Articles of group actions

Prove that a non-abelian group of order $pq$ ($p<q$) has a nonnormal subgroup of index $q$

So I’ve come up with a proof for the following question, and I’d like to know if it’s correct (as I couldn’t find anything online along the lines of what I did). Question Let $p$ and $q$ be primes with $p<q$. Prove that a non-abelian group of order $pq$ has a nonnormal subgroup of index […]

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 […]

Group Operations/ Group Actions

I’m currently taking my first abstract algebra course and am learning about group actions, orbits, and stabilizers. I’m reading the Artin textbook and I am not very clear of what exactly a group action allows us to do, what it looks like, and why it’s important. I know the two properties that must be satisfied […]

Algebra – Infinite Dihedral Group

Let $G$ be the set of bijections $\mathbb{R} \to \mathbb{R}$ which preserve the distance between pairs of points, and send integers to integers. Then $G$ is a group under composition of functions. The following two elements are obviously in $G$: the function $t$ (translation) where $t(x)=x+1$ for each $x \in \mathbb{R}$ and the function $r$ […]

Normal subgroup $H$ of $G$ with same orbits of action on $X$.

I have a somewhat broad question related to group actions and their restriction to a normal subgroup. If we have a group action $\sigma : G \times X \rightarrow X$ with orbits $G_x$, and a normal subgroup, $H$ of $G$, such that the restriction of the action $\sigma$ to $H$, $\sigma |_H : H \times […]

How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$?

Suppose the additive group $\mathbb Z^n$ acts on $\mathbb R^n$ through translation. How to show $\mathbb R^n/\mathbb Z^n$ is diffeomorphic to torus $\mathbb T^n$? The translation action is given by $$\psi_g:\mathbb R^n\rightarrow \mathbb R^n,\ x\mapsto g+x.$$

Complex projective line hausdorff as quotient space

I was wondering if there is a simple argument showing that the complex projective line defined as $\mathbb{CP^1} = \big(\mathbb{C}^2 \setminus \{0\}\big)/{\mathbb{C}^{\times}}$ is hausdorff when equipped with the quotient topology. So far I was picturing this scenario by analogy with $\big(\mathbb{R}^3 \setminus 0\big)/\mathbb{R}^{\times}$ and the 2-sphere therein. Imagining open, disjoint double cones surrounding distinct lines […]

Transitive subgroup of symmetric group

I’m working on the following question, and honestly have no idea how to begin. Any hints would be greatly appreciated! Let $H$ be a subgroup of $S_n$, the symmetry group of the set $\{1,2,\dots, n\}$. Show that if $H$ is transitive and if $H$ is generated by some set of transpositions, then $H=S_n$.

Isomorphic but not equivalent actions of a group G

This is in some sense a continuation of this problem. Given a group $G$ I would like to exhibit two actions of $G$ on a set $[n] =\{1,\ldots,n\}$ such that the two actions are isomorphic yet not equivalent. To recall we say that two group actions $\alpha,\beta : G \mapsto S_n$ are isomorphic if there […]

How are $G$-modules and linear group actions different

Let $M$ be an abelian group and let $G$ be a group acting on $M$ such that $M$ is a $G$-operator group, i.e. we have for $u, v \in M$ and $g,h \in G$ (1) $u\cdot 1_G = u$ (2) $(ug)h = u(gh)$ (3) $(u+v)g = ug + vg$ If $M$ is also a $\mathbb […]