List questions from group-actions

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Let $H \subset G$ be finite groups. Suppose that $H$ is a $p$-group and $p$ divides $[ G : H ]$. Prove that $p$ divides $[N_G(H) : H ]$.

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Consider the natural action of $G$ on the set $\{1,2,3,4\}$. Write down the orbits of the action and find the stabilizer of each point

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List Question

Show that for any $g\in G$, the stabiliser of $g.x$ is the subgroup $gG_xg^{-1}$

How block system are related to each other?

Show that a group action on the set of $1D$ subspaces is transitive

Action of $S_3$ on a set of subsets

Finite group acting freely on a smooth manifold (De Rham cohomology)

Confusions in group theory

Algorithm to computes the wreath product?

Abstract Algebra - Stabiliser of colored tetrahedron

Construct $\mathbb{RP}^n$

Group action on a completely regular space

Conjugation action and stabilizer group of order 24

Faithful group action via automorphisms

Let $H \subset G$ be finite groups. Suppose that $H$ is a $p$-group and $p$ divides $[ G : H ]$. Prove that $p$ divides $[N_G(H) : H ]$.

Completely reduciblity from a $G$-group to its normal subgroup

Consider the natural action of $G$ on the set $\{1,2,3,4\}$. Write down the orbits of the action and find the stabilizer of each point

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