Articles of group theory

Permutation isomorphic subgroups of $S_n$ are conjugate

Consider $G,H \leq S_n$ and their natural action on $[n] = \{1,\ldots,n\}.$ We say that $G$ and $H$ are permutation isomorphic if there is a bijection $\varphi:[n] \mapsto [n]$ and group isomorphism $f:G \mapsto H$ so that $$\varphi(g(o)) = f(g) (\varphi(o))$$ or in the standard notation involving group actions $\varphi(o^g) = \varphi(o)^{f(g)}.$ I would like […]

why is a polycyclic group that is residually finite p-group nilpotent?

I am trying to solve an exercise in D. Robinson’s book A Course in the Theory of Groups, which asks me to show that if $G$ is polycyclic and residually finite p-group for infinitely many prime p, then $G$ is nilpotent and finitely generated torsion-free. How do I show the nilpotent part? The hint given […]

Any periodic abelian group is the direct sum of its maximal p-subgroups

I have an exercise. I cannot solve. Please help me to solve it: Prove that any periodic abelian group is the direct sum of its maximal $p$-subgroups.

Proof of $G=N\rtimes H$ iff $G=NH$ and $N\cap H=\{1\}$

I have the following definition of a semi-direct product: Let $G$ be a group. Suppose $N\triangleleft G$ and $H<G$ such that every element of $G$ can be uniquely written $g=nh$. Then $G$ is the semi-direct product of $N$ and $H$. I have to prove the following lemma: Let $N \triangleleft G$ and $H<G$. Then $G=N\rtimes […]

Clarifications on the faithful irreducible representations of the dihedral groups over finite fields.

I would like to clarify a few things in the answer to this question: Faithful irreducible representations of cyclic and dihedral groups over finite fields 1) When a representation extends, what does the remaining generator map to under the representation? 2) Why does a representation extend if and only if $z^{-1} = z^{p^{d}}$ for some […]

Computing Invariant Subspaces of Matrix Groups

Does anyone have a program written in Mathematica (or SAGE or GAP) that computes the invariant subspace lattice of a matrix group?

In which of the finite groups, the inverse of Lagrange's Theorem is not correct?

This is a multiple choice for finite groups. For which one of the following groups, the converse of Lagrange’s Theorem is not generally satisfied? I know the converse is true for cyclic groups. 1) All abelian groups 2) All groups of order 8 3) The group $S_4$ 4) All groups of order 12 Thank you […]

Showing that the diagonal of $G \times G$ is maximal, where $G$ is simple

I have been trying to prove the following: Let $G$ be simple, and write $\Gamma=G \times G$. Let $D \le \Gamma$ be the diagonal subgroup, which consists of all elements of the form $(x,x)$, where $x \in G$. Show that $D$ is a maximal subgroup of $\Gamma$. As a hint I am given: Write $\Gamma=A […]

What are the finite order elements of $\mathbb{Q}/\mathbb{Z}$?

I need to find what are the at the group $\mathbb{Q}/\mathbb{Z}$. I think that any element at this group has a finite order, but I don’t know how to prove it… I’d like to get help with the proof writing… If I’m wrong, I’d like to to know it too… BTW: $\mathbb{Z}=(\mathbb{Z},+)$ $\mathbb{Q}=(\mathbb{Q},+)$ Thank you!

Cayley's Theorem – Questions on Proof Blueprint

Not a duplicate of this exquisite answer, the numbers in which I abide by here. Not querying the proof, hence please don’t discourse on it. Proof blueprint: Steps 1-2 in words. Left multiplication of every element in the group by a fixed element in the group constitutes a permutation of those elements. Steps 1-2 in […]