left\{\begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \textrm{ with } a=\pm 1, b \in \mathbb{Z}, c= \pm 1 \right\} \subset GL_2(\mathbb{R})$ and $H = \left\{\begin{pmatrix} a & ...
6
votes
0answers
484 views

Alternating and special orthogonal groups which are simple

I have an idle curiosity about a funny coincidence between two sequences of groups. It is well-known by those who know it well that the alternating group $A_d$ of degree $d$ is simple if and only if $...
6
votes
1answer
280 views

Is there a standard category-theoretic way to express a loop or quasigroup?

The standard way to encode a group as a category is as a "category with one object and all arrows invertible". All of the arrows are group elements, and composition of arrows is the group operation. ...
28
votes
2answers
6k views

Group of positive rationals under multiplication not isomorphic to group of rationals

A question that may sound very trivial, apologies beforehand. I am wondering why $( \mathbb{Q}_{>0} , \times )$ is not isomorphic to $( \mathbb{Q} , + )$. I can see for the case when $( \mathbb{Q} ,...
11
votes
1answer
625 views

Groups with order divisible by $d$ and no element of order $d$

It occurred to me that I somehow believe the following statement without actually knowing how to prove it: for every composite natural number $d$ there is a group whose order is divisible by $d$ yet ...
12
votes
3answers
1k views

Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$

Problem Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1$ then $p \equiv 1 \pmod{8}$ My attempt was, Since $p$ divides $n^4 + 1 \implies n^4 + 1 \equiv 0 \pmod{p} \...
3
votes
1answer
225 views

A problem from Isaacs' FGT

This is problem 3B.12. from the afore mentioned book. It asks to show that, if $G$ is a solvable Frattini-free group (that is, $Φ(G)=1$) and $H$ is a subgroup of some maximal subgroup $M$ of $G$, then ...
12
votes
1answer
4k views

Showing that a cyclic automorphism group makes a finite group abelian

From a bank of previous masters exams: Let $G$ be a finite group such that its automorphism group $\operatorname{Aut}(G)$ is cyclic. Prove that $G$ is abelian. Here's what I was thinking. Let $\...
2
votes
2answers
572 views

Isomorphism between HNN extension and semidirect product

Studying a course on geometry and groups, I fell on the following property (which is not given as an exercise, but rather as an observation). Let $A$ be any group, let $B \cong \langle t \rangle \...
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votes
2answers
1k views

Generalization of index 2 subgroups are normal

Let $G$ be a finite group and $H$ a subgroup of index $p$, where $p$ is a prime. If $\operatorname{gcd}(|H|, p-1)=1$, then $H$ must be normal. Does somebody have a quick proof of this?
5
votes
2answers
393 views

publishing an article which contains abstract math and programming on arxiv or journal

I have a soft question. I currently study abstract math which includes group theory. On the other hand, I have strong background in c++, data structures etc. So, most of the time I end up with an ...
6
votes
2answers
781 views

Conjugacy Classes of subgroups in GL(n,p)

What is the conjugacy class of subgroups of order $p$ in $GL(n,p)$? (Are all subgroups of order $p$ conjugate in $GL(n,p)$?)
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5answers
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Counting Elements of Order 2 in $\mathbb{Z}^{\times}_{n}$

The Euler totient function $\varphi(n) = |\mathbb{Z}^{\times}_{n}|$ is even on $\mathbb{N}_{>2}$, so it is feasible that the group $\mathbb{Z}_{n}^{\times}$ can support an element of order $2$. If $...
3
votes
2answers
225 views

Character theory questions

I am following the text by Isaacs on character theory and I have a few questions. From p. 10, it seems like an reducible representation is one whose matrix at each group element can be written in a ...
2
votes
1answer
775 views

All subgroups of a finite abelian p-group

Given a finite abelian p-group: $G = \displaystyle\prod_{i=1}^n p^{k_i}\mathbb{Z}_{p^k}$ for some integers $k,k_1,...,k_n$. Regarding elements of G as tuples $(x_1,...,x_n) \in G$, I can get ...
2
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1answer
554 views

