Show that every vector space has a basis using Zorn's lemma.
Show that for any positive integer $N$, the relation on $\mathbb{Z}$ given by $R_N=\{(a,b): a,b\in \mathbb{Z}, \text{ and } N \text{ divides } a-b\}$ is an equivalence relation.
Find an example of a relation which is:
(a) reflexive and symmetric but not transitive.
(b) symmetric and transitive but not reflexive.
(c) transitive and reflexive but not symmetric.
Let $f:X\to Y$ be a function. Define a relation $R$ of $X$ as follows: for $a,b\in X$, $aRb$ if $f(a)=f(b)$. Show that $R$ is an equivalence relation.
Let $\sim$ be an equivalence relation on a set $X$ and $Y=X/\sim$. Let $f:X\to Y$ be the function sending an element $a\in X$ to its equivalence class $[a]$. Let $R$ be the relation on $X$ associated to the function $f$ (as defined above). Show that $R=\sim$.
Let $(G,*,e)$ be a group containing exactly three elements. Show that $G$ is abelian.
Let $G$ be a group and $x,y\in G$ be such that $x\cdot y=y\cdot x$. Also assume that $<\{x\}>\cap<\{y\}>$ is the trivial subgroup. Show that $ord(xy)=lcm(ord(x),ord(y))$.
For $\sigma \in S_n$, show that $ord(\sigma)=lcm(n_1,\ldots,n_k)$ where $n_i=ord(\tau_i)$ and $\sigma=\tau_1\ldots\tau_k$ is a disjoint cycle decomposition of $\sigma$.
Let $G = \{ x \in \mathbb{R} : 0\le x<1\}$ and for $x, y \in G$ let $x * y$ be the fractional part of $x + y$ (i.e., $x * y = x + y - [x + y]$ where $[a]$ is the greatest integer less than or equal to $a$). Prove that $*$ is a well defined binary operation on $G$ and that $G$ is an abelian group under * (called the real numbers mod 1).
Let $\mu_n=\{z\in \mathbb{C}: z^n=1\}$ for $n\in \mathbb{N}$ and $G=\cup_{n\ge 1}\mu_n$. Show that $\mu_n$ for all $n\in \mathbb{N}$ and $G$ are groups under multiplication. Also show that $\mu_3\cup \mu_5$ is not a subgroup of $G$.
Let $(\mathbb{Z}/n)^*=\{[a]\in \mathbb{Z}/n:(a,n)=1\}$. Show that this is a group under multiplication. Compute the order of all the element of this group for $n=12$.
Let $\sigma$ and $\tau$ be the following: $\sigma=\begin{pmatrix} 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11\\ 3& 9& 7& 2& 11& 10& 6& 5& 1& 4& 8 \end{pmatrix}$ and $\tau=\begin{pmatrix} 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11\\ 11& 10& 9& 8& 7& 6& 5& 3& 4& 1& 2 \end{pmatrix}$. Compute disjoint cycle decompositions of $\sigma$, $\sigma^2$ and $\sigma\tau$. Also compute their orders.
Write down all the left coset and right coset in $S_3$ of the subgroups:
(i) $<(1\ 2)>$ and
(ii) $<(1\ 2\ 3)>$.
Show that if $f:G_1\to G_2$ and $g:G_2\to G_3$ are group homomorphisms then $g\circ f$ is also a group homomorphism.
Show that the dihedral group $D_8$ and the quaternion group $Q$ are not isomorphic.
Show that the group of automorphisms $Aut(\mathbb{Z}/n)$ is isomorphic to the group $(\mathbb{Z}/n)^*$.
Show that the order $n$ subgroup $H$ consiting of rotations of the dihedral group $D_{n}$ is a normal subgroup of $D_{n}$.
Show that the center of $S_n$ is the trivial group for $n\ge 3$ and the center of $GL(n,\mathbb R)$ is {$cI: c\in \mathbb R, c\ne 0$}.
Prove or disprove the following statements:
(a) Let $G$ be any group and $H\le G$. Then $C_G(H) \trianglelefteq G$.
(b) Let $G$ be any group and $H\le G$. Then $H\le C_{G}(H)$.
(c) Let $G$ be any group and $H\trianglelefteq G$ and $K\trianglelefteq H$ then $K\trianglelefteq G$.
In the dihedral group $D_4$, let $r$ be the order 4 element corresponding to rotation and $s$ be the reflection. Compute the normalizer $N_{D_8}(< s >)$.
Let $G$ be a group and $H$ a normal subgroup of $G$ of index $p$ for some prime number $p$. Let $K\le G$ be such that $K\not\subset H$ then show that $G=KH$ and the index $[K:K\cap H]=p$.
Let $A$ and $B$ be groups, $C\trianglelefteq A$ and $D\trianglelefteq B$ then $C\times D$ is a normal subgroup of $A\times B$ and $(A\times B)/(C\times D)=(A/C)\times (B/D)$.
Write down explicitly a subgroup of $S_8$ which is isomorphic to $Q_8$ (the quaternions).
Let $H=< s>$ be the subgroup of $D_{2n}$ generated by a reflection $s\in D_{2n}$. Show that the action of $D_{2n}$ on the set of left cosets of $H$ is faithful.
Let $G$ be a finite group and $G/Z(G)$ be cyclic group. Show that $G$ is abelian. Use this to conclude that every group of order $p^2$ is abelian for a prime number $p$.
For $i=1,2$, let $G_i$ act on a set $A_i$ faithfully and suppose $|A_i|=n_i$. Show that $G_1\times G_2$ is isomorphic to a subgroup of $S_{n_1+n_2}$.
Show that $X$ is a conjugacy class in $G_1\times G_2$ iff there exist $X_1$ and $X_2$ conjugacy classes in $G_1$ and $G_2$ respectively such that $X=X_1\times X_2$.
Compute all the conjugacy classes in $D_8$.
Prove that if $|G|=2907$ then $G$ is not a simple group.
Let $G$ be a non-abelian finite group of order $pq$ where $p > q$ are primes. Show that $G$ is isomorphic to a subgroup of $S_p$.
Let $G$ be a transitive subgroup of $S_n$.
a) Show that $G$ is 2-transitive iff $Stab_G(n)$ acts transitively on {$1,\ldots, n-1$}.
b) Show that if $G$ is two transitive then it is primitive.
Let $G$ be a group and $H$ be finite normal subgroup of $G$. Let $K$ be $p$-sylow subgroup of $H$. Show that if $K\trianglelefteq H$ then $K\trianglelefteq G$.
Show that if $|G|=231$ then 11-sylow and 7-sylow subgroups are normal and $Z(G)$ contains the 11-sylow subgroup.