Convention. All numbers here will be integers, and $p$ will be a prime number.

∆ Definition 1. A binary operation on a set $M$ is a map from the set of ordered pairs $M^2=\{ (a,b) | $ $ a \in M, b \in M\}$ into $M$. I.e. it is a way to establish a correspondence between a pair of elements from $M$ and an element from $M$. Image of pair $(a,b)$ is denoted by $a \cdot b $.

∆ Definition 2. A group is a pair $(G, \cdot)$ consisting of a set G and a binary operation ( $\cdot$ ) on it such that the following properties are satisfied: 1) $\forall a,b,c \in G: a \cdot (b \cdot c) =(a \cdot b) \cdot c$ (associativity); 2) $\exists e \in G\ \forall a \in G: a \cdot e =e \cdot a = a$ (the existance of the identily element); 3) $\forall a \in G\ \exists a^{-1} \in G : a \cdot a^{-1}= a^{-1} \cdot a = e$ (the existence of the inverse). A group $G$ is called finite group if there are finite number of elements in $G$. The number of elements in a finite group is called the order of a group and is denoted by $|G|$.

Convention. Associativity condition means that the result of multiplication does not depend on the way you place parentheses in an expression. From now on, we will not write any brackets in a multiplication of more than two elements.

Problem 1. Which of the following sets with operations are groups? a) Integers with addition $( \mathbb{Z} , + )$; b) Integers with substraction $( \mathbb{Z} , - )$; c) Natural numbers with addition $ ( \mathbb{N} , \cdot )$ ; d) Permutations with composition $ ( S_n , \circ )$; e) The set of even numbers with addition; f) The set of odd numbers with addition; g) The set of maps $X \rightarrow X$ with composition; h) The set $P(A)$ of all subsets of set $A$ with operation union $\cup$; i) $(P(A), \cap)$; j) $(P(A), \setminus)$; k) $(P(A), ∆)$, where $A ∆ B= (A \cup B) \setminus (A \cap B)$; l) $(\mathbb{Z}/ n \mathbb{Z} , +_n)$, where $\mathbb{Z}/ n \mathbb{Z} = \{ 0,1, \dots,n-1 \}$ and $a+_n b$ is the remainder when dividing $a+b$ by $n$; m) $(\mathbb{N}, \cdot)$, where $a \cdot b = a^b$, where $a\cdot_nb$ is the reminder when dividing $ab$ by $n$.; n) $(\mathbb{Z}/ n \mathbb{Z} \setminus \{0\} , \cdot_n)$; o) $(\mathbb{Z}/ n \mathbb{Z} ^{\times} , \cdot_n)$, where $\mathbb{Z}/ n \mathbb{Z} ^{\times} = \{ a \in\mathbb{Z}/ n \mathbb{Z} \ | \ GCD(a,n) =1 \}$. Send a solution

∆ Definition 3. Group $G$ is called commutative (or abelian) if for any $a,b \in G$, we have $ab=ba$.

Problem 2. Which groups from Problem 1 are commutative? Send a solution

Problem 3. Show that a) the identity element is unique; b) the inverse element is unique; c) $ba=e \ \Rightarrow b=a^{-1}$; d) $ba=a \ \Rightarrow b=e$; e) $ (a^{-1})^{-1}=a$. Send a solution

Problem 4. * Show that if in Definition 2 you substitute the properties of the existance of the identity element the existence of the inverse by: $1^{\circ}) \ \exists e \in G \ \forall a \in G : ea=a$ (left identity element); $2^{\circ}) \ \forall a \in G \ \exists a^{-1} \in G : a^{-1} a = e $ (left inverse), then you will get an equivalent definition of a group. Send a solution

∆ Definition 4. A map $f: G \rightarrow H$ from group $G$ to group $H$ is called isomorphism if it is bijective (one to one map) and preserves the group operation, i.e. $\forall x,y \in G: f(x \cdot y)=f(x) \cdot f(y)$. If such a map exists we say that groups $G$ and $H$ are isomorphic.

