Let $G$ be a simple group and let $X$ be a finite set.
Suppose $G$ acts nontrivially on $X$. That is, there exist $g\in G$ and $x \in X$ such that $g\cdot x \neq x$.
Then show that $G$ is a finite group and the order of $G$ divides $|X|!$.
Add to solve later Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$.
For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$.
Read solution
Add to solve later 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 $n$-dimensional vector space over $\F_p$. Therefore, $G_n$ acts on $\F_p^n$.
Let $e_n \in \F_p^n$ be the vector $(1,0, \dots,0)$.
(The so-called first standard basis vector in $\F_p^n$.)
Find the size of the $G_n$-orbit of $e_n$, and show that $\Stab_{G_n}(e_n)$ has order $|G_{n-1}|\cdot p^{n-1}$.
Conclude by induction that
\[|G_n|=p^{n^2}\prod_{i=1}^{n} \left(1-\frac{1}{p^i} \right).\]
Add to solve later Let $G$ be a finite group of odd order. Assume that $x \in G$ is not the identity element.
Show that $x$ is not conjugate to $x^{-1}$.
Read solution
Add to solve later Let $G$ be a finite group of order $n$ and suppose that $p$ is the smallest prime number dividing $n$.
Then prove that any subgroup of index $p$ is a normal subgroup of $G$.
Read solution
Add to solve later Determine whether a group $G$ of the following order is simple or not.
(a) $|G|=100$.
(b) $|G|=200$.
Read solution
Add to solve later Let $G$ be a finite abelian group of order $mn$, where $m$ and $n$ are relatively prime positive integers.
Then show that there exists unique subgroups $G_1$ of order $m$ and $G_2$ of order $n$ such that $G\cong G_1 \times G_2$.
Add to solve later Let $H$ be a subgroup of order $2$. Let $N_G(H)$ be the normalizer of $H$ in $G$ and $C_G(H)$ be the centralizer of $H$ in $G$.
(a) Show that $N_G(H)=C_G(H)$.
(b) If $H$ is a normal subgroup of $G$, then show that $H$ is a subgroup of the center $Z(G)$ of $G$.
Read solution
Add to solve later Let $G$ be a group of order $|G|=pqr$, where $p,q,r$ are prime numbers such that $p<q<r$.
Show that $G$ has a normal subgroup of order either $p,q$ or $r$.
Read solution
Add to solve later (a) Show that if a group $G$ has the following order, then it is not simple.
(b) Show that if the order of a group $G$ is equal to an even perfect number then the group is not simple.
Add to solve later In this post we review Sylow’s theorem and as an example we solve the following problem.
Show that a group of order $200$ has a normal Sylow $5$-subgroup.
Read solution
Add to solve later Determine all the conjugacy classes of the dihedral group
\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\]
of order $8$.
Add to solve later Let $D_8$ be the dihedral group of order $8$.
Using the generators and relations, we have
\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.\]
(a) Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$.
Prove that the centralizer $C_{D_8}(A)=A$.
(b) Show that the normalizer $N_{D_8}(A)=D_8$.
(c) Show that the center $Z(D_8)=\langle r^2 \rangle=\{1,r^2\}$, the subgroup generated by $r^2$.
Add to solve later Let $n$ be a positive integer. Let $D_{2n}$ be the dihedral group of order $2n$. Using the generators and the relations, the dihedral group $D_{2n}$ is given by
\[D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.\]
Put $\theta=2 \pi/n$.
(a) Prove that the matrix $\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta& \cos \theta
\end{bmatrix}$ is the matrix representation of the linear transformation $T$ which rotates the $x$-$y$ plane about the origin in a counterclockwise direction by $\theta$ radians.
(b) Let $\GL_2(\R)$ be the group of all $2 \times 2$ invertible matrices with real entries. Show that the map $\rho: D_{2n} \to \GL_2(\R)$ defined on the generators by
\[ \rho(r)=\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta& \cos \theta
\end{bmatrix} \text{ and }
\rho(s)=\begin{bmatrix}
0 & 1\\
1& 0
\end{bmatrix}\]
extends to a homomorphism of $D_{2n}$ into $\GL_2(\R)$.
(c) Determine whether the homomorphism $\rho$ in part (b) is injective and/or surjective.
Add to solve later Let $A$ and $B$ be normal subgroups of a group $G$. Suppose $A\cap B=\{e\}$, where $e$ is the unit element of the group $G$.
Show that for any $a \in A$ and $b \in B$ we have $ab=ba$.
Add to solve later Show that the center $Z(S_n)$ of the symmetric group with $n \geq 3$ is trivial.
Add to solve later Let $G$ be a group of order $|G|=pq$, where $p$ and $q$ are (not necessarily distinct) prime numbers.
Then show that $G$ is either abelian group or the center $Z(G)=1$.
Add to solve later An isomorphism from a group $G$ to itself is called an automorphism of $G$.
The set of all automorphism is denoted by $\Aut(G)$.
A subgroup $H$ of a group $G$ is called characteristic in $G$ if for any $\phi \in \Aut(G)$, we have $\phi(H)=H$. In words, this means that each automorphism of $G$ maps $H$ to itself.
Prove the followings.
(a) If $H$ is characteristic in $G$, then $H$ is a normal subgroup of $G$.
(b) If $H$ is the unique subgroup of $G$ of a given order, then $H$ is characteristic in $G$.
(c) Suppose that a subgroup $K$ is characteristic in a group $H$ and $H$ is a normal subgroup of $G$. Then $K$ is a normal subgroup in $G$.
Add to solve later Suppose the order of a group $G$ is $p^2$, where $p$ is a prime number.
Show that
(a) the group $G$ is an abelian group, and
(b) the group $G$ is isomorphic to either $\Zmod{p^2}$ or $\Zmod{p} \times \Zmod{p}$ without using the fundamental theorem of abelian groups.
Add to solve later Let $Z(G)$ be the center of a group $G$.
Show that if $G/Z(G)$ is a cyclic group, then $G$ is abelian.
Add to solve later 
