Group actions on sets

Let $\Phi$ and $\Phi^{\prime }$ be homomorphisms of a group $G$ into $S(\Omega )$ and $S\left(\Omega^{\prime }\right)$, respectively. Then the actions on $\Omega$ and on $\Omega^{\prime }$ that they define are called equivalent if there exists a bijective map $\sigma : \Omega \longrightarrow \Omega^{\prime }$, making the diagram

commutative for all $g \in G$. Thus, $\Phi_{g}^{\prime }=\sigma \Phi_{g} \sigma^{-1}$. Prove that every transitive action of a group $G$ is equivalent to the action of $G$ on the left cosets with respect to some subgroup $H$.

Relying on Theorem 2, prove that all groups of order $p^{2}$ ($p$ a prime number) are abelian.

Show that the center of the group $P$, given at the end of Example 1, has the form

Find the conjugacy classes of the group $P$.

Let $n$ be a natural number. Write it as a sum $n=n_{1}+n_{2}+\ldots +n_{m}$ with $n_{1} \geqslant n_{2} \geqslant \ldots \geqslant n_{m} \geqslant 1$. Denote by $p(n)$ the number of all such partitions with $m=1,2, \ldots$, so that $p(3)=3, p(4)=5$, etc. The decomposition $\pi =\pi_{1} \pi_{2} \ldots \pi_{m}$ of every permutation $\pi \in S_{n}$ into a product of independent cycles (see [ВА I, Ch. 1, § 8]) uniquely determines a partition of the number $n$. Show that the conjugacy classes of the group $S_{n}$ are in bijective correspondence with the partitions of the number $n$.

Let a permutation $\pi \in S_{n}$ be written as a product of $r$ cycles of length $1, s$ cycles of length 2, $t$ cycles of length 3, and so on, so that $n=r+2 s+3 t+\ldots$ Show that the cardinality of the conjugacy class in $S_{n}$ containing the permutation $\pi$ is given by the formula

Let a group $G$ act on a set $\Omega$. Let us call a subset $\Gamma \subset \Omega$ invariant with respect to $G$ (or $G$-invariant) if $g x \in \Gamma$ for all $g \in G$ and $x \in \Gamma$. For example, invariant sets under the action of $S O(2)$ on $\mathbb {R}^{2}$ are concentric rings.

Fig. 1

Show that every invariant subset of $\Omega$ is a union of orbits, and that the $G$-orbit of any element $x \in \Omega$ is precisely the smallest invariant subset containing $x$.

Show that for a group $G$ with a subgroup $H$, the action $H \times G \longrightarrow G$, defined by the shift $(h, g) \mapsto h g$, gives a partition of $G$ into right cosets of $G$ with respect to $H$.

By modifying the proof of Theorem 1, obtain the relation

where $r(G: \Omega )$ is the number of orbits of a permutation group $G$ acting on a set $\Omega$.

Alongside the centralizer $C(a)=\left\{ x \in G \mid x a=a x\right\}$ in a group $G$, one also considers the skew centralizer

which occurs in the theory of dynamical systems. In general, $D(a)$ is not a group.

Prove that:

1. $D(a)$ is a group $\Longleftrightarrow a^{2}=e$ and $D(a)=C(a)$;

2. the set $\mathbb {E}\left[a\right]=C(a) \cup D(a)$ is always a group.