Problem 5. Write down all pairwise not isomorphic groups consisting of a) 1,2,3; b) 4; c*) 13 elements. Send a solution

∆ Definition 5. A non-empty subset $H$ of a group $G$ is called a subgroup if it is closed under operations $\cdot$ and taking the inverse, i.e. the result of applying the operation to elements from $H$ gives an element from $H$ and the inverse of an element from $H$ belongs to $H$.

Problem 6. Is it true that: a) if $H$ is a subgroup of $G$, then $e \in H$; b) if $H$ is a subgroup of $G$, then $H$ is a group; c) if $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $K$ is a subgroup of $G$; d) the union of two subgroups is a subgroup; e) the intersection of two subgroups is a subgroup? Send a solution

Problem 7. Is it true that: a) $\mathbb{N}$ is a subgroup of $\mathbb{Z}$; b) $A_n$ is a subgroup of $S_n$, where $A_n$ is the set of even permutations on a set of order $n$. c) $S_n \setminus A_n$ is a subgroup of $S_n$? Send a solution

Problem 8. Describe all the subgroups of: a) $S_3$ b) $\mathbb{Z}$. Send a solution

∆ Definition 6. The order of element $a \in G$ is the smallest natural number $k$ such that $a^k=e$. Notation: $ord \ a$. If such a number does not exist we say that $ord \ a= 0$.

Problem 9. Show that for any element $a$ of a finite group, we have $ord \ a >0$. Send a solution

Problem 10. Show that $a^n=e$ if only if $ord \ a \ | \ n $, i.e. $n$ is divisible by the order of $a$. Send a solution

∆ Definition 7. The left coset of subgroup $H$ in group $G$ is the subset $aH=\{ax \ | \ x \in H \}$. Analogously, the right coset is the subset $Ha=\{xa \ | \ x \in H \}$.

Problem 11. Show that the left (right) cosets either do not intersect or coincide with each other. Send a solution

Problem 12. Write down the partition on left and right cosets of the following groups: a) $\mathbb{Z} / \ 2 \mathbb{Z}$; b) $S_n / \ A_n$; c) $S_3 / \ \langle (1,2) \rangle$, where $\langle (1,2) \rangle$ is the minimal subgroup of $S_3$ that has trasposition $(1,2) = \bigl(\begin{smallmatrix} 1 & 2 & 3 \\ 2 & 1 & 3 \end{smallmatrix}\bigr) \ $ in it. Send a solution

Problem 13. (Lagrange's Theorem). Show that for any finite group $G$ the order a subgroup $H$ divides the order of $G$, i.e. we have $| G | \mathrel{\vdots} | H |$. Send a solution

Problem 14. Show that the order of any element $a$ of finite group $G$ divides the order of $G$, i.e. we have $| G | \mathrel{\vdots} ord \ a$. Send a solution

∆ Definition 8. Denote by $\phi (n)$ the number of natural numbers that are less then $n$ and coprime with $n$. Function $\phi (n)$ is called Euler's function.

Problem 15. Find: a) $\phi(2)$; b) $\phi(6)$ c) $\phi(30)$; d) $\phi(p)$; e) $\phi(p^n)$. Send a solution

Problem 16. Show that for any coprime numbers $m$ and $n$, we have $\phi(mn)=\phi(m) \phi(n)$. Send a solution

Problem 17. Find $\phi( \ p_1^{k_1} \cdot \ldots \cdot p_n^{k_n} )$. Send a solution

Problem 18. (Euler's Theorem) Show that for any number $a$ that is coprime to $n$, we have $$ \mathrm{a^{\phi(n)} \equiv 1 \ \ mod(n) }.$$ Send a solution

Problem 19. * Describe the symmetry groups of: a) an equilateral triangle; b) a square; c) an equilateral $n$-gon (Dyhedral group $D_n$). Send a solution