Let $\calO(x)$ denote the orbit of $x\in X$ under the action of the group $P$.
Let $X^P=\{x_1, x_2, \dots, x_m\}$.
The orbits of an element in $X^p$ under the action of $P$ is the element itself, that is, $\calO(x_i)=\{x_i\}$ for $i=1,\dots, m$. Let $x_{m+1}, x_{m+2},\dots, x_n$ be representatives of other orbits of $X$.
Then we have the decomposition of the set $X$ into a disjoint union of orbits
\[X=\calO(x_1)\sqcup \cdots \sqcup \calO(x_m)\sqcup \calO(x_{m+1})\sqcup \cdots \sqcup \calO(x_n).\]
For $j=m+1, \dots, n$, the orbit-stabilizer theorem gives
\[|\calO(x_j)|=[P:\Stab_P(x_j)]=p^{\alpha_j}\]
for some positive integer $\alpha_j$. Here $\alpha_j \neq 0$ otherwise $x_j \in X^P$.
Therefore we have
\begin{align*}
|X|&=\sum_{i=1}^m|\calO(x_i)|+\sum_{j=m+1}^n|\calO(x_j)|\\
&=\sum_{i=1}^m 1 +\sum_{j=m+1}^n p^{\alpha_j}\\
&=|X^P|+\sum_{j=m+1}^n p^{\alpha_j}\\
&\equiv |X^P| \pmod{p}.
\end{align*}
This completes the proof.
Group of Invertible Matrices Over a Finite Field and its Stabilizer
Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number.
Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, […]
The Order of a Conjugacy Class Divides the Order of the Group
Let $G$ be a finite group.
The centralizer of an element $a$ of $G$ is defined to be
\[C_G(a)=\{g\in G \mid ga=ag\}.\]
A conjugacy class is a set of the form
\[\Cl(a)=\{bab^{-1} \mid b\in G\}\]
for some $a\in G$.
(a) Prove that the centralizer of an element of $a$ […]
A Subgroup of Index a Prime $p$ of a Group of Order $p^n$ is Normal
Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer.
Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$.
Then prove that $H$ is a normal subgroup of $G$.
(Michigan State University, Abstract Algebra Qualifying […]
Subgroup of Finite Index Contains a Normal Subgroup of Finite Index
Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.
Proof.
The group $G$ acts on the set of left cosets $G/H$ by left multiplication.
Hence […]
Subgroup Containing All $p$-Sylow Subgroups of a Group
Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$.
Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$.
Then show that $N$ contains all $p$-Sylow subgroups of […]
If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup
Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$.
Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$.
Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.
Hint.
It follows from […]
Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center $Z(G)$
Let $G$ be a finite group of order $231=3\cdot 7 \cdot 11$.
Prove that every Sylow $11$-subgroup of $G$ is contained in the center $Z(G)$.
Hint.
Prove that there is a unique Sylow $11$-subgroup of $G$, and consider the action of $G$ on the Sylow $11$-subgroup by […]