A proof of Sylow theorem

Having proved the Sylow theorem for general linear group over finite field, how to prove it for any finite group?
5
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1answer
351 views

Presentation of Borel subgroup of GL(2,p)

The Borel subgroup $B$ of GL(2,p) is the subgroup group of upper triangular matrices. It is easy to see that it is (internal) semi-direct product of two subgroups: $B=U\rtimes T$ where $U$ is the ...
9
votes
1answer
386 views

Isomorphism between $I_G/I_G^2$ and $G/G'$

Ok, this has been bugging me for a while, and I'm sure there's something obvious I'm missing. The references I've looked at for this result in an effort to resolve the issue didn't address it. $G$ is ...
10
votes
2answers
224 views

Non isomorphic groups who product with Z is isomorphic [duplicate]

Are there groups $G$ and $H$ such that $G$ and $H$ are not isomorphic but $G \times \mathbb Z$ and $H \times \mathbb Z$ are?
4
votes
1answer
208 views

Reference / Survey article on automorphisms of groups

can one suggest a survey article on automorphisms of $p$ groups, and automorphisms of abelian groups/ abelian $p$ groups?
4
votes
2answers
392 views

The Frattini subgroup contains a power of a certain subgroup

This is from an article by Hall from 1961; it's probably one of the most trivial observations in that article, but I can't get the reasoning. Let G be some group and let A be a normal abelian p-...
10
votes
3answers
457 views

Residual Finiteness of Fundamental Groups of Seifert Fibered Spaces

I'm trying to understand why, if $S$ is a Seifert fibered space, then $\pi_1(S)$ is residually finite. From theorems 12.2 and 11.10 in Hempel's "3-manifolds", we can work with a finite-sheeted ...
3
votes
2answers
996 views

Homomorphism of Groups and divisibility of orders

I've been studying for my exams and bumped across this one particular question that I've been having a tad difficulty on: Suppose that $\phi$ is a homomorphism from a finite group $G$ onto $H$ and ...
3
votes
2answers
301 views

How to prove a permutation group of degree n can be generated by (at most) n-1 elements?

Given a subgroup G of $S_n$. I want to prove that G can be generated by (at most) n-1 elements. All my ideas so far seem irrelevant. A hint would be greatly appreciated. My attempts so far: n-1 ...
4
votes
1answer
351 views

Semidirect Products of the form $(\mathbb{Z}_p \times \mathbb{Z}_p)\rtimes_{\phi}\operatorname{GL}(2,p)$

What are the different (non-isomorphic) semidirect products $(\mathbb{Z}_p \times \mathbb{Z}_p)\rtimes_{\phi}\operatorname{GL}(2,p)$, when $\phi \colon \operatorname{GL}(2,p)\rightarrow\operatorname{...
42
votes
3answers
57k views

Multiplication in Permutation Groups Written in Cyclic Notation

I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, if $$ a=(1\,3\,5\,2),\quad b=(2\,5\,6),\quad c=(1\,6\,3\,4), $$ then ...
1
vote
2answers
173 views

order of a group

let $D = <x, y | x^2, y^2, (xy)^n>$. What is the order of $D$? Thank you very much.
2
votes
1answer
300 views

questions about simple groups

how to show that there is no simple group of order $1755 = 3^3 \cdot 5 \cdot 13$? Thank you very much.
6
votes
3answers
453 views

Basic Representation Theory

I'm reading a paper that uses representation theory, and I'm stuck on something simple. Say $\Delta$ is an abelian group, and $\hat{\Delta}$ its group of irreducible characters. Say $M$ is a $\mathbb{...
6
votes
1answer
982 views

Symmetries of Cube

The group of orientation preserving isometries of Cube is $S_4$. But if we allow orientation reversing isometries also, then the group will be of order 48. What is this group (Structure)? ( Part of ...
4
votes
1answer
157 views

A bounded infinite cycle as a product of bounded involutions

Let $\sigma$ be a permutation of $\mathbf Q.$ We call $\sigma$ bounded (the term might be somewhat misleading, but however it is used in a couple of papers) if there is a real number $M$ such that $$ ...
2
votes
1answer
584 views

