Let $X$ be a set and $f:X\to X$ be a function. Show that the relation $R_f$ defined in class is reflexive if and only if $f$ is the identity function.

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 $P$ be a partition of $X$. Show that there exist an equivalence relation on $X$ such that $X/\sim \ =P$.

### Week 2

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 in problem (4) above). Show that $R=\sim$.

Let $G$ be a group and $a\in G$. Show that if $a\cdot a=a$ then $a=e$.

Let $G=\{e,a\}$ be a set of two elements. Define a binary operator $*$ on $G$ such that $(G,*,e)$ is a group. Show that $G$ is abelian.

Let $X$ be a set and $S(X)$ be the set of bijection from $X$ to itself. Show that composition is a binary operator on $S(X)$ which makes $S(X)$ into a group.

### Week 3

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 $G_n=\{z\in \mathbb{C}: z^n=1\}$ for $n\in \mathbb{N}$ and $G=\cup_{n\ge 1}G_n$. Show that $G_n$ for all $n\in \mathbb{N}$ and $G$ are groups under multiplication. Also show that $G_3\cup G_5$ is not a subgroup of $G$.

Compute the order of the element $[k]_n$ of the group $\mathbb{Z}/n$ and show that the order of $\sigma=(1 2)(3 4 5)\in S_5$ is 6 by computing all the powers of $\sigma$.

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 $G$ be a group and $x,y\in G$ be of finite order such that $xy=yx$. Show that order of $xy$ divides the lcm of $|x|$ and $|y|$. Show that this fails if $x$ and $y$ do not commute.

Show that the symmetric group $S_n$ is generated by the set of elements of order $2$.

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.

Show that the order of any element $\sigma$ in $S_n$ is the l.c.m. of the lengths of all the disjoint cycles in the disjoint cycle decomposition of $\sigma$.

### Week 4

Write down all the left coset and right coset in $S_3$ of the subgroups:

(i) $<(1\ 2)>$ and

(ii) $<(1\ 2\ 3)>$.Find all group homomorphisms $f:(\mathbb{Z}/n,+)\to (\mathbb{Z},+)$.

Show that every cyclic group is isomorphic to either $\mathbb{Z}/n$ for some $n\in \mathbb{N}$ or $\mathbb{Z}$.

Show that $\mathbb{Z}/6\mathbb{Z}$ is not isomorphic to $S_3$.

Show that $(\mathbb{Q},+)$ is not isomorphic to $(\mathbb{Z},+)$.

Show that $(\mathbb{C}^*,\cdot)$ is not isomorphic to $(\mathbb{R}^*,\cdot)$.

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.

### Week 5

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 $(\mathbb{Z}/n)^*$ of question 13.

Let $H\le G$ be groups. Let $i:H\to G$ be the inclusion map and for $g\in G$, let $\phi_g:G\to G$ be the function $\phi_g(x)=g^{-1}xg$. Show that $Im(\phi_g\circ i)=g^{-1}Hg$.

Show that the center of $S_n$ is the trivial group for $n\ge 3$.

Show that the order $n$ subgroup $H$ consiting of rotations of the dihedral group $D_{2n}$ is a normal subgroup of $D_{2n}$.

Let $\phi:G_1\to G_2$ be an epimorphism. If $H_1 \trianglelefteq G_1$ then $\phi(H_1) \trianglelefteq G_2$.

### Week 6

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 $D_8$, let $r$ be the order 4 element corresponding to rotation and $s$ be the reflection. Compute the normalizer $N_{D_8}(< s >)$.

Let $G$, $G_1$ and $G_2$ be groups. Let $p_1:G\to G_1$ and $p_2:G\to G_2$ be group homomorphisms. Show that the map $\phi:G\to G_1\times G_2$ given by $\phi(g)=(p_1(g),p_2(g))$ is a group homomorphism. Also show that $\pi_i\circ\phi=p_i$ for $i=1,2$ where $\pi_i:G_1\times G_2\to G_i$ are the projection maps (defined in class).

Let $G$ be a group and $H$, $K$ be subgroups of $G$. Show that $HK$ may not be a group.

Show that there exist a group $G$ with subgroups $H$ and $K$ such that $G$ isomorphic to the (external) direct product $H\times K$ but $HK\ne G$ and $H\cap G\ne \{e\}$.

