Задача 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 $\mathbb{Z}/ n \mathbb{Z} ^{\times} = \{ a \in\mathbb{Z}/ n \mathbb{Z} \ | \ GCD(a,n) =1 \}$.