Automorphisms of $\mathbb{R}^n$

It is well known that the only ring automorphism of $\mathbb{R}$ is the identity. This follows from the fact that all ring automorphisms of $\mathbb{R}$ must fix $0$, and be order preserving, and ...
0
votes
2answers
224 views

Stabilizers in SL(2,Z)

If A is a matrix in SL(2,Z) is it directly obvious the stabilizer of A , G_A = {B in SL(2,Z) | A.B = A}, is the set containig only the Identity? or is this not true?
2
votes
0answers
141 views

Representation theory question + SU(n)

Would you please help me in how to solve these questions : A- Let $H$ be a subgroup of a finite group $G$.Let $\alpha$ and $\beta$ be class function of $G$ and $H$ respectively. Prove that $$Ind^{G}...
4
votes
2answers
1k views

Subgroups whose order is relatively prime to the index of another subgroup

Suppose that $H, K$ are subgroups of a finite group $G$, with $|H|$ relatively prime to $|G:K|$. Does it necessarily follow that $H \leq K$, or is there a counterexample? This question arose from ...
3
votes
2answers
517 views

An 'opposite' or 'dual' group?

Let $(G, \cdot)$ be a group. Define $(G, *)$ as a group with the same underlying set and an operation $$a * b := b \cdot a.$$ What do you call such a group? What is the usual notation for it? I tried ...
0
votes
2answers
118 views

subgroup of $\mathbf{Z}_9$ generated by $8$

If I find the elements generated by $8$, can I say that the set of these elements is a subgroup of $\mathbf{Z}_9$ provided all these elements are in $\mathbf{Z}_9$? or to show that the set of these ...
2
votes
2answers
334 views

Isomorphic groups

I know that there is a formal definition isomorphism but for the purpose of this homework questions, I call two groups isomorphic if they have the same structure, that is group table for one can be ...
5
votes
2answers
296 views

A question about cyclic groups

How does one see that the cyclic group $C_n$ of order $n$ has $\phi(d)$ elements of order $d$ for each divisor $d$ of $n$? (where $\phi(d)$ is the Euler totient function)
2
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1answer
382 views

The existence of a $p$-Sylow subgroup

this question was posted on aops about 3 weeks ago, but never received an answer. Maybe it will fare better here? Let $G$ be a finite group and $H$ a subgroup. Let $P_H$ be a $p$-Sylow subgroup of $...
4
votes
1answer
199 views

Questions about p-adic representations

In a paper I'm currently reading, they have the following situation: $k$ is some number field that doesn't have a primitive $p^{th}$ root of unity, and $k(\zeta_p)$ a field above it with Galois group ...
2
votes
1answer
314 views

System of blocks form a partition?

I need help with next statement : Let $G$ be a group acting transitively on a set $A$. A nonempty subset $B$ of $A$ is called a block for $G$ if for each $x\in G$ either $B^x=B$ or $B^x \cap B=\...
7
votes
1answer
398 views

A torsion-free quotient?

Let $G$ be a group and $T$ the set of elements of finite order in $G$. If $T$ is a subgroup of $G$, then $G/T$ is a torsion-free group. Suppose $G$ is a compact Hausdorff topological group. Is it ...
3
votes
1answer
946 views

Proof that a group action is primitive in terms of point stabilisers

Can someone help me with proof of the next statement? Suppose that the group $G$ acts transitively on the set $A$, and let $H$ be the stabiliser of $a\in A$.Then $G$ acts primitively on $A$ if and ...
1
vote
1answer
243 views

Embedding of general linear groups

Can we embedd $GL(2n,q)$ into $GL(n,q^2)$ for $n\in \mathbb{N}$ and $q=p^m$, $p$ a prime? If yes, how?
2
votes
4answers
1k views

some short proofs on group theory

Let $G=\{a_2, a_2, a_3...a_n\}$ be a finite abelian group of order $n$. Let $x=a_1 a_2 a_3...a_n$. Prove that $x^2=e$. My solution: Well since the order of the group is finite, we know the order of ...