0
votes
0answers
98 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and every ...
0
votes
3answers
169 views

quasiprimitive non-solvable groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
5
votes
1answer
364 views

Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups?

Suppose $A$ and $B$ are finitely generated Abelian groups. Are all exact sequences of the form $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split? If not, is there an example? ...
3
votes
1answer
120 views

intersection growth of free profinite groups

Let $F_n$ be a free profinite group of finite rank $n$ and let $V_k$ denote the intersection of all open subgroups of $F_n$ of rank at most $k$ ($k \in \mathbb{N}$). My questions are: can I ...
2
votes
0answers
108 views

Hall's paper on the profinite groups and Andre Weils “voisinage” notion

I am reading through a classical paper A Topology for Free Groups and Related Groups by Marshall Hall Jr. in which profinite groups are defined for the first time. There he defines on p. 129: ...
-3
votes
1answer
127 views

finite index, self-normalizing subgroup of $F_2$ [closed]

Denote $F_2=\langle a, b\rangle$ to be the free group on two generators $a, b$. Let $H\leq F_2$ to be a subgroup with finite index $n$, so $H\cong F_{n+1}$ by Nielsen–Schreier theorem, recall that ...
2
votes
0answers
133 views

Abelian Subgroup in an infinite non-abelian 3-group

Does an infinite non-abelian 3-group of exponent greater than or equal to 9 has an infinite abelian subgroup? I know that 2-groups and 3-group of exponent 3 has an infinite abelian subgroup. I wonder ...
5
votes
1answer
220 views

growth rate of $\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$?

I am interested in the growth rate of this type of group: $G=\mathbb{Z}^2\rtimes_{\sigma} \mathbb{Z}$, where $\sigma(a)=\begin{pmatrix}x&y\\z&w\end{pmatrix}\in SL_2(\mathbb{Z})$, where $a$ is ...
1
vote
2answers
342 views

Semiring naturally associated to any monoid?

For any monoid $M$, we can naturally construct a semiring $S$ as follows: Let the additive monoid of $S$ be the free commutative monoid on $M$ Let the multiplicative monoid of $S$ be $M$ Then, if ...
1
vote
1answer
557 views

How to solve a system of equations over permutations?

Imagine you have a $n\times n$ matrix filled in with permutations over $n$ elements. Now you pick one permutation from each row randomly starting from the first row and by multiplying them get a ...
3
votes
1answer
308 views

Character theory of $2$-Frobenius groups.

This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there. Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and ...
1
vote
0answers
291 views

about union of conjugate proper subgroups in a math paper

My question is about the shaded area in this image. Does the symbol $L=\bigcup_{g \in G} T^{g}$ means that $L$ is a union of sets or $L=\langle T^{g}, g\in G \rangle$? If it means the first one, then ...
6
votes
3answers
1k views

Why are ring actions much harder to find than group actions?

I admit freely that the following question is a bit of a fishing expedition inspired by this lovely "definition" of a module as found on Wikipedia: A module is a ring action on an abelian group. ...
0
votes
1answer
464 views

Maximal subgroups of a finite p-group

I want to prove the following: Let $G$ be a finite abelian $p$-group that is not cyclic. Let $L \ne {1}$ be a subgroup of $G$ and $U$ be a maximal subgroup of L then there exists a maximal subgroup ...
1
vote
2answers
333 views

Abstract Commensurator Group of $\mathbb{Z}^n$ $Comm(\mathbb{Z}^n)\cong GL(n,\mathbb{Q})$?

Hello! In a paper I read that $\mathrm{Comm}(\mathbb{Z}^n)\cong \mathrm{GL}(n,\mathbb{Q})$. Why is that true? How can I find an isomorphism of this groups? I know that ...
0
votes
0answers
119 views

On X-s-permutable subgroups of a finite group

I want to prove Lemma 2.1(1) in the paper On X-s-Permutable Subgroups of a Finite Group by Min Bang SU, Yang Ming LI. It is on the web. This is my proof. . Since $H$ is $X−s−$permutable in $G$, then ...
9
votes
1answer
344 views

Given a rational number a/b does there exist a finite group G and an automorphism f s.t. f maps exactly a/b elements of G to their own inverses?

I was helping a friend prepare for his intro abstract final and he mentioned the professor had once asked the question name a group and an automorphism that takes 3/4 of the elements of the group to ...
5
votes
0answers
275 views

Can any group be realized as the multiplicative group of a ring? [duplicate]

Possible Duplicate: Ring with Z as its group of units? Given a group $G$, does there always exist a ring $R$ such that $R^\times \cong G$? I feel like this isn't true but that's just a hunch. ...
9
votes
2answers
894 views

Example of a Group which has $\text{SL}_{n}(\mathbb{Z})$ as the automorphism group

For the past one week, I have been trying to learn more about Automorphism groups of different groups. Very recently one of my friend asked this question to me: What is the Autmorphism group of ...
5
votes
2answers
1k views

Example of an infinite abelian but non-cyclic group whose automorphism group is cyclic.

Can anyone give me an example of: An infinite abelian but non-cyclic group whose automorphism group is cyclic.
3
votes
3answers
551 views

About solvable groups

Is it possible for a group (non-simple and non-abelian) that solvability of all of its proper subgroups leads the whole group to be solvable?
6
votes
3answers
838 views

Does “finitely presented” mean “always finitely presented”, considered in general

I'm wondering about the question, "If we have a finitely presented __, is it necessarily finitely presented with respect to any finite generating set for it?" I know this is true for groups and for ...
2
votes
2answers
250 views

Admissible finite groups

I want to know when an abelian group of even order is admissible (or has a complete map)? And when a nonabelian group of even order is admissible (or has a complete map)? Thanks.
4
votes
2answers
2k views

Sylow's theorem 3rd Proof Page 96 I.N.Herstein

I was just going through the 3rd Proof of Sylow's theorem given in the "Topics In Algebra" Book by I.N. Herstein. It looked very interesting and i really liked its Philosophy. My question what is its ...
1
vote
2answers
848 views

Extension problem

As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...
25
votes
3answers
2k views

Feit-Thompson Theorem: The Odd Order Paper

For reference, the Feit-Thompson Theorem states that every finite group of odd order is necessarily solvable. Equivalently, the theorem states that there exist no non-abelian finite simple groups of ...
7
votes
10answers
5k views

Learning Algebra & Group Theory on my own [closed]

I'm learning Algebra & Group Theory, casually, on my own. Professionally, I'm a computer consultant, with a growing interest in the mathematical and theoretical aspects. I've been amazed with ...
11
votes
3answers
913 views

Sign of infinite permutations?

Let $S_\infty$ the group of permutations of $\mathbb{N}$. It can be shown that there is no homomorphism $S_\infty \to \mathbf{Z}/2$ extending the sign on the finite symmetric groups. Is it possible to ...
39
votes
5answers
3k views

when is A isomorphic to A^3?

this is totally elementary, but I have no idea how to solve it: let $A$ be an abelian group such that $A$ is isomorphic to $A^3$. is then $A$ isomorphic to $A^2$? probably no, but how construct a ...