Let $G=S_3$ and $H=<(1\ 2)>$. Show that $(1)(2\ 3)H\ne (1\ 2)(1\ 3\ 2)H$ but $H=(1\ 2)H$ and $(2\ 3)H=(1\ 3\ 2)H$. Let $G$ be a group and $H\le G$. For $g_1,g_2,g_1',g_2'\in G$, suppose $g_1g_2H=g_1'g_2'H$ if $g_1H=g_1'H$ and $g_2H=g_2'H$. Show that $H$ is normal in $G$.

### Week 7

Let $G$ be a group. Show that the commutator subgroup $H=<\{xyx^{-1}y^{-1}:x,y\in G\}>$ is a normal subgroup of $G$.

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 $(A\times B)/(C\times D)=(A/C)\times (B/D)$.

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 $G=KH$ and the index $[K:K\cap H]=p$.

### Week 8

Let $\theta$ be an action of $G$ on a set $A$. Let $\phi_{\theta}$ be the induced homomorphism from $G$ to $S(A)$. Show that the action of $G$ on $A$ given by the homomorphism $\phi_{\theta}$ is same as $\theta$.

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 non-abelian finite group of order $pq$ where $p > q$ are primes. Show that $G$ is isomorphic to a subgroup of $S_p$.

Let a group $G$ acts on a set $S$ and $K$ be the kernel of the action. Let $H$ be a normal subgroup of $G$ contained in $K$. Show that the map $G/H\times S\to S$ sending $(gH,s)$ to $g.s$ is well-defined action of $G/H$ on $S$. Compute the kernel of this action.

### Week 9

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 $S_{n_1+n_2}$.

Compute all the conjugacy classes in $D_8$.

Let $G_1$ and $G_2$ be groups. 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 $X=X_1\times X_2$.

Let $p$ be a prime. Show that every group of order $p^2$ is abelian.

### Week 10

Let $\sigma\in S_n$ be the cycle $(1\ 2\ldots m)$. Show that the centralizer of $\sigma$ in $S_n$ is the subgroup generated by $\sigma$ and $S(\{m+1, m+2,\ldots,n\})$.

Compute the conjugacy classes in $A_4$.

Show that $A_4$ is not simple.

Show that $A_n$ is generated by 3-cycles for $n\ge 3$.

### Week 11

Let $q:G_1\to G_2$ be an epimorphism with $N=ker(q)$. Show that subgroups of $G_2$ are in natural bijection with subgroups of $G_1$ containing $N$, Under this bijection normal subgroups go to normal subgroups.

Show that a group of order 12 is not simple.

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 a group of order $pqr$ is not simple where $p$, $q$ and $r$ are primes.

Show that all the composition factors of a finite abelian group are of prime order.

Show that if $|G|=231$ then 11-sylow and 7-sylow subgroups are normal and $Z(G)$ contains the 11-sylow subgroup.

### Week 12

Show that $S_n$ is not a solvable group for $n\ge 5$.

Let $F$ be a field of $p$ elements $p$ a prime. Compute the order of $SL_2(F)$ and $PSL_2(F)$.

Let $G$ and $H$ be groups and $\phi:H\to Aut(G)$ be a group homomorphism. Let $G\ltimes H$ be the external semidirect product of $G$ and $H$ with respect to the action of $H$ on $H$ via $\phi$ (as defined in the class). Show that the maps $i_G:G\to G\ltimes H$ sending $g$ to $(g,e_H)$ and $i_H:H\to G\ltimes H$ sending $h$ to $(e_G,h)$ are injective group homomorphism. Let $G'=i_G(G)$ and $H'=i_H(H)$. Show that $G'$ is a normal subgroup of $G\ltimes H$ and $G\ltimes H$ is the internal semidirect product of $G'$ and $H'$.

Let $p$ be a prime. Show that $Aut((\mathbb{Z}/p)\times (\mathbb{Z}/p))=GL_2(\mathbb{Z}/p)$. Use it to construct a non-abelian group of order $p^3$ as a semidirect product.

Classify groups of order 28. (There are four isomorphism types)

### Week 13

Show that the map from the free group on the symbols $a_1,\ldots, a_n$ to a group $G$ given in the class is well-defined.